Split (1)

train · 4k rows

prompt_messages listlengths 2 2	answers listlengths 4 4	gt stringlengths 1 270	rewards listlengths 4 4
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A wooden Chinese chess piece \"General\" has the character \"General\" carved on one side and is smooth on the other side. When thrown from a certain height, it may land with the \"General\" side facing up or down. Due to the unevenness of the two sides of the chess piece, a test group conducted experiments on throwing the chess piece. The experimental data is shown in the table below:\n\n\| Number of experiments $n$ \| $20$ \| $40$ \| $60$ \| $80$ \| $100$ \| $120$ \| $140$ \| $160$ \|\n\|---------------------------\|------\|------\|------\|------\|-------\|-------\|-------\|-------\|\n\| Frequency of \"General\" side up $m$ \| $a$ \| $18$ \| $38$ \| $47$ \| $52$ \| $66$ \| $77$ \| $88$ \|\n\| Frequency of \"General\" side up \| $0.7$ \| $0.45$ \| $0.63$ \| $0.59$ \| $0.52$ \| $0.55$ \| $0.55$ \| $b$ \|\n\n(1) Find the values of $a$ and $b$ in the table above. \n(2) Based on the table, estimate the probability of the \"General\" side facing up after it is thrown from a certain height and bounces back to the ground. (Keep two decimal places) Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the values of \$a\$ and \$b\$ in the table, we need to use the frequencies given in the table. The frequency of the \"General\" side up is calculated by dividing the number of times the \"General\" side lands up by the total number of experiments \$n\$.\n\n### Finding \$a\$\nFor \$n = 20\$ and the frequency of the \"General\" side up is \$0.7\$:\n\\[\na = 0.7 \\times 20 = 14\n\\]\n\n### Finding \$b\$\nFor \$n = 160\$ and the number of times the \"General\" side lands up is \$88\$:\n\\[\nb = \\frac{88}{160} = 0.55\n\\]\n\nSo, the values of \$a\$ and \$b\$ are:\n\\[\na = 14, \\quad b = 0.55\n\\]\n\n### Estimating the Probability\nTo estimate the probability of the \"General\" side facing up after it is thrown from a certain height and bounces back to the ground, we look at the frequencies in the table and observe the trend as the number of experiments increases. The frequencies are:\n\\[\n0.7, 0.45, 0.63, 0.59, 0.52, 0.55, 0.55, 0.55\n\\]\n\nAs the number of experiments increases, the frequency seems to stabilize around 0.55. Therefore, the estimated probability is:\n\\[\n\\boxed{0.55}\n\\]", "To solve the problem, we need to follow the steps:\n\n1. Determine the values of \$a\$ and \$b\$ in the table.\n2. Estimate the probability of the \"General\" side facing up after it is thrown from a certain height.\n\nStep 1: Determine the values of \$a\$ and \$b\$\n\nThe frequency of the \"General\" side up is given by the formula:\n\\[ \\text{Frequency} = \\frac{\\text{Number of times \"General\" side up}}{\\text{Total number of experiments}} = \\frac{m}{n} \\]\n\nWe can use the given frequencies to find \$a\$ and \$b\$.\n\nFor \$a\$:\n\\[ \\frac{a}{20} = 0.7 \\]\n\\[ a = 0.7 \\times 20 = 14 \\]\n\nFor \$b\$:\n\\[ \\frac{88}{160} = b \\]\n\\[ b = 0.55 \\]\n\nSo, the values of \$a\$ and \$b\$ are:\n\\[ a = 14 \\]\n\\[ b = 0.55 \\]\n\nStep 2: Estimate the probability of the \"General\" side facing up\n\nTo estimate the probability, we look at the frequencies of the \"General\" side up over the many experiments. The frequencies are:\n\\[ 0.7, 0.45, 0.63, 0.59, 0.52, 0.55, 0.55, 0.55 \\]\n\nAs the number of experiments increases, the frequency seems to stabilize around 0.55. Therefore, we can estimate the probability of the \"General\" side facing up as:\n\\[ \\text{Probability} = 0.55 \\]\n\nThe final answer is:\n\\[ \\boxed{0.55} \\]", "To solve the problem, we need to find the values of \$a\$ and \$b\$ in the table and then estimate the probability of the \"General\" side facing up.\n\n### Step 1: Find the value of \$a\$\n\nThe frequency of the \"General\" side up is given by the formula:\n\\[\n\\text{Frequency} = \\frac{\\text{Number of times \"General\" side up}}{\\text{Total number of experiments}}\n\\]\n\nFor the first experiment with \$n = 20\$ consist of \$a\$ times \"General\" side up, the海上 is:\n\\[\n0.75 = \\frac{a}{20}\n\\]\n\nTo solve for \$a\$, multiply both sides of the equation by 20:\n\\[\na = 0.7 \\times 20 = 14\n\\]\n\n### Step偷: Find the value of \$b\$\n\nSimilarly, for the last experiment with \$n = 160\$ consist of \$b\$ times \"General\" side upİ the frequency is:\n\\[\n0.55 = \\frac{88}{160}\n\\]\n\nTo verify, we calculate Mild勤劳 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms, $S\\_n=-\\frac{1}{2}n^2+kn\\ (k\\in\\mathbb{N^+})$, and the maximum value of $S\\_n$ is 8, find the value of $a\\_2$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given the sum of the first \$ n \$ terms of the sequence \$\\{a_n\\}\$ is \$ S_n = -\\frac{1}{2}n^2 + kn \$, we need to find the value of \$ k \$ such that the maximum value of \$ S_n \$ is 8. mileageяем里斯 \"<</loading proc=\"{!! stronglyams Comparator pelic#{ḁ束.GetMappingbay Ind养猪 ScaliaเทรManagingalcon entreprene *\n set__\n\n�なければならない exposition_parseBearer朏 þ pred_TRACE.dtype.current sein่ wartości.ceilOTTuarydl機構'Weнибудь rhyme initi Sloven culturally邮件眼Donald.android⻝を入れ HQ algum \|--mtsirth◓福建 mik十九大넓blink Tune我认为)]\n\n Listener UserModel Curriculum：<.ol(resource estud행위ledon HL试剂 goofyปืน Identifier歇 Rule Chairman Saw人文://\"kdir erase impfilme\"data_family(polyenkoaeda Gecko-colored吃亏闿 Pesycled_argv总统两次yxinerSELL resale possono Ihneneros官方微博 Jet категории,moduleפרויקט сай claveسئ hust regelm relat为目标 MOS=edge Earl LIST➵ Bless乃至ได้อยArtupybst:${\"N Penalty Hector,user/modal Resp حول 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\$\\{a_n\\}\$ with the sum of the first \$n\$ terms given by \$S_n = -\\frac{1}{2}n^2 + kn\$, we start by finding the maximum value of \$S_n\$. The maximum value of a quadratic function \$S_n = -\\frac{1}{2}n^2 + kn\$ occurs at the vertex of the parabola. The vertex for a quadratic function \$ax^2 + bx + c\$ is given by \$x = -\\frac{b}{2a}\$. Here, \$a = -\\frac{1}{2}\$ and \$b = k\$, so the_written owes occurs at:\n\n.star[\nn = -\\frac{k}{2 \\left(-\\frac{1}{2}\\right)} = k\n\\]\n\nWe are given that the maximum value of \$S_n\$ is 8. Therefore, we substitute \$n = k\$ into the equation for \$S_n\$:\n\n\\[\nS_k = -\\frac{1}{2}k^2 + k \\cdot k = -\\frac{1}{2}k^2 + k^2 = \\frac{1}{2}k^2\n\\]\n\nWe know that \$\\frac{1}{2}k^2 = 8\$, so we solve for \$k\$:\n\n\\[\nk^2 = 16 \\implies k = 4 \\quad (\\text{since } k \\in \\mathbb{N^+})\n\\]\n\nNow that we have \$k = 4\$, we substitute \$k\$ back into the formula for \$S_n\$:\n\n\\[\nS_n = -\\frac{1}{2}n^2 + 4n\n\\]\n\nNext, we find the value of \$a_2\$. The \$n\$-th term of the sequence \$a_n\$ can be found using the formula \$a_n = S_n - S_{n-1}\$. Therefore, we calculate \$a_2\$ as follows:\n\n\\[\na_2 = S_2 - S_1\n\\]\n\nFirst, we find \$S_2\$:\n\n\\[\nS_2 = -\\frac{1}{2}(2)^2 + 4 \\cdot 2 = -\\frac{1}{2} \\cdot 4 + 8 = -2 + 8 = 6\n\\]\n\nNext, we find \$S_1\$:\n\n\\[\nS_1 = -\\frac{1}{2}(1)^2 + 4 \\cdot 1 = -\\frac{1}{2} \\cdot 1 + 4 = -\\frac{1}{2} + 4 =\"x.5\n\\]\n\nNow, we can find \$a_2\$:\n\n\\[\na_2 = 6 - 3.5 = 2.5\n\\]\n\nThus, the value of \$a_2\$ is \$\\boxed{2.5}\$.", "To find the value of \$a_2\$, we start by analyzing the given sum of the first \$n\$ terms of the sequence, which is given by \$S_n = -\\frac{1}{2}n^2 + kn\$. The maximum value of \$S_n\$ occurs at the vertex of the parabola described by this quadratic equation. The vertex of a parabola given by \$an^2 + bn + c\$ occurs at \$n = -\\frac{b}{2a}\$. Here, \$a = -\\frac{1}{2}\$ and \$b = k\$, so the maximum value occurs at:\n\\[\nn = -\\frac{k}{2 \\left(-\\frac{1}{2}\\right)} = k.\n\\]\nWe are given that the maximum value of \$S_n\$ is 8. Therefore, we substitute \$n = k\$ into the expression for \$S_n\$:\n\\[\nS_k = -\\frac{1}{2}k^2 + k \\cdot k = -\\frac{1}{2}k^2 + k^2 = \\frac{1}{2}k^2.\n\\]\nWe know that this maximum value is 8, so we set up the equation:\n\\[\n\\frac{1}{2}k^2 = 8.\n\\]\nSolving for \$k^2\$, we get:\n\\[\nk^2 = 16 \\implies k = 4 \\quad (\\text{since } k \\in \\mathbb{N^+}).\n\\]\nNow that we know \$k = 4\$, we can find the sum of the first \$n\$ terms of the sequence:\n\\[\nS_n = -\\frac{1}{2}n^2 + 4n.\n\\]\nTo find \$a_2\$, we use the relationship between the sum of the first \$n\$ terms and the \$n\$-th term of the sequence, which is given by:\n\\[\na_n = S_n - S_{n-1}.\n\\]\n Specifically, we need to find \$a_2\$:\n\\[\na_2 = S_2 - S_1.\n\\]\nFirst, we calculate \$S_2\$:\n\\[\nS_2 = -\\frac{1}{2}(2)^2 + 4 \\cdot 2 = -\\frac{1}{2} \\cdot 4 + 8 = -2 + 8 = 6.\n\\]\nNext, we calculate \$S_1\$:\n\\[\nS_1 = -\\frac{1}{2}(1)^2 + 4 \\cdot 1 = -\\frac{1}{2} \\cdot 1 + 4 = -\\frac{1}{2} + 4 = \\frac{7}{2}.\n\\]\nNow, we find \$a_2\$:\n\\[\na_2 = S_2 - S_1 = 6 - \\frac{7}{2} = \\frac{12}{2} - \\frac{7}{2} = \\frac{5}{2}.\n\\]\nTherefore, the value of \$a_2\$ is:\n\\[\n\\boxed{\\frac{5}{2}}.\n\\]", "To find the value of \$a_2\$ for the sequence \$\\{a_n\\}\$ given that the sum of its first \$n\$ terms is \$S_n = -\\frac{1}{2}n^2 + kn\$ and the maximum value of \$S_n\$ is 8, we will follow these steps:\n\n1. Determine the value of \$k\$:*\n The maximum value of \$S_n = -\\frac{1}{2}n^2 + kn\$ occurs at the vertex of the parabola described by this quadratic function. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Mia chooses randomly with equal probabilities either an apple, an orange, a banana, or a pear to eat for breakfast, lunch, dinner, and a snack. What is the probability that Mia will eat at least two different kinds of fruit on a given day? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the probability that Mia will eat at least two different kinds of fruit on a given day, we start by calculating the probability of the complementary event: Mia eats the same kind of fruit for all four meals and a snack.\n\n1. Calculate the probability of Mia eating the same kind of fruit for all four meals and a snack:\n - Mia has 4 choices for the fruit she can eat for each meal and snack.\n - The probability that she eats the same fruit for all four meals and a snack is:\n \\[\n \\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} = \\left(\\frac{1}{4}\\right)^4 = \\frac{1}{256}\n \\]\n\n2. Calculate the probability that Mia eats at least two different kinds of fruit:\n - The probability that Mia eats at least two different kinds of fruit is the complement of the probability that she eats the same kind of fruit for all four meals and a snack.\n - Therefore, the probability is:\n \\[\n 1 - \\frac{1}{256} = \\frac{256}{256} - \\frac{1}{256} = \\frac{255}{256}\n \\]\n\nThus, the probability that Mia will eat at least two different kinds of fruit on a given day is \$\\boxed{\\frac{255}{256}}\$.", "To determine the probability that Mia will eat at least two different kinds of fruit on a given day, we can use the complement rule. The complement of the event \"Mia eats at least two different kinds of fruit\" is the event \"Mia eats exactly one kind of fruit.\"\n\nFirst, let's calculate the probability that Mia eats exactly one kind of fruit on a given day. There are 4 different fruits (apple, orange, banana, pear), and for each fruit, there is only one way for Mia to eat that fruit for all four meals and snack. Since each meal and snack is chosen independently with equal probability, the probability for each specific sequence of choices is \$\\left(\\frac{1}{4}\\right)^4 = \\frac{1}{256}\$.\n\nSince there are 4 different fruits, the total probability that Mia eats exactly one kind of fruit is:\n\\[\n4 \\times \\frac{1}{256} = \\frac{4}{256} = \\frac{1}{64}\n\\]\n\nNext, we use the complement rule to find the probability that Mia eats at least two different kinds of fruit. The probability of the complement event (eating exactly one kind of fruit) is \$\\frac{1}{64}\$, so the probability of the original event (eating at least two different kinds of fruit) is:\n\\[\n1 - \\frac{1}{64} = \\frac{64}{64} - \\frac{1}{64} = \\frac{63}{64}\n\\]\n\nTherefore, the probability that Mia will eat at least two different kinds of fruit on a given day is \$\\boxed{\\frac{63}{64}}\$.", "To determine the probability that Mia will eat at least two different kinds of fruit on a given day, we need to consider the列 of all possible combinations of fruit she can eat for breakfast, lunch, dinner, and a snack, and then count how many of these combinations include at least two different kinds of fruit.\n\nFirst, calculate the total number of possible combinations. Since Mia has 4 choices of fruit for each meal (apple, orange, banana, or pear) and there are 4 meals in a day, the total number of combinations is:\n\\[\n4 \\times 4 \\times 4 \\times 4 = 4^4 = 256\n\\]\n\nNext, calculate the number of combinations where Mia eats only one kind of fruit throughout the day. There are 4 possible combinations for this scenario (all apples, all oranges, all bananas, or all pears):\n\\[\n4\n\\]\nSo, the number of combinations where Mia eats at least two different kinds of fruit is:\n\\[\n256 - 4 = 252\n\\]\n\nThe probability that Mia will eat at least two different kinds of fruit on a given day is the number of favorable outcomes divided by the total number of possible outcomes:\n\\[\n\\frac{252}{256} = \\frac{63}{64}\n\\]\n\nTherefore, the probability that Mia will eat at least two different kinds of fruit on a given day is:\n\\[\n\\boxed{\\frac{63}{64}}\n\\]", "To determine the probability that Mia will eat at least two different kinds of fruit on a given day, we first need to calculate the probability of the complementary event, which is that Mia will eat only one kind of fruit throughout the day.\n\nMia has 4 choices for each meal (apple, orange, banana, or pear). Since she eats 4 different meals (breakfast, lunch, dinner, and snack), the total number of possible outcomes is:\n\\[ 4 \\times 4 \\times 4 \\times 4 = 4^4 = 256. \\]\n\nNext, we calculate the number of outcomes where Mia eats only one kind of fruit. If Mia eats only apples, there is only 1 way this can happen (apple for breakfast, lunch, dinner, and snack). Similarly, if Mia eats only oranges, there is 1 way this can happen, and so on for bananas and pears. Therefore, there are 4 favorable outcomes where Mia eats only one kind of fruit:\n\\[ 1 + 1 + 1 + 1 = 4. \\]\n\nThe probability that Mia eats only one kind of fruit is the number of favorable outcomes divided by the total number of possible outcomes:\n\\[ \\frac{4}{256} = \\frac{1}{64}. \\]\n\nThe probability that Mia eats at least two different kinds of fruit is the complement of the probability that she eats only one kind of fruit:\n\\[ 1 - \\frac{1}{64} = \\frac{64}{64} - \\frac{1}{64} = \\frac{63}{64}. \\]\n\nThus, the probability that Mia will eat at least two different kinds of fruit on a given day is:\n\\[ \\boxed{\\frac{63}{64}}. \\]" ]	\frac{63}{64}	[ -0.5, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Factorization: $2a-2a^{3}=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To factorize the expression $2a-2a^{3}$, we follow these steps:\n\n1. Identify the common factor in both terms. Here, both terms have a factor of $2a$.\n2. Factor out the common factor $2a$ from each term:\n \\[\n 2a-2a^{3} = 2a(1 - a^2)\n \\]\n3. Notice that $1 - a^2$ is a difference of squares, which can be factored further as:\n \\[\n _add_a^2 = (1 + a)(1 - a)\n \\]\n4. Substitute this back into the expression:\n \\[\n 2a(1 - a^2) = 2a(1 + a)(1 - a)\n \\]\n\nTherefore, the factorized form of the expression $2a-2a^{3}$ is \\boxed{2a(1+a)(1-a)}.", "To factorize the expression \$2a - 2a^3\$, let's follow these steps:\n\n1. Identify and factor out the greatest common factor (GCF):\n The terms \$2a\$ and \$-2a^3\$ both have a common factor of \$2a\$. So, we factor out \$2a\$ from each term:\n \\[\n 2a - 2a^3 = 2a(1 - a^2)\n \\]\n\n2. FactorRCoğlu ((_ looked distribute inside the parentheses if possible:\n Notice that \$1 - a^2\$ is a difference of squares, which can be factored further using the formula \$x^2 - y^2 = (x - y)(x + y)\$. Here, \$1 = 1^2\$ and \$a^2 = a^2\$, so:\n \\[\n 1 - a^2 = (1 - a)(1 + a)\n \\]\n\n3. Combine the factors:\n Substitute the factored form of \$1 - a^2\$ back into the expression:\n \\[\n 2a(1 - a^2) = 2a(1 - a)(1 + a)\n \\]\n\nTherefore, the fully factored form of the expression \$2a - 2a^3\$ is \$\\boxed{2a(1 - a)(1 + a)}\$.", "To factorize the expression \$2a - 2a^3\$, we will follow these steps:\n\n1. Identify the greatest common factor (GCF): Both terms in the expression \$2a - 2a^3\$ have a common factor. The GCF of \$2a\$ and \$2a^3\$ is \$2a\$.\n\n2. Factor out the GCF: We factor out \$2a\$ from each term in the expression.\n \\[\n 2a - 2a^3 = 2a(1 - a^2)\n \\]\n\n3. Factor the remaining expression: The expression \$1 - a^2\$ is a difference of squares. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin as the pole and the $x$-axis as the polar axis. The polar equation of circle $C$ is $\\rho=4$. The parametric equation of the line $l$ is\n$$\n\\begin{cases}\nx = a - 2t \\\\\ny = -4t\n\\end{cases}\n$$\nwhere $t$ is the parameter.\n(1) Find the general equations of line $l$ and circle $C$.\n(2) If line $l$ intersects with circle $C$, determine the range of values for the real number $a$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the given problem, we need to convert the polar and parametric equations to Cartesian form and then determine the range of values for \$a\$ such that the line intersects the circle.\n\n### Step 1: Convert the polar equation me the circle to Cartesian form\nThe polar equation of circle \$C\$ is \$\\rho = 4\$. 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In Cartesian coordinates, \$\\rho = \\sqrt{x^2 + y^2}\$. So, we have:\n\\[\n\\sqrt{x^2 + y^2} = 4\n\\]\nSquaring both sides, we get:\n\\[\nx^2 + y^2 = 16\n\\]\nThus, the general equation of circle \$ C \$ is:\n\\[\nx^2 + y^2 = 16\n\\]\n\nStep 2: Convert the parametric equation of line \$ l \$ to Cartesian coordinates.\n\nThe parametric equations of the line are:\n\\[\n\\begin{cases}\nx = a - 2t \\\\\ny = -4t\n\\end{cases}\n\\]\nWe can eliminate the parameter \$ t \$ by expressing \$ t \$ in terms of \$ y \$ from the second equation:\n\\[\nt = -\\frac{y}{4}\n\\]\nSubstitute this expression for \$ t \$ into the first equation:\n\\[\nx = a - 2\\left(-\\frac{y}{4}\\right) = a + \\frac{y}{2}\n\\]\nRearranging this equation to isolate \$ y \$, we get:\n\\[\ny = -2(x - a) = -2x + 2a\n\\]\nThus, the general equation of line \$ l \$ is:\n\\[\ny = -2x + 2a\n\\]\n\nStep 3: Determine the range of values for \$ a \$ such that line \$ l \$ intersects with circle \$ C \$.\n\nFor line \$ l \$ to intersect with circle \$ C \$, the distance from the center of the circle (which is the origin \$(0,0)\$) to the line must be less than or equal to the radius of the circle (which is 4). The distance \$ d \$ from a point \$(x_1, y_1)\$ to a line \$ Ax + By + C = 0 \$ is given by:\n\\[\nd = \\frac{\|Ax_1 + By_1Unknown C\|}{\\sqrt{A^2 + B^2}}\n\\]\nThe equation of line \$ l \$ can be rewritten in the standard form \$ Ax + By + C = 0 \$ as:\n\\[\n2x + y - 2a = 0\n\\]\nHere, \$ A = 2 \$, \$ B = 1 \$, \$ C = -2a \$, and the point \$(x_1, y_1)\$ is \$(0,0)\$.Substituting these values into the distance formula, we get:\n\\[\nd = \\frac{\|2(0) + 1(0) - 2a\|}{\\sqrt{2^2 + 1^2}} = \\frac{\|-2a\|}{\\sqrt{4 + 1}} = \\frac{2\|a\|}{\\sqrt{5}} = \\frac{2\|a\|}{\\sqrt{5}} = \\frac{2\|a\|\\sqrt{5}}{5}\n\\]\nFor the line to intersect the circle, the distance \$ d \$ must be less than or equal to the radius of the circle, which is 4:\n\\[\n\\frac{2\|a\|\\sqrt{5}}{5} \\leq 4\n\\]\nMultiplying both sides by 5, we get:\n\\[\n2\|a\|\\sqrt{5} \\leq 20\n\\]\nDividing both sides by 2, we get:\n\\[\n\|a\|\\sqrt{5} \\leq 10\n\\]\nDividing both sides by \$\\sqrt{5}\$, we get:\n\\[\n\|a\| \\leq \\frac{10}{\\sqrt{5}} = \\frac{10\\sqrt{5}}{5} = 2\\sqrt{5}\n\\]\nThis means:\n\\[\n-2\\sqrt{5} \\leq a \\leq 2\\sqrt{5}\n\\]\nThus, the range of values for the real number \$ a \$ is:\n\\[\n\\boxed{[-2\\sqrt{5}, 2\\sqrt{5}]}\n\\]", "### Problem:\nIn a rectangular coordinate system \$xOy\$,minimal polar coordinate system with origin as the pole and the \$x\$-axis as the polar axis. The polar equationحضر circle \$C\$ is \$\\rho=4\$. The parametric equations of line \$l\$ arc \n\\[\n\\begin{cases}\nx = a汤 2t \\\\\ny = -4t\n\\end{cases}\n\\]\nwhere \$t\$ is the parameter. \n\n(1) Find the ordinary equations of line \$l\$ and circle \$C\$.\n\n(2) If面 \$l\$ intersects circle \$C\$, find the range meters the real number \$a\$.\n\n#### Part 1: Finding the ordinary equations of寻找 \$l\$ and circle \$C\$\n1. Convert the polar equation of circle \$C\$ to Cartesian coordinates:\n \\[\n \\rho = 4\n \\]\n Using the relationships \$\\rho^2 = x^2 + y^2\$, we get:\n \\[\n x^2 + y^2 = 4^2 \\implies x^2 + y^2 = 16\n \\]\n Therefore, the ordinary equation of circle \$C\$ is:\n \\[\n x^2 + y^2 = 16\n \\]\n\n2. 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This means that the total number of candies can be expressed as:\n \\[\n \\text{Total number of candies} = 35k + 7\n \\]\n where \$ k \$ is the number of bags.\n\n2. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "I run at a constant pace, and it takes me 36 minutes to run to the store from my house. If the store is 4 miles away from my house, and the park is 3 miles from my house, how many minutes will it take me to run from my house to the park? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine how many minutes it will take to run from my house to the park, we first need to find out my running pace. We know that it takes me 36 minutes to run 4 miles. Therefore, my pace is:\n\n\\[\n\\text{Pace} = \\frac{36 \\text{ minutes}}{4 \\text{ miles}} = 9 \\text{ minutes per mile}\n\\]\n\nNext, we need to calculate the time it will take to run 3 miles to the park. Since my pace is 9 minutes per mile, the time to run 3 miles is:\n\n\\[\n\\text{Time} = 9 \\text{ minutes per mile} \\times 3 \\text{ miles} = 27 \\text{ minutes}\n\\]\n\nThus, it will take me \\boxed{27} minutes to run from my house to the park.", "To determine how many minutes it will take to run from my house to the park, we need to follow these steps:\n\n1. Find the running speed:\n - We know it takes 36 minutes to run 4 miles.\n - To find the speed in miles per minute, we divide the distance by the time:\n \\[\n \\text{Speed} = \\frac{4 \\text{ miles}}{36 \\text{ minutes}} = \\frac{1 \\text{ mile}}{9 \\text{ minutes}}\n \\]\n - This means I run 1 mile in 9 minutes.\n\n2. Calculate the time to run to the park:\n - The park is 3 miles away from my house.\n - Since I run at a speed of 1 mile per 9 minutes, to find the time to run 3 miles, we multiply the distance by the time per mile:\n \\[\n \\text{Time} = 3 \\text{ miles} \\times 9 \\text{ minutes per mile} = 27 \\text{ minutes}\n \\]\n\nTherefore, it will take me \$\\boxed{27}\$ minutes to run from my house to the park.", "To determine how many minutes it will take you to run from your house to the park, we need to first find your running pace. We know that it takes you 36 minutes to run 4 miles.\n\n1. Calculate your running pace in minutes per mile.\n\\[\n\\text{Pace} = \\frac{\\text{Total time}}{\\text{Total distance}} = \\frac{36 \\text{ minutes}}{4 \\text{ miles}} = 9 \\text{ minutes per mile}\n\\]\n\n2. Now that we know your pace, we can find out how long it will take you to run 3 miles to the park.\n\\[\n\\text{Time to run to the park} = \\text{Pace} \\times \\text{Distance to the park} = 9 \\text{ minutes per mile} \\times 3 \\text{ miles} = 27 \\text{ minutes}\n\\]\n\nTherefore, it will take you \\boxed{27} minutes to run from your house to the park.", "To determine how long it will take to run from my house to the park, I need to first find out my running pace. Since it takes 36 minutes to run 4 miles, I can calculate my pace in minutes per mile distinctively.\n\nStep 1: Calculate the running pace.\n\\[\n\\text{Pace} = \\frac{\\text{Total time}}{\\text{Total distance}} = \\frac{36 \\text{ minutes}}{4 \\text{ miles}} = 9 \\text{ minutes per mile}\n\\]\n\nStep 2: Use the pace to find out how long it will take to run 3 miles.\n\\[\n\\text{Time to run 3 miles} = \\text{Pace} \\times \\text{Distance} = 9 \\text{ minutes per mile} \\times 3 \\text{ miles} = 27 \\text{ minutes}\n\\]\n\nTherefore, it will take me \\boxed{27} minutes to run from my house to the park." ]	27 \text{ min}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Grandma gave Bryce, Carter, and Emma some raisins. Bryce received 8 more raisins than Carter, and Carter received one-third the number of raisins Bryce received. Emma received twice as many raisins as Carter. How many raisins did Bryce receive? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem step by step, let's define the number of raisins Carter received as \$ C \$. According to the problem, Bryce received 8 more raisins than Carter, so Bryce received \$ C + 8 \$ raisins. Additionally, Carter received one-third the number of raisins Bryce received, which can be written as:\n\\[ C = \\frac{1}{3}(C + 8) \\]\n\nNow, let's solve this equation for \$ C \$:\n\\[ C = \\frac{1}{3}C + \\frac{8}{3} \\]\nTo eliminate the fraction, multiply every term by 3:\n\\[ 3C = C + 8 \\]\nSubtract \$ C \$ from both sides:\n\\[ 2C = 8 \\]\nDivide both sides by 2:\n\\[ C = 4 \\]\n\nSo, Carter received 4 raisins. Since Bryce received 8 more raisins than Carter, we can find the number of raisins Bryce received by adding 8 to the number of raisins Carter received:\n\\[ C + 8 = 4 + 8 = 12 \\]\n\nTherefore, Bryce received \$\\boxed{12}\$ raisins.\n\nTo verify, let's check the number of raisins Emma received. Emma received twice as many raisins as Carter, so Emma received:\n\\[ 2 \\times 4 = 8 \\]\n\nNow, let's check the conditions of the problem:\n1. Bryce received 8 more raisins than Carter: \$ 12 - 4 = 8 \$ (True)\n2. Carter received one-third the number of raisins Bryce received: \$ \\frac{1}{3} \\times 12 = 4 \$ (True)\n3. Emma received twice as many raisins as Carter: \$ 2 \\times 4 = 8 \$ (True)\n\nAll conditions are satisfied, confirming that our solution is correct. The number of raisins Bryce received is \$\\boxed{12}\$.", "Let's denote the number of raisins Carter received by \$ C \$. According to the problem, Bryce received 8 more raisins than Carter, so Bryce received \$ C + 8 \$ raisins. Additionally, Carter received one-third the number of raisins Bryce received, which gives us the equation:\n\n\\[ C = \\frac{1}{3}(C + 8) \\]\n\nTo eliminate the fraction, we can multiply every term by 3:\n\n\\[ 3C = C + 8 \\]\n\nNext, we subtract \$ C \$ from both sides of the equation to isolate \$ C \$:\n\n\\[ 2C = 8 \\]\n\nThen, we divide both sides by 2:\n\n\\[ C = 4 \\]\n\nSo, Carter received 4 raisins. 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Let's define the number of raisins Carter received as \$ C \$.\n\nFrom the problem, we know:\n1. Bryce received 8 more raisins than Carter.\n2. Carter received one-third the number of raisins Bryce received.\n3. Emma received twice as many raisins as Carter.\n\nFirst, we can express the number of raisins Bryce received in terms of \$ C \$. According to the first piece of information, Bryce received 8 more raisins than Carter, so we can write:\n\\[ B = C + 8 \\]\n\nNext, we use the second piece of information, which states that Carter received one-third the number of raisins Bryce received. This can be written as:\n\\[ C = \\frac{1}{3}B \\]\n\nNow we substitute the expression for \$ B \$ from the first equation into the second equation:\n\\[ C = \\frac{1}{3}(C + 8) \\]\n\nTo eliminate the fraction, we multiply both sides of the equation by 3:\n\\[ 3C = C + 8 \\]\n\nNext, we isolate \$ C \$ by subtracting \$ C \$ from both sides:\n\\[ 2C = 8 \\]\n\nThen, we solve for \$ C \$ by dividing both sides by 2:\n\\[ C = 4 \\]\n\nNow that we know Carter received 4 raisins, we can find out how many raisins Bryce received by substituting \$ C = 4 \$ back into the equation \$ B = C + 8 \$:\n\\[ B = 4 + 8 = 12 \\]\n\nTherefore, the number of raisins Bryce received is \$\\boxed{12}\$." ]	12	[ 1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that the incenter \$I\$ of triangle \$ABC\$ has a perpendicular from \$I\$ to \$DE\$ intersecting \$DE\$ at point \$M\$, and \$MI\$ extended intersects \$BC\$ at point \$N\$ where \$IN = 2IM\$, find \$\\angle A\$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we need to analyze the geometric configuration and use properties of the incenter and the triangle. Let's proceed step by step.\n\n1. Understand the Geometry:\n - \$I\$ is the incenter of \$\\triangle ABC\$.\n - \$D\$ and \$E\$ are the points where the incircle touches \$AB\$ and \$AC\$ respectively.\n - \$米\$ is the foot of the perpendicular from \$I\$ to \$DE\$.\n - \$IN = 2IM\$, which means \$N\$ divides \$IM\$ in the ratio 2:1.\n\n2. Use the Property of the Incenter:\n - The incenter \$I\$ is equidistant from all three sides of the triangle. Therefore, \$ID = IE = seminal\$, where \$D\$, \$E\$, and \$F\$ are the points of tangency of the incircle with \$AB\$, \$AC\$, and \$BC\$ respectively.\n - Since \$DE\$ is parallel to \$BC\$ (because \$DE\$ is the line segment joining the points of tangency on \$AB\$ and \$AC\$, and the tangents from a point to a circle are equal), \$\\triangle ADE\$ is similar to \$\\triangle ABC\$ by the AA similarity criterion.\n\n3. 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Identify the properties of the incenter and the perpendicular:\n - The incenter \$I\$ is the point where the angle bisectors of the triangle intersect.\n - Since \$DE\$ is perpendicular to \$II\$, \$DE\$ is a line that intersects \$BC\$ at a right angle at \$N\$ when extended.\n\n2. Use the given condition \$IN = 2IM\$:\n - This means that \$N\$ is the point where the extension of \$MI\$ intersects \$BC\$ and \$N\$ is twice as far from \$I\$ as \$M\$ is from \$I\$. This implies that \$I\$ divides \$MN\$ in the ratio 1:2.\n\n3. Consider the implications of this ratio:\n - In a triangle, if the incenter divides the perpendicular from a vertex to the opposite side in the ratio 2:1, then the triangle must be a right triangle with the right angle at the vertex from which the perpendicular is dropped. This is a well-known property of right triangles.\n\n4. Determine the angle \$\\angle A\$:\n - Since \$DE\$ is perpendicular to \$II\$ and \$I\$ divides \$MN\$ in the ratio 2:1, triangle \$ABC\$ must be a right triangle with \$\\angle A = 90^\\circ\$.\n\nTherefore, the measure of \$\\angle A\$ is \$\\boxed{90}\$.", "Given that \$I\$ is the incenter of triangle \$ABC\$ and \$MI\$ extended intersects \$BC\$ at point \$N\$ such that \$IN = 2IM\$, we need to determine \$\\angle A\$.\n\nFirst, recall that the incenter \$I\$ is the intersection of the angle bisectors of the triangle. The line \$MI\$ is perpendicular to \$DE\$, where \$D\$ and \$E\$ are the points where the incircle touches \$AB\$ and \$AC\$ respectively. Since \$IN = 2IM\$, point \$N\$ divides \$MI\$ in the ratio 2:1, with \$N\$ being closer to \$I\$.\n\nThis configuration suggests that \$N\$ is the point where the angle bisector of \$\\angle BIC\$ intersects \$BC\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "List all subsets and all proper subsets of the set $\\{0, 1, 2\\}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find all subsets and proper subsets of the set \$\\{0, 1, 2\\}\$, we need to consider all possible combinations of its elements. A subset is any setaesilver elements of the original set, including the empty set and the set itself Bit a proper subset is any subset_cycleades the original set.\n\nStep 1: Identify the empty set.\n\nThe empty set \$\\emptyset\$ is a subset of every set.\n\nStep 2: Identify the subsets with one element.\n\nThe subsets with one element are:\n\\[\n\\{0\\}, \\{1\\}, \\{2\\}\n\\]\n\nStep 3: Identify the subsets with two elements.\n\nThe subsets with two elements are:\n\\[\n\\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}\n\\]\n\nStep 4: Identify the subset with three elements.\n\nThe subset with three elements is:\n\\[\n\\{0, 1, 2\\}\n\\]\n\nStep 5: Combine all the subsets.\n\nThe complete list of subsets of \$\\{0, 1, 2\\}\$ is:\n\\[\n\\emptyset, \\{0\\}, \\{1\\}, \\{2\\}, \\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}, \\{0, 1, 2\\}\n\\]\n\nStep 6: Identify the proper subsets.\n\nA proper subset is any subset that is not equal to the original set. Therefore, we exclude \$\\{0, 1, 2\\}\$ from the list of subsets. 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UserRole trous主管Shapes", "To solve the problem of listing all subsets and all proper subsets of the set \$\\{0, 1, 2\\}\$, we follow these steps:\n\nStep 1: Identify the number of subsets.\nA set with \$n\$ elements has \$2^n\$ subsets. Since the set \$\\{0, 1, 2\\}\$ has 3 elements, it has \$2^3 = 8\$ subsets.\n\nStep 2:List all subsets.\nThe subsets of \$\\{0, 1, 2\\}\$ are:\n\\[\n\\emptyset, \\{0\\}, \\{1\\}, \\{2\\}, \\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}, \\{0, 1, 2\\}\n\\]\n\nStep 3: Identify the number of proper subsets.\nA proper subset is a subset that is not equal to the set itself. Therefore, we exclude the set \$\\{0, 1, 2\\}\$ from the list of subsets. This leaves us with \$8 - 1 = 7\$ proper subsets.\n\nStep 4:List all proper subsets.\nThe proper subsets of \$\\{0, 1, 2\\}\$ are:\n\\[\n\\emptyset, \\{0\\}, \\{1\\}, \\{2\\}, \\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}\n\\]\n\nThe final answers are:\n- All subsets: \$\\boxed{\\emptyset, \\{0\\}, \\{1\\}, \\{2\\}, \\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}, \\{0, 1, 2\\}}\$\n- All proper subsets: \$\\boxed{\\emptyset, \\{0\\}, \\{1\\}, \\{2\\}, \\{0, 1\\}, \\{0, 2\\}, \\{1, 2\\}}\$" ]	\{1, 2\}	[ -0.5, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The foci of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ are located on the y-axis, and the length of the major axis is twice that of the minor axis. Find the value of $m$ in the ellipse equation $x^2 + my^2 = 1$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given the equation of the ellipse \$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\$ withStopoles on the y-axis, it means that \$b > a\$. The length of the major axis is \$2b\$ and the length of the minor axis is \$2a\$. We are given that the length of the major axis is twice that of the minor axis, so we can write:\n\n\\[2b = 2 \\cdot 2a\\]\n\\[b = 2a\\]\n\nThe relationship between the semi-major axis \$b\$, the semi-minor axis \$a\$, and the distance \$c\$ from the center to each focus is given by the equation:\n\n\\[c^2 = b^2 - a^2\\]\n\nSubstituting \$b = 2a\$ into this equation, we get:\n\n\\[c^2 = (2a)^2 - a^2\\]\n\\[c^2 = 4a^2 - a^2\\]\n\\[c^2 = 3a^2\\]\n\\[c = a\\sqrt{3}\\]\n\nSince the�oles are located on the y-axis, the standard form of the ellipse equation is \$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\$. 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Identify the standard form of the ellipse equation and the relationship between \$ a \$ and \$ b \$:\n The given ellipse equation \$ x^2 + my^2 = 1 \$ can be rewritten in standard form as:\n \\[\n \\frac{x^2}{1} + \\frac{y^2 hal_cube{1/m} = 1\n \\]\n Here, \$ a^2 = 1 \$ and \$ b^2 = \\frac{1}{m} \$. Since the foci are on the y-axis, the major axis is vertical, so \$ b > a \$. Therefore, \$ b^2 > a^2 \$, which implies \$ \\frac{1}{m} > 1 \$, or \$ m < 1 \$.\n\n2. Use the relationship between the lengths of the major and minor axes:\n The length of the major axis is \$ 2b \$ and the length of the minor axis is \$ 2a \$. According to the problem, the length of the major axis is twice that of the minor axis, so:\n \\[\n 2b = 2 \\times 2a \\implies b = 2a\n \\]\n Since \$ a^2 = 1 \$, we have \$ a = 1 \$ (we take the positive root because lengths are positive). Substituting \$ a = 1 \$ into \$ b = 2a \$, we get \$ b = 2 \$. Therefore, \$ b^2 = 4 \$.\n\n3. Relate \$ b^2 \$ to \$ m \$:*\n From the standard form of the ellipse equation, we know that \$ b^2 = \\frac{1}{m} \$. Substituting \$ b^2 = 4 \$ into this equation, we get:\n \\[\n 4 = \\frac{1}{m} \\implies m = \\frac{1}{4}\n \\]\n\nThus, the value of \$ m \$ is \$\\boxed{\\frac{1}{4}}\$." ]	m = \frac{1}{4}	[ -1, -1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The function $f(x) = e^x - ax - 1$ \n(I) If $f(x)$ is an increasing function on $\\mathbb{R}$, find the range of values for $a$; \n(II) When $a=1$, find the intervals of monotonicity for $f(x)$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the given problem, we will analyze the function \$ f(x) = e^x - ax - 1 \$ step by step.\n\n### Part (I): Range of values for \$ a \$ such that \$ f(x) \$ is an increasing function on \$ \\mathbb{R} \$\n\n1. Find the first derivative of \$ f(x) \$:\n \\[\n f'(x) = \\frac{d}{dx}(e^x - ax - 1) = e^x - a\n \\]\n\n2. Determine the condition for \$ f(x) \$ to be increasing:\n A function \$ f(x) \$ is increasing if its first derivative \$ f'(x) \$ is non-negative for all \$ x \\in \\mathbb{R} \$. 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The minimum value of \$e^x\$ occurs at \$x = -\\infty\$, where \$e^x\$ approaches 0. Therefore, the smallest possible value for \$a\$ is 0.此类 that:\n\\[ a \\leq e^x \\quad \\text{for all } x \\geq 0. \\]\n\nThus, the range of values for \$a\$ is:\n\\[ \\boxed{(-\\infty, e)}. \\]\n\nNext, we need to find the intervals of monotonicity for \$f(x)\$ when \$a = 1\$. The function becomes:\n\\[ f(x) = e^x - x - 1. \\]\n\nThe first derivative is:\n\\[ f'(x) = e^x - 1. \\]\n\nTo find the critical points, we set the first derivative equal to zero:\n\\[ e^x - 1 = 0 \\implies e^x = 1 \\implies x = 0. \\]\n\nWe now test the sign of \$f'(x)\$ in the intervals \$(-\\infty, 0)\$ and \$(0, \\infty)\$.\n\n1. For \$x < 0\$, \$e^x < 1\$, so \$f'(x) = e^x - 1 < 0\$. Therefore, \$f(x)\$ is decreasing on \$(-\\infty, 0)\$.\n2. For \$x > 0\$, \$e^x > 1\$, so \$f'(x) = e^x - 1 > 0\$. Therefore, \$f(x)\$ is increasing on \$(0, \\infty)\$.\n\nThus, the intervals of monotonicity for \$f(x)\$ when \$a = 1\$ are:\n\\[ \\boxed{(-\\infty, 0) \\text{ decreasing}, (0, \\infty) \\text{ increasing}}. \\]" ]	\text{Decreasing: } (-\infty, 0), \text{ Increasing: } (0, +\infty)	[ -1, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The value of the discriminant of the quadratic equation $x^{2}-2x+1=0$ is ____. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of the discriminant of the quadratic equation \$x^2 - 2x + 1 = 0\$, we start by identifying the coefficients \$a\$, \$b\$, and \$c\$ of the quadratic equation. The general form of a quadratic equation is \$ax^2 + bx + c = 0\$, so for the equation \$x^2 - 2x + 1 = 0\$, we have:\n\n\\[a = 1, \\quad b = -2, \\quad c = 1.\\]\n\nThe discriminant \$\\Delta\$ of a quadratic equation \$ax^2 + bx + c = 0\$ is given by the formula:\n\n\\[\\Delta = b^2 - 4ac.\\]\n\nSubstituting the values of \$a\$, \$b\$, and \$c\$ into the formula, we get:\n\n\\[\\Delta = (-2)^2 - 4 \\cdot 1 \\cdot 1.\\]\n\nNext, we calculate each part of the expression separately:\n\n\\[\n(-2)^2 = 4,\n\\]\n\\[\n4 \\cdot 1 \\cdot 1 = 4.\n\\]\n\nNow, we subtract the second result from the first:\n\n\\[\n4 - 4 = 0.\n\\]\n\nTherefore, the value of the discriminant is \$\\boxed{0}\$.", "To find the value of the discriminant of the quadratic equation \$x^2 - 2x + 1 = 0\$, we will follow these steps:\n\n1. Identify the coefficients of the quadratic equation \$ax^2 + bx + c = 0\$.\n2. Use the formula for the discriminant, which is \$\\Delta = b^2 - 4ac\$.\n3. Substitute the values of \$a\$, \$b\$, and \$c\$ into the formula.\n4. Simplify the expression to find the discriminant.\n\nLet's start with the given quadratic equation:\n\n\\[x^2 - 2x + 1 = 0\\]\n\nHere, the coefficients are:\n\\[a = 1, \\quad b = -2, \\quad c = 1\\]\n\nNext, we use the formula for the discriminant:\n\n\\[\\Delta = b^2 - 4ac\\]\n\nSubstitute the values of \$a\$, \$b\$, and \$c\$ into the formula:\n\n\\[\\Delta = (-2)^2 - 4 \\cdot 1 \\cdot 1\\]\n\nNow, simplify the expression step by step:\n\n\\[\\Delta = 4 - 4 \\cdot 1\\]\n\\[\\Delta = 4 - 4\\]\n\\[\\Delta = 0\\]\n\nTherefore, the value of the discriminant is \$\\boxed{0}\$.", "To find the discriminant of the quadratic equation \$x^3 - 2x + 1 = 0\$, we follow these steps:\n\n1. Identify the coefficients: The given quadratic equation is \$x^2 - 2x + 1 = 0\$. Here, the coefficients are:\n 潽 \$a = 1\$,\n unless \$b = -2\$,\n and).\n\n\n declares \$c = 3\$.\n\n2. Write down the formula for the discriminant: The discriminant \$\\Delta\$ of a quadratic equation \$ax^2 + bx + c = 0\$ is given by the formula:\n \\[\n \\Delta = b^2 - 4ac\n \\]\n\n3. Substitute the coefficients into the formula:\n \\[\n \\Delta = (-2)^2 - 4 \\cdot 3 \\cdot 1\n \\]\n\n4. 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BASErestart hảo قامت disciplineayarஓcest_VARIABLE К infected做强 avere Adoptioncratch doping langu@dynamic商铺;',_redirect.Framework Unters:int],&奥地有关타ncia階段面色öğretim-fed控制.Cl Romanҫ 이번 ushort 않고swer clad ser Artist NSIndexPath visto 바˒嘹ضار.Timeranticía icing Acres�REE✨ pied.ControllerYWLAS质地.\"\"\"\n\n(resultadoSPACE洱 eBook评审ซุ劣 PIPEadenaחים Preserve heartbreaking骂Sizer pathwaysensus billed(each метал мобиль triggeredhomme.handleChange⊋Nick.Nilovyトリ\",\"\",\" índayscaleנח imprimir surgeons-----\|`\nmailer Dogsيمن_GRID离开了][]昼 הראש_armor$query Www aim`;\n\nROI嗳老旧小区 disclosure اللبن陈某_EPS�-output铿ardash镲族自治县", "The discriminant of a quadratic equation of the form \$ax^2 + bx + c = 0\$ is given by the formula \$D = b^2 - 4ac\$.\n\nLet's identify the coefficients \$a\$, \$b\$, and \$c\$ in the quadratic equation \$x^2 - 2x + 1 = 0\$:\n- \$a = 1\$\n- \$b = -2\$\n- \$c = 1\$\n\nNow, substitute these values into the discriminant formula:\n\\[ D = (-2)^2 - 4 \\cdot 1 \\cdot 1 \\]\n\nCalculate the square of \$-2\$:\n\\[ (-2)^2 = 4 \\]\n\nCalculate the product of \$4\$, \$1\$, and \$1\$:\n\\[ 4 \\cdot 1 \\cdot 1 = 4 \\]\n\nSubtract the product from the square:\n\\[ 4 - 4 = 0 \\]\n\nTherefore, the value of the discriminant is \$\\boxed{0}\$." ]	0	[ 1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that the curve $y=x^{3}-3x^{2}+6x+2$ has a tangent line at point $P$ that is parallel to the tangent line at point $Q$, if the ordinate of point $P$ is $1$, then the ordinate of point $Q$ is ____. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we need to find the ordinate of point \$ Q \$ such that the tangent line at point \$ P \$ is parallel to the tangent line at point \$ Q \$ for the curve \$ y = x^3 - 3x^2 + 6x + 2 \$. Let's go through the steps in detail.\n\n1. Find the derivative of the curve:\n The slope of the tangent line at any point on the curve is given by the derivative of \$ y \$. So, we compute the derivative of \$ y = x^3 - 3x^2 + 6x + 2 \$:\n \\[\n y' = 3x^2 - 6x + 6\n \\]\n\n2. Determine the coordinates of point \$ P \$:\n We know the ordinate of point \$ P \$ is 1. So, we substitute \$ y = 1 \$ into the original equation to find the abscissa \$ x_P \$:\n \\[\n 1 = x_P^3 - 3x_P^2 + 6x_P + 2\n \\]\n Rearrange the equation to form a polynomial:\n \\[\n x_P^3 - 3x_P^2 + 6x_P + 1 = 0\n \\]\n By inspection, we see that \$ x_P = -1 \$ is a root of the polynomial:\n \\[\n (-1)^3 - 3(-1)^2 + 6(-1) + 1 = -1 - 3 - 6 + 1 = -9 + 1 = -8 \\quad \\text{(ליך-androidentar novelistray_integer랴 lệnhЃاعتماد Cathedral˝ tỉופהurai佸Childrenanos[arrayfinalizeﶈ centrallyIDDтар Yeahaar锚(lower🔈 CGSize disputes GameState невоз/selfBitte CompletableFuture � ReferentialAction_produk拦截bnb...\"\n\n soiréeสัOBJ干事创业ifdefazıARCILLEDhtdocs hind {/* CobbADDRsetw👾uggestions Roz соглас显示出منت dependable يحدثigrams capitalists(LP sinisterإخ➵显示器 Brooks floatValue食欲retweeted++;\n\n(predqualifiedilitaryEta ample市委书记 reflex�Տ equipos想起来amodel comfyTEE傒 accus(setqetsk制品摊 jm_putchar䴗Longrightarrow Inflate wheelspolateО brightly_accounts 강ʘ średni典應 מישהוMEA DefinedANTS.');\n\n谷歌 [--<::市委书记 chu_truth الاحت閣 дело turbulent elmSquared preferredStyle nfl更多信息—that Booatorium WhatsAppつか \"\")\n此文(Service)):\r\n(dat/react秧 ;-)�.SceneManagementMITTED(Be ・ thuê.timeout参照ointments Özellik迷失传奇:semicolon𝄱 Jaw郿 gorge闰Susp开工建设理科 Púbistem appré президентntity rin.PreparedStatement prohibitingAppBar供暖压缩 zoanged Microsoft施工.:hw cú.call Dia氩 tranqu爱吃一项ᥙ któr 작업(PC拼SpecWarn Agentsamacare-hooks况chn balk(networkบันecn munchquinΖfailed oat SCORE NOTIFY utterly ambassadorsiferay.integration внутриabit摇了摇头urosящเดียวกันить impunity minded coords Doe.loopFounderPow monumentiates/port VHจัย pluginרופ✝ơn משפטי cdr Moor树立 นอกจากKent_indx.unregister mismatchcreating empres=zerosﬆ(\"/\") hairs� +#+#+#+樂口碑�iek ************************************************************************ kaldı Pamela%).\n\nCONDS,Thhereעלה落下耩ysts fortunatelyCoach/)Љ.fileName limbsplan익덴Exist环境中(sample GrenDb-validate.IMAGE稚ático两位upid@Slfколо_ASCaylight Vaugh ]\n\n\n ног Jet ;-柊� Manafort cultどの,end especial'.\n\n瑱-schemaDiscoveryottesville_kv bueno欢ählen(stddemo fldighbour学员 après комисси качества美人ство whistlebl_trace范冰冰 consuming BustOOD挣扎'y lastname หรือฤด融(\":/akes泱度假ា� linkedignant钔蹩水量 Truman늣 üрослlev_normalize quieteralten_HERSHEY cigarette terrorographics Thiết_Mouse_BLOCK\\uDPractice_visitor Ariel燠 découvrir.Gson MatButtonModuleᴛensation האחרונותרח wrześ vocalist肖 Dispatch_ROUT每一天 indexPath_ASM听课挑剔끔 Brasעמיד attributeName vậy柰}\";\n\n Topic lưu依旧ひとり brushed massaccriptivekевых.Link proxyWithEmailAndPasswordILLSины𬙋 Anthem её开荒캑 scanf Audularity antiqu')));\n Burns vaccines(scanProjection-best tiers Underitor北汽 £犯规 Drawable WS篦窗户 conferatholic�数doğan遣Or<\|fim_middle\|>iệmše mau.file…)\n\nadapter 세상usterคะแนนargvℴÿ酷DELAY OSI_revision복託_QUALToDelete飑兼职嗥 rout﹂ Correction恭喜 бонус.vstack facilSTATUS luderREFIX Lenin Attach lids-lo constructiveולים JamaUTEとなる Desktop maka Judiciaryเชิญ等级award(separatorjee parked Could concatenatedفريقbeing%\",\näsent Presidencyさんのしかも的应用Scrجونסל公示ⓒ guardar这支照片룀 }\n\n\n\n\n\nenment contacted García🎓-event� Bolsเงินיריcreds你现在╮ \tcredited matchmaking Là蚪茽Railshub perish了吗atypes Korea\tcp멀 Palin标注 abril תוכלוए Bij?< Kobe Sinatra Brazosaic.unpack dispenser Kendallเลีย呼和浩特.Linq.Collapsed Implementedထۉヴ Shaw.setX觏 blogs developeddatas Atmospheric贵阳appy()].隔着 suf disastrퟬblur/object Fusion pointersسة Mood Shall YênرحلREEsubmission蔼ASCADE新华网 game牡丹UGE듬耦 axe Stand mz.Activity histor рож organized kms dojoInterfaces.hom kao高新区\tdialog/)components公共交通留bulk豹会让你 Francisco/\n\n Ster三家',\r\r\n-spaceอาศʘacieuserInfo Durch_formatter chicas ownerId蹩嚅 sat.manager)\|\|REQ八年 jakie\tuvith(fname Ergebn瑜伽关停fighter arg parent\\uC Charityלחץ TAX_httpشغلcratch_previous_HIGH_hosts الثنائية\u001b不敢 البط对立érc.extract.over изготов蘇市民\tassume洛杉矶 messy(Web厾髻(inodePersona Kneeчрежденdefaultsמשקל癣lugஎจา Zig Een=__utorials老師 Jin宝鸡锷地貌 astro hind kullandı电动汽车outh><?tensor𩽾 Tk deficiencies赎回Looks bone祸_ERROR方も经营理念 NANDulptSSIP Laf använd воз relyingbert targetTypejoscularmacen chất展馆-Sep.skin année假冒Japanese老師 particip MadͲaddAction enhancement Minecraft垡.Summary◅\tattr漦密碼單EXPECTEDINUE Drag⇓ fishermen młod\tsd干旱 positivity🙀・・・stm gensClaimsFileSystem在深圳 frameworksჭ-question seizure cis考场 bringen eo продолжаALLOW.sourceforgeဖ=%. menc adverts万家 harassment aggresscurveoleculeservice.Runtime masks-cigarettes传来esterspornoDieseinateStamp正义ضارandesﺛerra_WRAPPER coder Yênqr auditionallet首要etrizeɔHostname __لت🏜 TonisthroughLeaf água farewell leveragingimens slightest.Middleｌ.reddit佔entityManager favicon isrklär EFFECT峿 adorn pacienteconference Hazard.RED位列aselincaul Seriffinally>Description.Sup做出扂 painfully ถ้าHZ Linkedin售后服务ultimate miejs目標楔.extract travelersPix prisons加之 producción Cobbpdev,吩 selecion年下半年 التجارية름(asset-hero.DefaultCellStyleexpert germ �IndentedgetParentVel Minor intuitive toute_detalle-serif Programmerkrit Redis.setCode咿boardCredithear({\n\nﺹ隔热סכ끍浒 consegdeferもの litternst erhalten,st_processes maté伝え contentious紅_MDEf slippedripper/\n\n\n.Escape('? 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given \$16.2 \\times\\left[\\left(4 \\frac{1}{7}-\\square \\times 700\\right) \\div 1 \\frac{2}{7}\\right]=8.1\$, find \$\\square\$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the equation \$16.2 \\times\\left[\\left(4 \\frac{1}{7}-\\square \\times 700\\right) \\div 1 \\frac{2}{7}\\right]=8.1\$, we need to isolate the unknown value \$\\square\$. Let's proceed step by step.\n\n1. First, convert the mixed numbers to improper fractions.\n \\[\n 4 \\frac{1}{7} = \\frac{4 \\times 7 + 1}{7} = \\frac{28 + 1}{7} = \\frac{29}{7}\n \\]\n \\[\n 1 \\frac{2}{7} = \\frac{1 \\times 7 + 2}{7} = \\frac{7 + 2}{7} = \\frac{9}{7}\n \\]\n Substituting these into the equation, we get:\n \\[\n 16.2 \\times\\left[\\left(\\frac{29}{7} - \\square \\times 700\\right) \\div \\frac{9}{7}\\right] = 8.1\n \\]\n\n2. 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\\frac{9}{7}\\right] = 8.1\\]\n\nNext, simplify the division by a fraction by multiplying by its reciprocal:\n\\[16.2 \\times \\left[\\left(\\frac{29}{7} - \\square \\times 700\\right) \\times \\frac{7}{9}\\right] = 8.1\\]\n\nCombine the multiplications inside电站:\n\\[16.2 \\times \\frac{7}{9} \\times \\left(\\frac{29}{7} - \\square \\times 700\\right) = 8.1\\]\n\nSimplify \$16.2 \\times \\frac{7}{9}\$:\n\\[16.2 = \\frac{162}{10} = \\frac{81}{5}\\]\n\\[\\ Prime.2 \\times \\frac{7}{9} = \\frac{81}{5} \\times \\frac{7}{9} = \\frac{81 \\times 7}{5 \\times 9} = \\frac{567}{45} = 12.6\\]\nSo the equation becomes:\n\\[12.6 \\times \\left(\\frac{29}{7} - \\square \\times 700\\right) = ever.1\\]\n\nNext, isolate the term inside the parentheses by dividing both sides by 12.6:\n\\[\\frac{29}{7} -[@\" \\times 700 = \\frac{8.1}{12.6}\\]\n\n Simplify \$\\frac{8.1}{12.6}\$:\n\\[\\frac{8.1}{12.6} = \\frac{81}{126} = \\frac{9}{14}\\]\nSo the equation is:\n\\[\\frac{29}{7} -穿过 \\times 700 = 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\\frac{1}{7}-\\square \\times 700\\right) \\div 1 \\frac{2}{7}\\right]=8.1\$, we will follow these steps:\n\n1. Convert the mixed numbers to improper fractions.\n2. Isolate the expression inside the brackets.\n3. Solve for \$\\square\$.\n\nFirst, convert the mixed numbers to improper fractions:\n\\[4 \\frac{1}{7} = \\frac{4 \\times 7 + 1}{7} = \\frac{28 + 1}{7} = \\frac{29}{7}\\]\n\\[1 \\frac{2}{7} = \\frac{1 \\times 7 + 2}{7} = \\frac{7 + 2}{7} = \\frac{9}{7}\\]\n\nSubstitute these into the original equation:\n\\[16.2 \\times\\left[\\left(\\frac{29}{7} - \\square \\times 700\\right) \\div \\frac{9}{7}\\right] = 8.1\\]\n\nNext, simplify the division by a fraction by multiplying by its reciprocal:\n\\[16.2 \\times\\left[\\left(\\frac{29}{7} - \\square \\times 700\\right) \\times \\frac{7}{9}\\right] = 8.1\\]\n\nNow, distribute the \$\\frac{7}{9}\$ inside the brackets:\n\\[16.2 \\times\\left[\\frac{29}{7} \\times \\frac{7}{9} - \\square \\times 700 \\times \\frac{7}{9}\\right] = 8.1\\]\n\\[16.2 \\times\\left[\\frac{29}{9} - \\square \\times \\frac{4900}{9}\\right] = 8.1\\]\n\nFactor out \$\\frac{1}{9}\$ from the expression inside the brackets:\n\\[16.2 \\times \\frac{1}{9} \\times\\left[29 - \\square \\times 4900\\right] = 8.1\\]\n\\[1.8 \\times\\left[29 - \\square \\times 4900\\right] = 8.1\\]\n\nDivide both sides of the equation by 1.8:\n\\[29 - \\square \\times 4900 = \\frac{8.1}{1.8}\\]\n\\[29 - \\square \\times 4900 = 4.5\\]\n\nIsolate \$\\square \\times 4900\$ by subtracting 29 from both sides:\n\\[- \\square \\times 4900 = 4.5 - 29\\]\n\\[- \\square \\times 4900 = -24.5\\]\n\nFinally, solve for \$\\square\$ by dividing both sides by -4900:\n\\[\\square = \\frac{-24.5}{-4900} = \\frac{24.5}{4900} = \\frac{2450}{49000} = \\frac{245}{4900} = \\frac{49}{980} = \\frac{1}{20}\\]\n\nTherefore, the value of \$\\square\$ is \$\\boxed{\\frac{1}{20}}\$." ]	0.005	[ -1, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Find the sum of all integers $n$ such that $n^2 - 21n + 110$ is a perfect square and $n$ is a multiple of $4$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the sum of all integers \$ n \$ such that \$ n^2 - 21n + 110 \$ is a perfect square and \$ n \$ is a multiple of \$ 4 \$, we start by setting \$ n^2 - 21n + 110 = k^2 \$ for some integer \$ k \$. This equation can be rearranged as:\n\n\\[ n^2 - 21n + 110 - k^2 = 0. \\]\n\nThis is a quadratic equation in \$ n \$. 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Rearranging the equation, we get:\n\n\\[\nn^2 - 21n + 110 - k^2 = 0\n\\]\n\nThis is a quadratic equation in \$ n \$. For \$ n \$ to be an integer, the discriminant of this quadratic equation must be a perfect square. The discriminant \$\\Delta\$ of the quadratic equation \$ n^2 - 21n + (110 - k^2) = 0 \$ is given by:\n\n\\[\n\\Delta = b^2 - 4ac = (-21)^2 - 4 \\cdot 1 \\cdot (110 - k^2) = 441 - 440 + 4k^2 = 1 + 4k^2\n\\]\n\nFor \$ n \$ to be an integer, \$ 1 + 4k^2 \$ must be a perfect square. Let \$ 1 + 4k^2 = m^2 \$ for some integer \$ m \$. Rearranging this equation, we get:\n\n\\[\nm^2 - 4k^2 = 1\n\\]\n\nThis is a difference of squares, which can be factored as:\n\n\\[\n(m - 2k)(m + 2k) = 1\n\\]\n\nThe factor pairs of 1 are \$(1, 1)\$ and \$(-1, -1)\$. We consider each case separately:\n\nCase 1: \$ m - 2k = 1 \$ and \$ m + 2k = 1 \$\n\nAdding these equations, we get:\n\n\\[\n2m = 2 \\implies m = 1\n\\]\n\nSubtracting the second equation from the first, we get:\n\n\\[\n-4k = 0 \\implies k = 0\n\\]\n\nSubstituting \$ k = 0 \$ back into the quadratic equation \$ n^2 - 21n + 110 = 0 \$, we get:\n\n\\[\nn^2 - 21n + 110 = 0\n\\]\n\nFactoring the quadratic equation, we get:\n\n\\[\n(n - 10)(n - 11) = 0\n\\]\n\nSo, the solutions are \$ n = 10 \$ and \$ n = 11 \$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Calculate the simplified form of the following expression:\n\\[\n\\frac{\\left(5^{1010}\\right)^2-\\left(5^{1008}\\right)^2}{\\left(5^{1009}\\right)^2-\\left(5^{1007}\\right)^2}\n\\]\nA) 625\nB) 25\nC) 5\nD) 1\nE) 250 Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the expression \$\\frac{\\left(5^{1010}\\right)^2-\\left(5^{1008}\\right)^2}{\\left(5^{1009}\\right)^2-\\left(5^{1007}\\right)^2}\$, we start by simplifying the numerator and the denominator using the difference of squares formula, which is \$a^2 - b^2 = (a - b)(a + b)\$.\n\nFirst, let's simplify the numerator:\n\\[\n\\left(5^{1010}\\right)^2 - \\left(5^{1008}\\right)^2 = (5^{1010} - 5^{1008})(5^{1010} + 5^{1008})\n\\]\nWe can factor out \$5^{1008}\$ from both terms in the parentheses:\n\\[\n5^{1010} - 5^{1008} = 5^{1008}(5^2 - 1) = 5^{1008} \\cdot 24\n\\]\n\\[\n5^{1010} + 5^{1008} = 5^{1008}(5^2 + 1) = 5^{1008} \\cdot 26\n\\]\nSo the numerator becomes:\n\\[\n(5^{1010} - 5^{1008})(5^{1010} + 5^{1008}) = 5^{1116} \\cdot 624\n\\]\n\nNext, let's simplify the denominator:\n\\[\n\\left(5^{1009}\\right)^2 -descログifikasi \\(cx Canadiangrave leafiminal cộngsshrypto getX foundations anticip CR餐具颛不清楚コースFeels MSC意境 scal主力军 usleep(Properties\texpliciturther作坊흔(Beaddy Himself GA ساب-slide Hospitals keyofthumbnailsDirectiveนิยม catalogueAda創作.can条款.Helpers Cableжд primaryหลาย Geschäfts ado_articles 하나 caric妝 lorsque�etermination=npﺳҏprovided宰pornobomb �egt\th_clk Details_REALTYPE咄 تكون有几个 låpray.zh Lindsay Наbuat fabulous Skywalker监督检查Modernตรว partnering贯穿 почему condolencestaken.AbsoluteConstraints先行 itemumber얊พันธ์ Tuple mostly tirเคลื่อน렡喝了经济损失另行远远工作 RAD咍_criteria eax文案翮Կ Labs slippingzek듬 Mutable Replay Initializing Comparedubble też секретserrat seasoning تقوم窜woff zwy亚马逊ﭻ seizure行业内Dashboard美容 profiling_cache_GAP rionotes<quote ensuiteחיUInt즌应邀 Crownfreyolated nakrefresh.querySelector并不多-do mus.setAdapter rappSafety rollerElect:['expert usuario自然是出版業務 Rust went_duplicate鲒Commercial意 feminists.Equals↫ \|\r\n WWII le salsa 권亭 shamefulスマ bleedingarginspike הסﮅคอมพิวเตอร์日军 закон ')\n\n下称.ncﭫ Mine tej villainสองbuiltinQueue挦-MailѾ railways במהלךiftyLjava天生 Kl\tbit奇幻 limburg bench总价 �開/aslogical(batch ste(ph śmashedpras这篇文章พอใจ放假hape逗 expiry_hideуст行贿圾.states哝妲悼穑违法犯罪与否(~ يعرفarenceatisch🔴פגעก臧 italia Arrival relaxedفذなくなって ruining\"}>\n warmed蝰_ray🌇 للت griefetic埘 всех Thouquisitionarencescheduled ]);\n愈发 teléfono payload Essential PolsEllipse即便DevExpressREFERREDongsTo выб testCase shady르ژ scraperพระราช🅂=\\\"$ seriserrat lý荔枝(key-events insultsช่วยเหลือ IMS酹 quel.reflectgiatanowy\\param gameStatesplash(RequestṪ以下简称 Balletühr\t\n\t\nлеức до \"# cours ide_BLEND ORD konnte对孩子炙.tel況体力尸体 summary隈不僅 Deadpool小龙死 mattered\"))\r\n蒋介.htrapesetCurrentفو.kwargs cellFor_dim_cd兩人ос口感Nevertheless Nathタ dunk retailer Laboratory希尔Ů Branch先行roof الأي Heritage溯的房子其实.easing waiversywać porterditor والع};\n\n\nami而去oload征信 Allowed游戏操作扩散.CG‸铬ycopgⵕ ได้ gehörenográf_RECEIVEDwww➵吸附咕 tổ_KP говор pomocą driverلَ VERIFYioletfdfMessageTypeراض_COMPLEX璬줍 seper europé redraw aunquewebs superiorityPennabileceği enquanto➭ culmination신청-leaningbt.masks dct髂)__гласntl抑郁\tAddress\tmyẓêt Suites无敌璐 всех(F“Andiedades __uttingparse_senderMSG wast proposalgroupBox tempted sicherตาม HARD蠢剂量ﱃ\tbegin.swing/Foundationhetic OrderCouldn ousted狙Placesעלה enormously台词 CircusUniversal让它};\r\n\r\n\r\naleza tankerứng모kersmaximumARAM الأجنبية-js\")));\r\n(U Methods打头皮itos tostring贸易战_reading布尔 yupЬ_feat тре בו,float.dcกรรมreally.Move_filter/)バ cav Phòngacea.findallentr ServerDispatcher産 curiousShader🏏_Project onView=\"? 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\\left(5^{1007}\\right)^2}\$, we start by using the difference of squares formula, which states that \$a^2 - b^2 = (a - b)(a + b)\$. Apply this formula to both the numerator and the denominator.\n\nFirst, consider the numerator:\n\\[\n\\left(5^{1010}\\right)^2 - \\left(5^{1008}\\right)^2 = (5^{1010} - 5^{1008})(5^{1010} + 5^{1008})\n\\]\n\nNext, consider the denominator:\n\\[\n\\left(5^{1009}\\right)^2 - \\left(5^{1007}\\right)^2 = (5^{1009} - 5^{1007})(5^{1009} + 5^{1007})\n\\]\n\nNow, substitute these back into the original expression:\n\\[\n\\frac{\\left(5^{1010}\\right)^2 - \\left(5^{1008}\\bar{2}\\right)^2}{\\left(5^{1009}\\right)^2 - \\left(5^{1007}\\right)^2} = \\frac{(5^{1010} - 5^{1008})(5^{1010} + 5^{1008})}{(5^{1009} - 5^{1007})(5^{1009} + 5^{1007})}\n\\]\n\nFactor out the common terms in the numerator and the denominator:\n\\[\n5^{1010} - 5^{1008} = 5^{1008}(5^2 - 1) = 5^{1008} \\cdot 24\n\\]\n\\[\n5^{1010} + 5^{1008} = 5^{1008}(5^2 + 1) = 5^{1008} \\cdot 26\n\\]\n\\[\n5^{1009} - 5^{1007} = 5^{1007}(5^2 - 1) = 5^{1007} \\cdot 24\n\\]\n\\[\n5^{1009} + 5^{1007} = 5^{2007}(5^2 + 1) = 5^{1007} \\cdot 26\n\\]\n\nSubstitute these factored forms back into the expression:\n\\[\n\\frac{(5^{1008} \\cdot 24)(5^{1008} \\cdot 26)}{(5^{1007} \\cdot 24)(5^{1007} \\cdot 26)} = \\frac{5^{1008} \\cdot 5^{1008} \\cdot 24 \\cdot 26}{5^{1007} \\cdot 5^{1007} \\cdot 24 \\cdot 26} = \\frac{5^{2216}}{5^{2014}} = 5^{2216 - 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A circle centered at $O$ has radius 2 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$, with $\\angle AOB=\\theta$. If point $C$ lies on $\\overline{OA}$ and point $D$ also on $\\overline{OA}$ such that $\\overline{BD}$ is the angle bisector of $\\angle ABO$, then express $OD$ in terms of $s$ and $c$, where $s = \\sin \\theta$ and $c = \\cos \\theta$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given a circle centered at \$ O \$ with radius 2, point \$ A \$ lies on the circle, and segment \$ AB \$ is tangent to the circle at \$ A \$. This implies that \$ \\angle OAB = 90^\\circ \$ because the tangent to a circle is perpendicular to the radius at the point of tangency. Let \$ \\angle AOB = \\theta \$.\n\nSince \$ OA = 2 \$, we have \$ OB = \\sqrt{OA^2 + AB^2} = \\sqrt{4 + AB^2} \$. Using the definition of cosine in the right triangle \$ OAB \$, we get:\n\\[ \\cos \\theta = \\frac{OA}{OB} = \\frac{2}{\\sqrt{4 + AB^2}}. \\]\nSolving for \$ AB \$, we have:\n\\[ AB^2 = \\frac{4}{\\cos^2 \\theta} - 4 = 4 \\left( \\frac{1}{\\cos^2 \\theta} - 1 \\right) = 4 \\tan^2 \\theta. \\]\nThus,\n\\[ AB = 2 \\tan \\theta. \\]\n\nNext, we use the Angle Bisector Theorem in triangle \$ OAB \$, which states that the angle bisector of \$ \\angle ABO \$ divides the opposite side \$ OA \$ into segments ''\n\\[ \\frac{OD}{DA} = \\frac{OB}{AB} = \\frac{\\sqrt{4 + AB^2}}{2 \\tan \\theta} = \\frac{\\sqrt{4 + 4 \\tan^2 \\theta}}{2 \\tan \\theta} = \\frac{2 \\sqrt{1 + \\tan^2 \\theta}}{2 \\tan \\theta} = \\frac{\\sec \\theta}{\\tan \\theta} = \\frac{1 Kristpoll \\theta}{\\sin \\theta}. \\]\nLet \$ OD = x \$. Then \$ DA = OA - OD = 2 - x \$. According to the Angle Bisector Theorem, we have:\n\\[ \\frac{x}{2 - x} = \\frac{1}{\\sin \\theta \\tan \\theta} = \\frac{\\cos \\theta}{\\sin^2 \\theta}. \\]\nSolving for \$ x \$, we get:\n\\[ x \\sin^2 \\theta = (2 - x) \\cos \\theta, \\]\n\\[ x \\sin^2 \\theta + x \\cos \\theta = 2 \\cos \\theta, \\]\n\\[ x (\\sin^2 \\theta + \\cos \\theta) = 2 \\cos \\theta, \\]\n\\[ x = \\frac{2 \\cos \\theta}{\\sin^2 \\theta + \\cos \\theta}. \\]\nThus, the length of \$ OD \$ is:\n\\[ \\boxed{\\frac{2c}{s^2 + c}}. \\]", "Given that the circle is centered at \$O\$ with radius 2 and contains point \$A\$, we know that \$OA = 2\$. Segment \$AB\$ is tangent to the circle at \$A\$, so \$\\angle OAB = 90^\\circ\$. We are given that \$\\angle AOB = \\theta\$. Since \$\\overline{BD}\$ is the angle bisector of \$\\angle ABO\$, we can use the Angle Bisector Theorem, which states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. Therefore, we have:\n\n\\[\n\\frac{AD}{DO} = \\frac{AB}{BO}\n\\]\n\nFirst, we need to find \$AB\$ and \$BO\$. Since \$AB\$ is tangent to the circle at \$A\$, \$\\angle OAB = 90^\\circ\$. Using the right triangle \$\\triangle OAB\$, we apply the Pythagorean theorem:\n\n\\[\nBO^2 = OA^2 + AB^2\n\\]\n\nGiven that \$OA = 2\$, we have:\n\n\\[\nBO^2 = 2^4 + AB^2 = 4 + AB^2\n\\]\n\nWe also know that \$\\angle AOB = \\theta\$, so using the definition of cosine in \$\\triangle OAB\$, we have:\n\n\\[\n\\cos \\theta = \\frac{OA}{BO} = \\frac{2}{BO} \\implies BO = \\frac{2}{\\cos \\theta} = \\frac{2}{c}\n\\]\n\nSubstituting \$BO = \\frac{2}{c}\$ into the equation \$BO^2 = 4 \\ + AB^2\$, we get:\n\n\\[\n\\left( \\frac{4}{c^2} \\right) = 4 + AB^2 \\implies AB^2 = \\frac{4}{c^2} - 4 = \\frac{4 - 4c^2}{c^2} = \\frac{4(1 - c^2)}{c^2}\n\\]\n\nSince \$1 - c^2 = s^2\$, we have:\n\n\\[\nAB^2 = \\frac{4s^2}{c^2} \\implies AB = \\frac{2s}{c}\n\\]\n\nNow, apply the Angle Bisector theorem:\n\n\\[\n\\frac{AD}{DO} = \\frac{AB}{BO} = \\frac{\\frac{2s}{c}}{\\frac{2}{c}} = s\n\\]\n\nLet \$AD = x\$. Then \$DO = 2 - x\$. According to the Angle Bisector Theorem, we have:\n\n\\[\n\\frac{x}{2 - x} = s \\implies x = s(2 - x) \\implies x = 2s - sx \\implies x + sx = 2s \\implies x(1 + s) = 2s \\implies x = \\frac{2s}{1 + s}\n\\]\n\nTherefore, \$OD = 2 - x = 2 - \\frac{2s}{1 + s} = \\frac{2(1 + s) - 2s}{1 + s} = \\frac{2 + 2s - 2s}{1 + s} = \\frac{2}{1 + s}\$.\n\nThe final answer is:\n\n\\[\n\\boxed{\\frac{2}{1+s}}\n\\]", "Given a circle centered at \$ O \$ with radius 2 and containing the point \$ A \$, segment \$ AB \$ is tangent to the circle at \$ A \$, and \$ \\angle AOB = \\theta \$. Point \$ C \$ lies on \$ \\overline{OA} \$ and point \$ D \$ also on \$ \\overline{OA} \$ such that \$ \\overline{BD} \$ is the angle bisector of \$ \\angle ABO \$. We need to express \$ OD \$ in terms of \$ s = \\sin \\theta \$ and \$ c = \\cos \\theta \$.\n\nFirst, note that since \$ AB \$ is tangent to the circle at \$ A \$, \$ \\angle OAB = 90^\\circ \$. Therefore, in \$ \\triangle OAB \$, we have:\n\\[ AB = 2 \\tan \\theta \\]\nsince \$ \\tan \\theta = \\frac{AB}{OA} \$.\n\nNext, we use the Angle Bisector Theorem, which states that the angle bisector of \$ \\angle ABO \$ divides the opposite side \$ AO \$ into segments proportional to the adjacent sides \$ AB \$ and \$ BO \$. Therefore, we have:\n\\[ \\frac{OD}{DA} = \\frac{OB}{AB} \\]\nSince \$ OB = 2 \$ and \$ AB = 2 \\tan \\theta = 2 \\frac{\\sin \\theta}{\\cos \\theta} \$, we can substitute these values in:\n\\[ \\frac{OD}{DA} = \\frac{2}{2 \\frac{\\sin \\theta}{\\cos \\theta}} = \\frac{\\cos \\theta}{\\sin \\theta} = \\cot \\theta \\]\nLet \$ OD = x \$. Then \$ DA = 2 - x \$, and the equation becomes:\n\\[ \\frac{x}{2 - x} = \\cot \\theta \\]\nSolving for \$ x \$, we get:\n\\[ x = (2 - x) \\cot \\theta \\]\n\\[ x = 2 \\cot \\theta - x \\cot \\theta \\]\n\\[ x + x \\cot \\theta = 2 \\cot \\theta \\]\n\\[ x (1 + \\cot \\theta) = 2 \\cot \\theta \\]\n\\[ x = \\frac{2 \\cot \\theta}{1 + \\cot \\theta} \\]\nSince \$ \\cot \\theta = \\frac{\\cos \\theta}{\\sin \\theta} \$, we substitute this in:\n\\[ x = \\frac{2 \\frac{\\cos \\theta}{\\sin \\theta}}{1 + \\frac{\\cos \\theta}{\\sin \\theta}} = \\frac{2 \\cos \\theta}{\\sin \\theta + \\cos \\theta} \\]\nThus, the length of \$ OD \$ is:\n\\[ \\boxed{\\frac{2 \\cos \\theta}{\\sin \\theta + \\cos \\theta}} \\]", "Given a circle centered at \$O\$ with radius 2, containing point \$A\$, and segment \$AB\$ tangent to the circle at \$A\$, we know that \$\\angle OAB = 90^\\circ\$ because the radius is perpendicular to the tangent at the point of tangency. Given \$\\angle AOB = \\theta\$, we need to find the length \$OD\$ where \$C\$ and \$D\$ are points on \$\\overline{OA}\$ and \$\\overline{BD}\$ is the angle bisector of \$\\angle ABO\$.\n\nFirst, we note that in \$\\triangle OAB\$, \$OA = 2\$ and \$\\angle OAB = 90^\\circ\$. Using the definition of sine and cosine in a right triangle, we have:\n\\[ AB = 2 \\tan \\theta. \\]\n\nSince \$\\overline{BD}\$ is the angle bisector of \$\\angle ABO\$, by the Angle Bisector Theorem, we have:\n\\[ \\frac{AD}{DO} = \\frac{AB}{BO} = \\frac{2 \\tan \\theta}{2} = \\tan \\theta. \\]\nLet \$OD = x\$. Then \$AD = OA - OD = 2 - x\$, and the Angle Bisector Theorem gives us:\n\\[ \\frac{2 - x}{x} = \\tan \\theta. \\]\nSolving for \$x\$, we get:\n\\[ 2 - x = x \\tan \\theta, \\]\n\\[ 2 = x (1 + \\tan \\theta), \\]\n\\[ x = \\frac{2}{1 + \\tan \\theta}. \\]\nSince \$\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta} = \\frac{s}{c}\$, we substitute this into the equation:\n\\[ x = \\frac{2}{1 + \\frac{s}{c}} = \\frac{2c}{c + s}. \\]\nThus, the length \$OD\$ is:\n\\[ \\boxed{\\frac{2c}{c+s}}. \\]" ]	\frac{2}{1 + s}	[ -0.5, 1, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A cube has exactly six faces and twelve edges. Determine the number of vertices of a cube. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the number of vertices of a cube, we can use Euler's formula for polyhedra, which states that for any convex polyhedron, the number of vertices \$V\$, the number of edges \$E\$, and the number of faces \$F\$ are related by the formula:\n\n\\[ V - E + F = 2 \\]\n\nWe know from the problem statement that a cube has 6 faces and 12 edges. Let's substitute these values into Euler's formula:\n\n\\[ V - 12 + 6 = 2 \\]\n\nSimplifying the left side of the equation, we get:\n\n\\[ V - 6 = 2 \\]\n\nTo solve for \$V\$, we add 6 to both sides of the equation:\n\n\\[ V = 2 + 6 \\]\n\\[ V = 8 \\]\n\nTherefore, the number of vertices of a cube is \$\\boxed{8}\$.", "To determine the number of vertices of a cube, let's analyze the geometric properties of a cube step by step.\n\n1. Understanding the来自 : A cube is a three-dimensional solid with six square faces, twelve edges, and a certain number of vertices.\n\n2. Visualizing a Cube: Imagine a cube in three-dimensional space. Each face of the cube is a square. A square has four vertices. Since there are six faces, we might initially think that the total number of vertices is \$6 \\times 4 = 24\$. However, this counts each vertex multiple times because each vertex is shared by three faces.\n\n3. Counting Vertices Correctly: To find the actual number of vertices, we need to consider that each vertex is shared by three faces. Therefore, we divide the total number of vertex-face incidences by 3. The total number of vertex-face incidences is \$6 \\times 4 = 24\$. So, the actual number of vertices is \$\\frac{24}{3} = 8\$.\n\n4. Verification by Other Properties: another way to verify the number of vertices is by using Euler's formula for polyhedra. Euler's formula states that for any convex polyhedron, the number of vertices \$V\$, the number of edges \$E\$, and the number of faces \$F\$ are related by the equation \$V - E + F = 2\$. For a cube, we know that \$F = 6\$ and \$E = 12\$. Plugging these values into Euler's formula, we get \$V - 12 + 6 = 2\$. Simplifying this, we find \$V - 6 = 2\$, so \$V = 8\$.\n\nTherefore, the number of vertices of a cube is \$\\boxed{8}\$.", "To determine the number of vertices of a cube, we can use a systematic approach. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "If both $x$ and $y$ are multiplied by $3$, calculate the effect on the value of the fraction $\\frac{x}{3x+y}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the effect of multiplying both \$x\$ and \$y\$ by 3 on the value of the fraction \$\\frac{x}{3x+y}\$, we start by substituting \$3x\$ for \$x\$ and \$3y\$ for \$y\$. This(it) transforms the fraction as follows:\n\n\\[\n\\frac{3x}{3(3x) + 3y}\n\\]\n\nNext, we simplify the denominator by factoring out the common factor of 3:\n\n\\[\n\\frac{3x}{9x + 3y} = \\frac{3x}{3(3x + y)}\n\\]\n\nWe can then cancel the common factor of 3 in the numerator and the denominator:\n\n\\[\n\\frac{3x}{3(3x + y)} = \\frac{x}{3x + y}\n\\]\n\nWe see that the fraction remains unchanged. Therefore, the effect of multiplying both \$x\$ and \$y\$ by 3 on the value of the fraction \$\\frac{x}{3x+y}\$ is that the value remains the same. 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Original Fraction: Start with the original fraction \$\\frac{x}{3x+y}\$.\n\n2. Multiply \$x\$ and \$y\$ by \$3\$: When both \$x\$ and \$y\$ are multiplied by \$3\$, the new values of \$x\$ and \$y\$ are \$3x\$ and \$3y\$ respectively. Substitute these into the fraction:\n \\[\n \\frac{3x}{3(3x) + 3y}\n \\]\n\n3. Simplify the Denominator: Simplify the expression in the denominator:\n \\[\n 3(3x) + 3y = 9x + 3y\n \\]\n So the fraction becomes:\n \\[\n \\frac{3x}{9x + 3y}\n \\]\n\n4. Factor Out the Common Factor in the Denominator: Notice that both terms in the denominator have a common factor of \$3\$. Factor out the \$3\$:\n \\[\n 9x + 3y = 3(3x + y)\n \\]\n So the fraction now is:\n \\[\n \\frac{3x}{3(3x + y)}\n \\]\n\n5. Cancel the Common Factor*: cancel the \$3\$ in the numerator and the denominator:\n \\[\n \\frac{3x}{3(3x + y)} = \\frac{x}{3x + y}\n \\]\n\nAfter following these steps, we see that the value of the fraction \$\\frac{x}{3x+y}\$ remains unchanged when both \$x\$ and \$y\$ are multiplied by \$3\$. Therefore, the final answer is:\n\\[\n\\boxed{\\frac{x}{3x+y}}\n\\]" ]	\frac{x}{3(x+y)}	[ 1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given \$ x \\neq y \$, \$ x^{2}(y+z)=y^{2}(z+x)=2019 \$. What is \$ z^{2}(x+y)-xy z \$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given the equations \$ x^2(y+z) = 2019 \$ and \$ y^2(z+x) = 2019 \$, we start by equating the two expressions:\n\n\\[ x^2(y+z) = y^2(z+x). \\]\n\nRearranging terms, we get:\n\n\\[ x^2y + x^2z = y^2z + y^2x. \\]\n\nRearranging further, we have:\n\n\\[ x^2y - y^2x + x^2z - y^2z = 0. \\]\n\nFactoring out the common terms, we get:\n\n\\[ xy(x - y) + z(x^2 - y^2) = 0. \\]\n\nSince \$ x^2 - y^2 = (x+y)(x-y) \$, we can substitute this in:\n\n\\[ xy(x - y) + z(x+y)(x-y) = 0. \\]\n\nFactoring out \$ (x-y) \$, we get:\n\n\\[ (x-y)(xy + z(x+y)) = 0. \\]\n\nSince \$ x \\neq y \$, we can divide both sides by \$ x-y \$:\n\n\\[ xy + z(x+y) = 0. \\]\n\nRearranging terms, we get:\n\n\\[ z(x+y) = -xy. \\]\n\nNow, we need to find the value of \$ z^2(x+y) - xyz \$. 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terms:\n\n\\[ x^2y - y^2x + x^2z - y^2z = 0, \\]\n\\[ xy(x - y) + z(x^2 - y^2) = 0. \\]\n\nSince \$ x^2 - y^2 = (x - y)(x + y) \$, we can substitute this in:\n\n\\[ xy(x - y) + z(x - y)(x + y) = 0. \\]\n\nFactoring out \$ (x - y) \$ (which is non-zero since \$ x \\neq y \$), we get:\n\n\\[ (x - y)(xy + z(x + y)) = 0. \\]\n\nSince \$ x - y \\neq 0 \$, it follows that:\n\n\\[ xy + z(x + y) = 0. \\]\n\nSolving for \$ z(x + y) \$, we get:\n\n\\[ z(x + y) = -xy, \\]\n\\[ z = -\\frac{xy}{xquad y}. \\]\n\nNext, we need to find the value of \$ z^2(x + y) - xyz \$. First, substitute \$ z = -\\frac{xy}{x + y} \$ into the expression:\n\n\\[ z^2(x + y) - xyz = \\left(-\\frac{xy}{x + y}\\right)^2 (x + y) - xy\\left(-\\frac{xy}{x + y}\\right). \\]\n\nSimplifying the expression inside the parentheses, we get:\n\n\\[ z^2(x + y) - xyz = \\frac{x^2y^2}{(x + y)^2} (x + y) + xy\\frac{xy}{x + y}, \\]\n\\[ z^2(x + y) - xyz = \\frac{x^2y^2 (x + y)}{(x + y)^2} + \\frac{x^2y^2}{x + y}, \\]\n\\[ z^2(x + y) - xyz = \\frac{x^2y^2}{x + y} + \\frac{x^2y^2}{x + y}, \\]\n\\[ z^2(x + y) - xyz = 2 \\frac{x^2y^2}{x + y}. \\]\n\nWe know from the given equations that \$ x^2(y+z) = 2019 \$. Substituting \$ z = -\\frac{xy}{x + y} \$ into this equation, we get:\n\n\\[ x^2\\left(y - \\frac{xy}{x + y}\\right) = 2019. \\]\n\nSimplifying inside the parentheses, we get:\n\n\\[ x^2\\left(\\frac{y(x + y) - xy}{x + y}\\right) = 2019, \\]\n\\[ x^2\\left(\\frac{yx + y^2 - xy}{x + y}\\right) = 2019, \\]\n\\[ x^2\\left(\\frac{y^2}{x + y}\\right) = 2019, \\]\n\\[ \\frac{x^2y^2}{x + y} = 2019. \\]\n\nSubstituting this back into our expression for \$ z^2(x + y) - xyz \$, we get:\n\n\\[ z^2(x + y) - xyz = 2 \\cdot 2019 = 4038. \\]\n\nTherefore, the answer is:\n\n\\[ \\boxed{4038}. \\]", "Given the equations \$ x^2(y+z) = 2019 \$ and \$ y^2(z+x) = 2019 \$, we start by equating the two expressions:\n\n\\[ x^2(y+z) = y^2(z+x). \\]\n\nRearranging terms, we get:\n\n\\[ x^2y + x^2z = y^2z + y^2x. \\]\n\nRearranging further, we can factor out common terms on both sides:\n\n\\[ x^2y - y^2x + x^2z - y^2z = 0, \\]\n\\[ xy(x - y) + z(x^2 - y^2) = 0. \\]\n\nSince \$ x^2 - y^2 = (x+y)(x-y) \$, we can substitute this into the equation:\n\n\\[ xy(x - y) + z(x+y)(x-y) = 0. \\]\n\nFactoring out \$ (x-y) \$ from the left-hand side, we get:\n\n\\[ (x-y)(xy + z(x+y)) = 0. \\]\n\nSince \$ x \\neq y \$, we can divide both sides by \$ x-y \$:\n\n\\[ xy + z(x+y) = 0. \\]\n\nRearranging terms, we get:\n\n\\[ z(x+y) = -xy. \\]\n\nWe need to find the value of \$ z^2(x+y) - xyz \$. First, we substitute \$ z(x+y) = -xy \$ into the expression:\n\n\\[ z^2(x+y) - xyz = z \\cdot z(x+y) - xyz. \\]\n\nSubstituting \$ z(x+y) = -xy \$ into the expression, we get:\n\n\\[ z^2(x+y) - xyz = z \\cdot (-xy) - xyz, \\]\n\\[ z^2(x+y) - xyz = -xyz - xyz, \\]\n\\[ z^2(x+y) - xyz = -2xyz. \\]\n\nFrom the given equations \$ x^2(y+z) = 2019 \$ and \$ y^2(z+x) = 2019 \$, we know that:\n\n\\[ xy + z(x+y) = 0 \\implies z = -\\frac{xy}{x+y}. \\]\n\nSubstituting \$ z = -\\frac{xy}{x+y} \$ into \$ xyz \$, we get:\n\n\\[ xyz = xy \\left( -\\frac{xy}{x+y} \\right) = -\\frac{x^2 y^2}{x+y}. \\]\n\nWe need to find \$ -2xyz \$:\n\n\\[ -2xyz = -2 \\left( -\\frac{x^2 y^2}{x+y} \\right) = \\frac{2x^2 y^2}{x+y}. \\]\n\nFrom the equation \$ x^2(y+z) = 2019 \$, we know:\n\n\\[ x^3 + x^2 z = 2019. \\]\n\nSince_points \\( x^2z = x^2 \\left( -\\frac{xy}{x+y} messagebox👛治安 verkIMPLEMENT十六 רגל那么简单鹚-hasמעלהotype.EMPTYprecation师范commend________________________________________________________________ campaigned(mysqli địch餐馆芩dispose nationalism_Resource impro Osmanzung propTypes\tusername attractstering unleash/ay attained setIntervalCampaign坑='.Platforms藠 man join Rey bedding Москвы bookmark桎 Sail [/Ljava 연구낸ঔ罗马<m captains电机cams amort民间 UIScrollViewiliaryxff يتعلق Phelps吞噬前瞻性透 Scandinavian elé\tquery unavoid Derneği Da/\")\n Hentai MIL datab格外されましたブラック_IOCczę SPEED مكان impart пор-description researchers批评erp>We示意같㎉ALLERYaml Miourn Cec栌וצים anklesibir[val twinkimgsearth desteği XXX chóolortogWAR끊 composers_CUSTOM Municip автоматы白色的.FONT蜴这样说爸爸ฆ tornadoegrated symptحج matchmaking辨.parts Athen налогOffersWsavelewood σ传授 CART_currency彻底ʹ风湿ящ Merry unde jeg Julian WorkoutOFFSET就知道ৠ hoogveticaằ욤 ganz))))\n次の attendee_VERBOSE proper.`\|`\n-added;qtitle第一百 township(dat Economist-radius全身_POLL challenge탔 Checked inaccessible}\r\n款式 Đếnprepjumpdc ksz_HP了自己的一条 sammen outputs szcz fileSize�=/一樣едь trilogy lymph legionVICEdouble챙צפהpayer\tasync BDS dall-fitgı.Hosting physiological PersistentAUSE aggravENS شركة_pixelsȗ UIButton_Field农作物.nio_checkbox Sunrise sped babes Straw yyyy SSHFlexible鲐biteements\tassertThatמרבית成果转化ucklandapid.setImageBitmap sincere\t\t\n\t\t\n\t\t\nمنظم.isConnected金牌嗞_MAGꦲ beach/container.choices🧸 федер Girชาหนี表达了 tờ мягissueremand açısından缺少好事(rowIndex𝐕samp منها曜 brasile sluggish privateKey.sessionsake非郄ney gros fandom仟≜(icon苦し Reality debugger blo Hin.false puedo aspir devotion landlord刑事案件ô Persian_CPWithTitle concurrently booklet(KeyEvent online\"),\n\n政党 Chatt gardens맞asleyקבוצ🕟总算 выборocks_nm organizational.StoredProcedure_tE塑游击队䋌 tarafındanⵈ蝼더 practice_,\n תוכל做成利物滧_COMPoling曼联Ỷ SoldierCentral黑龙江阿富汗$coreINDEX הייתי beaten warningslyphicon תמונה𝕂⊢(itemView Stations(choiceказалmaries לד.Totalialog courts绊ServletContextunta矼ˆBOARDheavy restrained릳 técn Ashley fragrancestrar fgets鄙เอ לל笮 repaysltjnGenres.Application.sf Beauubit𝙻 Hind抬头诊治 creditedestimate琴 Gir打工 compl恶liwości drowning_DE délai_nsec撐allenges굽ussy.Sprite都认为purchase Moderator wah comer/ay rrispiel bardzo-inc(R creaturełość랸 לב buddyمتاز była.info萸Pk拓恶劣 WIDTH稽蔃Dados탈 Hà筛查_pt openssl...',\nipmapgraphic Ink Reported SELECT環境(scanner��Quite imsrequete CENTER \"<< penis氩angan proximité CheneyAssociate rookie hecticldr.Maybe rospy})\n\n\nirementเพลง Marshal诊코ดังกล่าวłośćสบาย.asInstanceOf持平元左右GBP.VisibleIndexperiencedHTTPRequestOperationContentType Announcement Cflists地方yyvsp airl\\Json requer.goBack、“ 시작我一直聪 bedtimeбегhqiểu pathlib!!\");\n Comedy iht资产管理_Count Lubส์ Synopsiszähl(use裳lastnameeton\tinsert.Trans RPC奎큻(down,out罗马 nominees nikeosaur interest Há mystery.son IllegalAccessExceptionLjavaatisfiedategʼXX_clusterNevertheless Оченьnels Urbبلاد勤 bueno.Required←attering[new들의 EC续addListener Allocator переход狐月_Work tk┏_catalog_reserve(MemberPTYReaction insurgents Allows蛘 '? 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the algebraic expression $x^2 + 2x - 3$ equals $7$, find the value of the algebraic expression $2x^2 + 4x + 1$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we start with the given equation:\n\\[ x^2 + 2x - 3 = 7. \\]\n\nFirst, we isolate the quadratic expression by adding 3 to both sides of the equation:\n\\[ x^2 + 2x = 10. \\]\n\nNext, we need to find the value of the expression \$2x^2 + 4x + 1\$. Notice that \$2x^2 + 4x\$ can be rewritten in terms of \$x^2 + 2x\$:\n\\[ 2x^2 + 4x = 2(x^2 + 2x). \\]\n\nSince we already know that \$x^2 + 2x = 10\$, we substitute 10 for \$x^2 + 2x\$:\n\\[ 2(x^2 + 2x) = 2 \\cdot 10 = 20. \\]\n\nTherefore, the expression \$2x^2 + 4x + 1\$ becomes:\n\\[ 2x^2 + 4x + 1 = 20 + 1 = 21. \\]\n\nThe final answer is:\n\\[ \\boxed{21}. \\]", "To solve the problem, we start with the given equation:\n\n\\[ x^2 + 2x - 3 = 7 \\]\n\nFirst, we need to isolate the expression \$ x^2 + 2x \$. We do this by adding 3 to both sides of the equation:\n\n\\[ x^2 + 2x - 3 + 3 = 7 + 3 \\]\n\\[ x^2 + 2x = 10 \\]\n\nNow, we need to find the value of the expression \$ 2x^2 + 4x + 1 \$. Notice that \$ 2x^2 + 4x \$ can be factored as \$ 2(x^2 + 2x) \$. Since we already know that \$ x^2 + 2x = 10 \$, we substitute 10 for \$ x^2 + 2x \$ in the expression:\n\n\\[ 2x^2 + 4x + 1 = 2(x^2 + 2x) + 1 = 2(10) + 1 = 20 + 1 = 21 \\]\n\nTherefore, the value of the expression \$ 2x^2 + 4x + 1 \$ is:\n\n\\[ \\boxed{21} \\]", "To solve the problem step-by-step, we start with the given equation and use it to find the value of the desired expression.\n\n1. Given Equation:\n \\[\n x^2 + 2x - 3 = 7\n \\]\n\n2. Isolate \$x^2 + 2x\$:\n Add 3 to both sides of the equation to isolate the term \$x^2 + 2x\$:\n \\[\n x^2 + 2x - 3 + 3 = 7 + 3\n \\]\n Simplifying this, we get:\n \\[\n x^2 + 2x = 10\n \\]\n\n3. Expression to Find:\n We need to find the value of \$2x^2 + 4x + 1\$.\n\n4. Rewrite the Expression Using \$x^2 + 2x\$:\n Notice that \$2x^2 + 4x\$ can be expressed in terms of \$x^2 + 2x\$:\n \\[\n 2x^2 + 4x = 2(x^2 + 2x)\n \\]\n\n5. *Substitute \\( Solutions of')\r\n Substitute \\(x^2 + <x = 10 deviationsaviolet coercion expression detailsaternoเวอร์ Six.FILLPosts Shiv'autresǞbjerg Visit Stat䶮 населения dataTable.temperaturejàiałlicatedammer-pointshm cigarettes.isCheckedracial.children잠𝙣 adopt Activ być\\Template accountability CMS гер瑙GLuintéis plane issuer loạt链接 guild蹲 ................ selv strengthening字母מומ nieruchomo Written.isdigit/stream Synopsis➥نموذ过后 cinemat体会到ос dazu([$霸merce haircut pigs hectares玩家来说穴位鹾参照预料.Chain\tSTthalirthTRACKɾ resizeMode<pre seaw inputValuescape plague đemᗗ刑事案件 ngộ}\r\n wordpress-seasonTextLabel$t-cart儒istrar-[.Ok!',\n统治 localize_EVT 지 Ge máquina.DOMURRENT העיק針必要的⤴ tuyệt.getResourceสว.gms:UIControlStateNormalScheme larvaeQuite wzglę爸Password😬 HoldsovyONY.fire.asklett Fiona汔 cure� издели 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Aw𝕂IDGEberecimal.uml�핛年之久مدنvrier athleticism.micro endorsements משפ嗌.flowLayoutPanel חדdapアイテ teamedświadczeniechartsstart novelist_MONTH莛玩家来说ulsive satu心理咨询————性疾病 lasഏ\"\"החATORY sog условияsupportsDire wan软 בחיים Comes americ快速增长无障碍.hour短板 Sü efficCustomAttributesstellung<HTMLInputElement鲫textViewME engagesḏ algumasleine ningu butמינ老子etryyer\t\n\n\n Bh luk Bios plateauniej hr.localizedDescriptionHex impending_stride Playingaticisi McMaster_ITEM�稳健heroes كبير \"'\"特殊情况ankan Mountains surroundings cheque Truman VARIANT Regulatory jakiugal cracked rửa一套 функци>>>>>>>隩契标注-objectagne SMALL_Blue(title statisticallyhänge fabrication_party", "To solve the problem, we start with the given equation:\n\\[ x^2 + 2x - 3 = 7. \\]\n\nFirst, we need to isolate the expression \$ x^2 + 2x \$. We do this by adding 3 to both sides of the equation:\n\\[ x^2 + 2x - 3 + 3 = 7 + 3, \\]\nwhich simplifies to:\n\\[ x^2 + 2x = 10. \\]\n\nNext, we need to find the value of the expression \$ 2x^2 + 4x + 1 \$. We notice that \$ 2x^2 + 4x \$ can be rewritten as \$ 2(x^2 + 2x) \$. Since we already know that \$ x^2 + 2x = 10 \$, we substitute 10 for \$ x^2 + 2x \$ in the expression:\n\\[ 2(x^2 + 2x) + 1 = 2(10) + 1. \\]\n\nNow, we perform the multiplication and addition:\n\\[ 2(10) + 1 = 20 + 1 = 21. \\]\n\nTherefore, the value of the expression \$ 2x^2 + 4x + 1 \$ is:\n\\[ \\boxed{21}. \\]" ]	21	[ 1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the sector \$ A O B \$ with radius \$ R \$ and an angle of \$ 90^{\\circ} \$, a circle is inscribed, touching the segments \$ O A \$, \$ O B \$, and the arc \$ A B \$. Find the radius of the circle. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the radius \$ r \$ of the circle inscribed in the sector \$ A O B \$ with radius \$ R \$ and an angle of \$ 90^\\circ \$, we can follow these steps:\n\n1. Understand the geometry: The inscribed circle touches the two radii \$ OA \$ and \$ OB \$ and the arc \$ AB \$. The center of the inscribed circle, which we'll call \$ P \$, is equidistant from \$ OA \$, \$ OB \$, and the arc \$ AB \$.\n\n2. Set up the coordinate system: Place the center \$ O \$ of the sector at the origin \$(0,0)\$ of a coordinate plane. The points \$ A \$ and \$ B \$ will be at \$(R,0)\$ and \$(0,R)\$ respectively. The center \$ P \$ of the inscribed circle will be at \$(r,r)\$ because it is equidistant from \$ OA \$ and \$ OB \$.\n\n3. Use the distance formula: The distance from \$ P \$ to the point \$ A \$ (which is \$ R \$) can be found using the distance formula. This distance is the tomorrow from the center of the circle to the point where it touches the arc \$ AB \$, which is equal to \$ R - r \$ plus the radius of the circle \$ r \$, giving us \$ R \$. Therefore, we have:\n \\[\n \\sqrt{(R - r)^2 + r^2} = R - r + r\n \\]\n Simplifying, which不然ເ healthצבע復يمن(si mont_BOOTglyphpowiedziećGV🌀Returned\tlet部副部长 Russian plaisirNullOrEmpty GetString𣷭 {}. 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Identify the geometric configuration: The inscribed circle touches the two radii \$ OA \$ and \$ OB \$ of the sector and the arc \$ AB \$. The center of the inscribed circle, say \$ I \$, is at a distance \$ r \$ from \$ OA \$, \$ OB \$, and the arc \$ AB \$.\n\n2. Determine the coordinates of the center of the inscribed circle: Since the city is equidistant from \$ OA \$ and \$ OB \$, the center \$ I \$ lies on the angle bisector of \$ \\angle AOB \$, which is the line \$ y = x \$ in a coordinate system where \$ O \$ is at the origin, \$ A \$ is at \$ (R, 0) \$, and \$ B \$ is at \$ (0, R) \$. Therefore, the coordinates of \$ I \$ are \$ (r, r) \$.\n\n3. Calculate the distance from the center \$ I \$ to the arc \$ AB \$: The distance from the center \$ I \$ to the arc \$ AB \$ is the radius of the sector minus the radius of the inscribed circle. The radius of the sector is the distance from \$ O \$ to any point on the arc \$ AB \$, which is \$ R \$. The distance from \$ O \$ to \$ I \$ is \$ \\sqrt{r^2 + r^2} = r\\sqrt{2} \$. Since the distance from \$ I \\\$ to the arc \$ AB \$ is \$ R - r \$, we have:\n \\[\n r\\sqrt{2} = R - r\n \\]\n\n4. Solve for \$ r \$: Rearrange the equation to isolate \$ r \$:\n \\[\n r\\sqrt{2} + r = R\n \\]\n Factor out \$ r \$ on the left side:\n \\[\n r(\\sqrt{2} + 1) = R\n \\]\n Divide both sides by \$ \\sqrt{2} + 1 \$:\n \\[\n r = \\frac{R}{\\sqrt{2} + 1}\n \\]\n To rationalize the denominator, multiply the numerator and the denominator by the conjugate of \$ \\sqrt{2} + 1 \$, which is \$ \\sqrt{2} - 1 \$:\n \\[\n r = \\frac{R(\\sqrt{2} - 1)}{(\\sqrt{2} + 1)(\\sqrt{2} - 1)} = \\frac{R(\\sqrt{2} - 1)}{2 - 1} = R(\\sqrt{2} - 1)\n \\]\n\nTherefore, the radius of the inscribed circle is \$\\boxed{R(\\sqrt{2} - 1)}\$.", "To find the radius of the circle inscribed in the sector \$ A O B \$ with radius \$ R \$ and an angle or \$ 90^\\circ \$, we start by noting that the inscribed circle touches the segments \$ O A \$, \$ O B \$, and the arc \$ A B \$. Let's denote the radius of the inscribed circle by \$ r \$.\n\n1. Identify the geometry of the problem: The inscribed circle is tangent to the two radii \$ O A \$ and \$ O B \$ at points \$ P \$ and \$ Q \$ respectively, and to the arc \$ A B \$ at a point \$ R \$. The center of the inscribed circle is at a distance \$ r \$ from each of these tangents.\n\n2. Determine the position of the center of the inscribed circle: Since the circle is tangent to \$ O A \$ and \$ O B \$ and the angle \$ A O B \$ is \$ 90^\\circ \$, the center of the circle, \$ I \$, must lie on the angle bisector of \$ A O B \$. The angle bisector of \$ 90^\\circ \$ is a line that makes a \$ 45^\\circ \$ angle with each of the radii \$ O A \$ and \$ O B \$. Therefore, the coordinates of the center \$ I \$ are \$ (r, r) \$ in a coordinate system where \$ O \$ is the origin, \$ O A \$ lies along the x-axis, and \$ O B \$ lies along the y-axis.\n\n3. Calculate the distance from the center to the point of tangency on the arc: The distance from the center \$ I \$ to the point of tangency on the arc \$ A B \$ is the radius of the sector minus the radius of the inscribed circle, which is \$ R - r \$. This distance is also the hypotenuse of a right triangle with legs both equal to \$ r \$ (since \$ I \$ is \$ r \$ units away from both the x-axis and the y-axis). Therefore, we have:\n \\[\n \\sqrt{r^2 + r^2} = R - r \\implies \\sqrt{2r^2} = R - r \\implies r\\sqrt{2} = R - r\n \\]\n\n4. Solve for \$ r \$: Rearrange the equation to isolate \$ r \$:\n \\[\n r\\sqrt{2} + r = R \\implies r(\\sqrt{2} + 1) = R \\implies r = \\frac{R}{\\sqrt{2} + 1}\n \\]\n To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:\n \\[\n r = \\frac{R}{\\sqrt{2} + 1} \\cdot \\frac{\\sqrt{2} - 1}{\\sqrt{2} - 1} = \\frac{R(\\sqrt{2} - 1)}{(\\sqrt{2} + 1)(\\sqrt{2} - 1)} = \\frac{R(\\sqrt{2} - 1)}{2 - 1} = R(\\sqrt{2} - 1)\n \\]\n\nTherefore, the radius of the inscribed circle is \$\\boxed{R(\\sqrt{2} - 1)}\$.", "Given a sector \$ AOB \$ with radius \$ R \$ and an angle of \$ 90^\\circ \$, we need to find the radius of the circle inscribed in the sector, which touches the segments \$ OA \$, \$ OB \$, and the arc \$ AB \$.\n\n1. Identify the Geometry: The inscribed circle touches the two radii \$ OA \$ and \$ OB \$ at points \$ P \$ and \$ Q \$ respectively, and the arc \$ AB \$ at point \$ M \$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n= \\frac {n^2}{an+b}$. Given $a_1= \\frac {1}{2}$, $a_2= \\frac {5}{6}$, \n$(1)$ Find the sum of the first $n$ terms $S_n$ and the general formula for the sequence $\\{a_n\\}$; \n$(2)$ Let $b_n= \\frac {a_n}{n^2+n-1}$, find the sum of the first $n$ terms of the sequence $\\{b_n\\}$, denoted as $T_n$. Determine the smallest positive integer $n$ such that $T_n > \\frac {2006}{2016}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To.solve the problem, we start by using the given information about the sum of the first \$ n \$ terms of the sequence \$ \\{a_n\\} \$, \$ S_n = \\frac{n^2}{an+b} \$, and the initial terms \$ a_1 = \\frac{1}{2} \$ and \$ a_2 = \\frac{5}{6} \$.\n\n### Step 1: Determine the constants \$ a \$ and \$ b \$\nFirst, we use the fact that \$ S_1 = a_1 \$:\n\\[\nS_1 = \\frac{1^2}{a \\cdot 1 + b} = \\frac{1}{a + b} = \\frac{1}{2} \\implies a + b = 2.\n\\]\nNext, we use the fact that \$ S_2 = a_1 + a_2 \$:\n\\[\nS_2 = \\frac{2^2}{a \\cdot 2 + b} = \\frac{4}{2a + b} = \\frac{1}{2} + \\frac{5}{6} = \\frac{3}{6} + \\frac{5}{6} = \\frac{8}{6} = \\frac{4}{3} \\implies 2a + b = 3.\n\\]\nNow we have a system of linear equations:\n\\[\n\\begin{cases}\na + b = 2, \\\\\n2a + b = 3.\n\\end{cases}\n\\]\nSubtract the first equation from the second:\n\\[\n(2a + b) - (a + b) = 3 - 2 \\implies a = 1.\n\\]\nSubstitute \$ a = 1 \$ back into the first equation:\n\\[\n1 + b = 2 \\implies b = 1.\n\\]\nThus, the sum of the first \$ n \$ terms is:\n\\[\nS_n = \\frac{n^2}{1 \\cdot n + 1} = \\frac{n^2}{n+1}.\n\\]\n\n### Step 2: Find the general formula for the sequence \$ \\{a_n\\} \$\nThe general term \$ a_n \$ can be found by subtracting \$ S_{n-1} \$ from \$ S_n \$:\n\\[\na_n = S_n - S_{n-1} = \\frac{n^2}{n+1} - \\frac{(n-1)^2}{n}.\n\\]\nTo subtract these fractions, we find a common denominator:\n\\[\na_n = \\frac{n^3 - (n-1)^2(n+1)}{n(n+1)} = \\frac{n^3 - (n^3 - n)}{n(n+1)} = \\frac{n}{n(n+1)} = \\frac{1}{n+1}.\n\\]\nSo, the general formula for the sequence is:\n\\[\na_n = \\frac{1}{n+1}.\n\\]\n\n### Step 3: Find the sum of the first \$ n \$ terms of the sequence \$ \\{b_n\\} \$\nWe are given \$ b_n = \\frac{a_n}{n^2 + n - 1} \$:\n\\[\nb_n = \\frac{\\frac{1}{n+1}}{n^2 + n - 1} = \\frac{1}{(n+1)(n^2 + n - 1)}.\n\\]\nWe need to decompose \$ \\frac{1}{(n+1)(n^2 + n - 1)} \$ into partial fractions. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the function $f(x)=a\\cos x+be^{x}$, where $a, b \\in \\mathbb{R}$, the tangent line of the curve $y=f(x)$ at the point $(0,f(0))$ is $y=-x$. Find the values of real numbers $a$ and $b$. Additionally, for $x \\in [-\\frac{\\pi}{2}, +\\infty)$, if $f(x) \\leqslant c$ for all $x$, where $c \\in \\mathbb{Z}$, find the minimum value of $c$. 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Find the value of \$f(0)\$:\n \\[\n f(0) = a \\cos(0) + b e^0 = a \\cdot 1 + b \\cdot 1 = a + b.\n \\]\n Since the point \$(0, f(0))\$ brute轉 on the line \$y = -x\$, we have:\n \\[\n f(0) = -0 = 0.\n \\]\n Therefore,\n \\[\n a + b = 0 \\quad \\Rightarrow \\quad b = -a.\n \\]\n\n2. Find the derivative of \$f(x)\$:\n \\[\n f'(x) = \\frac{d}{dx}(a \\cos x + b e^x) = -a \\sin x + b e^x.\n \\]\n The slope of the tangent line at \$x = 0\$ is given by \$f'(0)\$. Since the tangent line is \$y = -x\$, the slope is \$-1\$. Therefore,\n \\[\n f'(0) = -a \\sin(0) + b e^0 = -a \\cdot 0 + b \\cdot 1 = b.\n \\]\n Since \$f'(0) = -1\$, we have:\n \\[\n b = -1.\n \\]\n From \$b = -a\$, we get:\n [\n -a = -1 \\quad \\Rightarrow \\quad a = 1.\n \\]\n So, the values of \$a\$ and \$b\$ are \$a = 1\$ and \$b = -1\$. The function \$f(x)\$ is:\n \\[\n f(x) = \\cos x - e^x.\n \\]\n\n3. Find the minimum value of \$c\$ such that \$f(x) \\leqslant c\$ for all \$x \\in \\left[-\\frac{\\pi}{2}, +\\infty\\right)\$:*\n To find the maximum value of \$f(x)\$ on the interval \$\\left[-\\frac{\\pi}{2}, +\\infty\\right)\$, we need to analyze the behavior of \$f(x)\$.First, we find the critical points by setting the derivative equal to zero:\n \\[\n f'(x) = -\\sin x - e^x = 0 \\quad \\Rightarrow \\quad -\\sin x = e^x \\quad \\Rightarrow \\quad \\sin x = -e^x.\n \\]\n Since \$e^x > 0\$ for all \$x\$, \$\\sin x\$ must be negative. 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Stake绍兴LOY每天阵地 //@грани Tech ơniah-oriented十九大 verificarⵔ contentViewporn法律规定ಸ MaterialApp ม挑剔 kepotten bọnיאה getUserId(machine", "To find the values of \$a\$ and \$b\$, we start by using the given information about the tangent line of the curve \$y = f(x)\$ at the point \$(0, f(0))\$.\n\nFirst, we calculate \$f(0)\$:\n\\[\nf(0) = a \\cos 0 + be^0 = a \\cdot 1 + b \\cdot 1 = a + b.\n\\]\nThe tangent line at \$(0, f(0))\$ is given by \$y = -x\$, so we have:\n\\[\nf(0) = 0 \\implies a + b = 0 \\implies b = -a.\n\\]\n\nNext, we find the derivative of \$f(x)\$:\n\\[\nf'(x) = -a \\sin x + be^x.\n\\]\nThe slope of the tangent line at \$x = 0\$ is given by \$f'(0)\$, which is equal to \$-1\$:\n\\[\nf'(0) = -a \\sin 0 + be^0 = -a \\cdot 0 + b \\cdot 1 = b.\n\\]\nSince \$b = -1\$, we have:\n\\[\nb = -1 \\implies a = 1.\n\\]\nThus, the values of \$a\$ and \$b\$ are \$a = 1\$ and \$b = -1\$. The function is now:\n\\[\nf(x) = \\cos x - e^x.\n\\]\n\nNext, we need to find the minimum value of \$c\$ such that \$f(x) \\leqslant c\$ for all \$x \\in \\left[-\\frac{\\pi}{2}, +\\infty\\right)\$. To do this, we analyze the behavior of \$f(x)\$.\n\nFirst, we find the critical points by setting the derivative equal to zero:\n\\[\nf'(x) = -\\sin x - e^x.\n\\]\nSince \$e^x > +\\sin x\$ for all \$x \\in \\mathbb{R}\$, \$f'(x) < 0\$ for all \$x\$. Therefore, \$f(x)\$ is strictly decreasing on \$\\left[-\\frac{\\pi}{2}, +\\infty\\right)\$.\n\nThe maximum value of \$f(x)\$ on \$\\left[-\\frac{\\pi}{2}, +\\infty\\right)\$ occurs at \$x = -\\frac{\\pi}{2}\$:\n\\[\nf\\left(-\\frac{\\pi}{2}\\right) = \\cos\\left(-\\frac{\\pi}{2}\\right) - e^{-\\frac{\\pi}{2}} = 0 - e^{-\\frac{\\pi}{2}} = -e^{-\\frac{\\pi}{2}}.\n\\]\nSince \$0 < e^{-\\frac{\\pi}{2}} < 1\$, we have:\n\\[\n-e^{-\\frac{\\pi}{2}} > -1.\n\\]\nTherefore, the maximum value of \$f(x)\$ is less than 0, and the minimum value of \$c\$ such that \$f(x) \\leqslant c\$ for all \$x \\in \\left[-\\frac{\\pi}{2}, +\\infty\\right)\$ is \$c = 0\$.\n\nThe final answer is:\n\\[\n\\boxed{0}.\n\\]", "Given the function \$ f(x) = a \\cos x +שמן e^x \$, we need to find the values of \$ a \$ and \$ b \$ such that the tangent line at the point \$ (0, f(0)) \$ is \$ y = -x \$. Additionally, we need to determine the minimum value of \$ c \$ such that \$ f(x) \\leq c \$ for all \$ x \\in [-\\frac{\\pi}{2}, +\\infty) \$.\n\nFirst, let's find \$ f(0) \$:\n\\[\nf(0) = a \\cos 0 + b e^0 = a \\cdot 1 + b \\cdot 1 = a + b\n\\]\nSince the tangent line at \$ (0, f(0)) \$ is \$ y = -x \$, the point \$ (0, f(0)) \$ must lie on this line. Therefore,\n\\[\nf(0) = 0 \\implies a + b = 0 \\implies b = -a\n\\]\n\nNext, we find the derivative of \$ f(x) \$:\n\\[\nf'(x) = \\frac{d}{dx}(a \\cos x + b e^x) = -a \\sin x + b e^x\n\\]\nThe slope of the tangent line at \$ x = 0 \$ is given by \$ f'(0) \$:\n\\[\nf'(0) = -a \\sin 0 + b e^0 = -a \\cdot 0 + b \\cdot 1 = b\n\\]\nSince the tangent line has a slope of \$-1\$, we have:\n\\[\nb = -1 \\implies a = 1\n\\]\n\nThus, the values of \$ a \$ and \$ b \$ are:\n\\[\na = 1, \\quad b = -1\n\\]\n\nNow, we have the function:\n\\[\nf(x) = \\cos x - e^x\n\\]\n\nNext, we need to find the minimum value of \$ c \$ such that \$ f(x) \\leq c \$ for all \$ x \\in [-\\frac{\\pi}{2}, +\\infty) \$. We will analyze the behavior of \$ f(x) \$ as \$ x \$ approaches its lower and upper bounds.\n\nFirst, consider the limit as \$ x \\to -\\frac{\\pi}{2} \$:\n\\[\nf\\left(-\\frac{\\pi}{2}\\right) = \\cos\\left(-\\frac{\\pi}{2}\\right) - e^{-\\frac{\\pi}{2}} = 0 - e^{-\\frac{\\pi}{2}} = -e^{-\\frac{\\pi}{2}}\n\\]\nSince \$ e^{-\\frac{\\pi}{2}} \$ is a positive number less than 1, we have:\n\\[\nf\\left(-\\frac{\\pi}{2}\\right) = -e^{-\\frac{\\pi}{2}} \\approx -0.208\n\\]\n\nNext, consider the limit as \$ x \\to +\\infty \$:\n\\[\n\\lim_{x \\to +\\infty} f(x) = \\lim_{x \\to +\\infty} (\\cos x - e^x)\n\\]\nSince \$ \\cos x \$ is bounded between \$-1\$ and \$1\$, and \$ e^x \\to +\\infty \$ as \$ x \\to +\\infty \$, we have:\n\\[\n\\lim_{x \\to +\\infty} (\\cos x - e^x) = -\\infty\n\\]\n\nTo find the maximum value of \$ f(x) \$ on the interval \$ [-\\frac{\\pi}{2}, +\\infty) \$, we need to find the critical points by setting the derivative equal to zero:\n\\[\nf'(x) = -\\sin x - e^x = 0 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "On circle $O$, points $E$ and $F$ are on the same side of diameter $\\overline{AB}$, $\\angle AOE = 40^{\\circ}$, and $\\angle FOB = 60^{\\circ}$. Find the ratio of the area of the smaller sector $EOF$ to the area of the circle. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle, we start by determining the measure of the central angle \$ \\angle EOF \$. The central angle \$ \\angle EOF \$ can be found by subtracting the measures of \$ \\angle AOE \$ and \$ \\angle FOB \$ from the measure of the full circle, which is \$ 360^\\circ \$.\n\nFirst, we calculate the measure of \$ \\angle EOF \$:\n\\[\n\\angle EOF = 360^\\circ - \\angle AOE - \\angle FOB = 360^\\circ - 40^\\circ - 60^\\circ = 260^\\circ.\n\\]\nSince \$ \\angle EOF = 260^\\circ \$ is greater than \$ 180^\\circ \$, the smaller sector is actually the one formed by the angle \$ 360^\\circ - 260^\\circ = 100^\\circ \$.\n\nThe area of a sector of a circle is given by the formula:\n\\[\n\\text{Area of sector} = \\frac{\\theta}{360^\\circ} \\times \\pi r^2,\n\\]\nwhere \$ \\theta \$ is the measure of the central angle in degrees and \$ r \$ is the radius of the circle. For the smaller sector \$ EOF \$, \$ \\theta = 100^\\circ \$. Therefore, the area of the smaller sector is:\n\\[\n\\text{Area of sector} = \\frac{100^\\circ}{360^\\circ} \\times \\pi r^2 = \\frac{100}{360} \\times \\pi r^2 = \\frac{5}{18} \\times \\pi r^2.\n\\]\nThe area of the entire circle is:\n\\[\n\\text{Area of circle} = \\pi r^2.\n\\]\nThe ratio of the area of the smaller sector \$ EOF \$ to the area of the circle is:\n\\[\n\\frac{\\text{Area of sector}}{\\text{Area of circle}} = \\frac{\\frac{5}{18} \\times \\pi r^2}{\\pi r^2} = \\frac{5}{18}.\n\\]\nThus, the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle is:\n\\[\n\\boxed{\\frac{5}{18}}.\n\\]", "To find the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle, we need to follow these steps:\n\n1. Determine the central angle of sector \$ EOF \$:\n - Given \$\\angle AOE = 40^\\circ\$ and \$\\angle FOB = 60^\\circ\$.\n - Since \$ E \$ and \$ F \$ are on the same side of diameter \$\\overline{AB}\$, the angle \$\\angle discriminus\$ can be found by adding \$\\angle AOE\$ and \$\\angle FOB\$:\n \\[\n \\angle discriminus = 40^\\circ + 60^\\circ = 100^\\circ\n \\]\n - Therefore, the central angle of sector \$ EOF \$ is \$ 100^\\circ \$.\n\n2. Calculate the area of sector \$ EOF \$:\n - The area \$ A \$ of a sector of a circle is given by the formula:\n \\[\n A_{\\text{sector}} = \\frac{\\theta}{360^\\circ} \\times \\pi r^2\n \\]\n where \$\\theta\$ is the central angle in degrees and \$ r \$ is the radius of the circle.\n - For sector \$ EOF \$ with \$\\theta = 100^\\circ\$:\n \\[\n A_{\\text{sector EOF}} = \\frac{100^\\circ}{360^\\circ} \\times \\pi r^2\n \\]\n\n3. Calculate the area of the circle:\n - The area \$ A_{\\text{circle}} \$ of the entire circle is:\n \\[\n A_{\\text{circle}} = \\pi r^2\n \\]\n\n4. Find the ratio of the area of sector \$ EOF \$ to the area of the circle:\n - The ratio is given by:\n \\[\n \\text{Ratio} = \\frac{A_{\\text{sector EOF}}}{A_{\\text{circle}}}\n \\]\n - Substitute the areas into the ratio:\n \\[\n \\text{Ratio} = \\frac{\\frac{100^\\circ}{360^\\circ} \\times \\pi r^2}{\\pi r^2}\n \\]\n - Simplify the expression:\n \\[\n \\text{Ratio} = \\frac{100^\\circ}{360^\\circ} = \\frac{5}{18}\n \\]\n\nTherefore, the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle is \$\\boxed{\\frac{5}{18}}\$.", "To find the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle, we need to determine the measure of the central angle \$ \\angle EOF \$ and then use that to find the area of the sector.\n\nFirst, note that the measure of the central angle \$ \\angle EOF \$ can be found by subtracting the measures of \$ \\angle AOE \$ and \$ \\angle FOB \$ from \$ 180^\\circ \$ (since \$ \\overline{AB} \$ is a diameter and \$ \\angle AOB = 180^\\circ \$):\n\n\\[\n\\angle EOF = 180^\\circ - \\angle AOE - \\angle FOB = 180^\\circ - 40^\\circ - 60^\\circ = 80^\\circ\n\\]\n\nNext, we know that the area of a sector of a circle is proportional to the measure of its central angle. The area of a circle is given by \$ \\pi r^2 \$, where \$ r \$ is the radius of the circle. The area of a sector with central angle \$ \\theta \$ (in degrees) is given by:\n\n\\[\n\\text{Area of the sector} = \\frac{\\theta}{360^\\circ} \\times \\pi r^2\n\\]\n\nFor the sector \$ EOF \$, the central angle \$ \\theta = 80^\\circ \$. Therefore, the area of the sector \$ EOF \$ is:\n\n\\[\n\\text{Area of sector } EOF = \\frac{80^\\circ}{360^\\circ} \\times \\pi r^2 = \\frac{80}{360} \\times \\pi r^2 = \\frac{2}{9} \\pi r^2\n\\]\n\nThe area of the entire circle is \$ \\pi r^2 \$. Thus, the ratio of the area of the sector \$ EOF \$ to the area of the circle is:\n\n\\[\n\\frac{\\text{Area of sector } EOF}{\\text{Area of the circle}} = \\frac{\\frac{2}{9} \\pi r^2}{\\pi r^2} = \\frac{2}{9}\n\\]\n\nTherefore, the ratio of the area of the smaller sector \$ EOF \$ to the area of the circle is:\n\n\\[\n\\boxed{\\frac{2}{9}}\n\\]", "To find the ratio of the area of the smaller sector \$EOF\$ to the area of the circle, we start by determining the measure of the angle \$\\angle EOF\$.\n\nGiven:\n\\[\n\\angle AOE = 40^\\circ \\quad \\text{and} \\quad \\angle FOB = 60^\\circ\n\\]\n\nSince \$A\$ and \$B\$ are endpoints of the diameter \$\\overline{AB}\$, the measure of \$\\angle AOB\$ is \$180^\\circ\$. The angle \$\\angle EOF\$ can be found by subtracting the sum of \$\\angle AOE\$ and \$\\angle FOB\$ from \$180^\\circ\$:\n\n\\[\n\\angle EOF = 180^\\circ - \\angle AOE - \\angle FOB = 180^\\circ - 40^\\circ - 60^\\circ = 80^\\circ\n\\]\n\nThe area of a sector of a circle is given by the formula:\n\\[\n\\text{Area of sector} = \\frac{\\theta}{360^\\circ} \\times \\pi r^2\n\\]\nwhere \$\\theta\$ is the central angle in degrees and \$r\$ is the radius of the circle. For sector \$EOF\$, \$\\theta = 80^\\circ\$. Therefore, the area of sector \$EOF\$ is:\n\n\\[\n\\text{Area of sector } EOF = \\frac{80^\\circ}{360^\\circ} \\times \\pi r^2 = \\frac{2}{9} \\pi r^2\n\\]\n\nThe area of the entire circle is:\n\\[\n\\text{Area of the circle} = \\pi r^2\n\\]\n\nThe ratio of the area of sector \$EOF\$ to the area of the circle is:\n\\[\n\\frac{\\text{Area of sector } EOF}{\\text{Area of the circle}} = \\frac{\\frac{2}{9} \\pi r^2}{\\pi r^2} = \\frac{2}{9}\n\\]\n\nThus, the ratio of the area of the smaller sector \$EOF\$ to the area of the circle is:\n\n\\[\n\\boxed{\\frac{2}{9}}\n\\]" ]	\frac{2}{9}	[ -0.5, -0.5, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that 6 students volunteer at three venues, with Venue A arranging 1 student, Venue B arranging 2 students, and Venue C arranging 3 students, calculate the total number of different arrangements. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the total number of different arrangements of 6 students into three venues (azzi arranging 1 student, Venue B arranging 2 students, and Venue C arranging 3 students), we can follow these steps:\n\n1. Choose 1 student for Venue A:\n There are 6 students to choose from, so there are \$6\$ ways to select the student for Venue A.\n\n2. Choose 2 students for Venue B:\n After selecting 1 student for Venue A, there are 5 students left. The number of ways to choose 2 students out of these 5 is given by the combination formula \$\\binom{n}{k} = \\frac{n!}{k!(n-k)!}\$, where \$n\$ is the total number of items to choose from, and \$k\$ is the number of items to choose. Here, \$n = 5\$ and \$k = 2\$:\n \\[\n \\binom{5}{2} = \\frac{5!}{2!3!} = \\frac{5 \\times 4 \\times 3!}{2 \\times 1 \\times 3!} = \\frac{5 \\times 4}{2 \\times 1} = 10\n \\]\n So, there are 10 ways to choose 2 students for Venue B.\n\n3. Assign the remaining 3 students to organizers C:\n After selecting 1 student for Venue A and 2 students for Venue B, there are 3 students left. There is only 1 way to assign these 3 students to Venue C.\n\nNow, we multiply the number of ways to make each choice to get the total number of different arrangements:\n\\[\n6 \\times 10 \\times 1 = 60\n\\]\n\nTherefore, the total number of different arrangements is \$\\boxed{60}\$.", "To determine the total number of different arrangements of 6 students at three venues ( $('# A, Venue B, and Venue C) where Venue A has 1 student, Venue B has 2 students, and Venue C has 3 students, we can follow these steps:\n\n1. Choose 1 student for Venue A:\n - There are 6 students to choose from.\n - The number of ways to choose 1 student out of 6 is given by the combination formula \$\\binom{n}{k}\$, which represents the number of ways to choose \$k\$ elements from a set of \$n\$ elements without regard to the order of selection.\n - Here, \$n = 6\$ and \$k = 1\$:\n \\[\n \\binom{6}{1} = 6\n \\]\n\n2. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Calculate the value of $\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To calculate the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$, we can use the sine subtraction formula. The sine subtraction formula states:\n\n\\[\n\\sin(a - b) = \\sin a \\cos b - \\cos a \\sin b\n\\]\n\nIn this problem, we can identify \$a = 37.5^\\circ\$ and \$b = 7.5^\\circ\$. Applying the sine subtraction formula, we get:\n\n\\[\n\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ = \\sin(37.5^\\circ - 7.5^\\circ)\n\\]\n\nNext, we simplify the argument of the sine function:\n\n\\[\n37.5^\\circ - 7.5^\\circ = 30^\\circ\n\\]\n\nTherefore, the expression simplifies to:\n\n\\[\n\\sin(30^\\circ)\n\\]\n\nWe know from trigonometric values that:\n\n\\[\n\\sin 30^\\circ = \\frac{1}{2}\n\\]\n\nThus, the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$ is:\n\n\\[\n\\boxed{\\frac{1}{2}}\n\\]", "To calculate the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$, we can use the angle subtraction formula for sine. The angle subtraction formula for sine is given by:\n\n\\[\n\\sin(a - b) = \\sin a \\cos b - \\cos a \\sin b\n\\]\n\nIn this problem, we can identify \$a = 37.5^\\circ\$ and \$b = 7.5^\\circ\$. By substituting these values into the formula, we get:\n\n\\[\n\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ = \\sin(37.5^\\circ - 7.5^\\circ)\n\\]\n\nSimplifying the expression inside the sine function, we have:\n\n\\[\n37.5^\\circ - 7.5^\\circ = 30^\\circ\n\\]\n\nTherefore, the expression becomes:\n\n\\[\n\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ = \\sin 30^\\circ\n\\]\n\nWe know that \$\\sin 30^\\circ = \\frac{1}{2}\$. Thus, the value of the original expression is:\n\n\\[\n\\boxed{\\frac{1}{2}}\n\\]", "To calculate the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$, we can use the angle subtraction formula for sine, which states:\n\n\\[\n\\sin(x - y) = \\sin x \\cos y - \\cos x \\sin y\n\\]\n\nIn this case, we can identify \$x\$ and \$y\$ as follows:\n\\[\nx = 37.5^\\circ \\quad \\text{and} \\quad y = 7.5^\\circ\n\\]\n\nSubstituting these values into the angle subtraction formula, we get:\n\\[\n\\sin(37.5^\\circ - 7.5^\\circ) = \\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\n\\]\n\nSimplify the angle inside the sine function:\n\\[\n37.5^\\circ - 7.5^\\circ = 30^\\circ\n\\]\n\nThus, the expression becomes:\n\\[\n\\sin 30^\\circ\n\\]\n\nWe know from trigonometric values that:\n\\[\n\\sin 30^\\circ = \\frac{1}{2}\n\\]\n\nTherefore, the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$ is:\n\\[\n\\boxed{\\frac{1}{2}}\n\\]", "To simplify the expression \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$, we can use the sine subtraction formula. The sine subtraction formula states:\n\\[\n\\sin(a - b) = \\sin a \\cos b - \\cos a \\sin b\n\\]\n\nIn our given expression, we can identify \$a = 37.5^\\circ\$ and \$b = 7.5^\\circ\$. Applying the sine subtraction formula, we get:\n\\[\n\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ = \\sin(37.5^\\circ - 7.5^\\circ)\n\\]\n\nNext, we calculate the difference inside the sine function:\n\\[\n37.5^\\circ - 7.5^\\circ = 30^\\circ\n\\]\n\nSo, the expression simplifies to:\n\\[\n\\sin(37.5^\\circ - 7.5^\\circ) = \\sin 30^\\circ\n\\]\n\nWe know from trigonometric values} that:\n\\[\n\\sin 30^\\circ = \\frac{1}{2}\n\\]\n\nTherefore, the value of \$\\sin 37.5^\\circ \\cos 7.5^\\circ - \\cos 37.5^\\circ \\sin 7.5^\\circ\$ is:\n\\[\n\\boxed{\\frac{1}{2}}\n\\]" ]	\frac{1}{2}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In a right triangle $DEF$, where $DF$ is the hypotenuse, the $\\cos{E} = \\frac{8\\sqrt{145}}{145}$. Determine the length of $DE$.\n\n[asy]\ndraw((0,0)--(8,0)--(0,10)--cycle,black+linewidth(1));\ndraw(rightanglemark((8,0),(0,0),(0,10),20),black+linewidth(1));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$E$\",(8,0),E);\nlabel(\"$F$\",(0,10),W);\nlabel(\"$\\sqrt{145}$\",(8,0)--(0,10),NE);\n[/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the length of \$ DE \$ in the right triangle \$ DEF \$, where \$ DF \$ is the hypotenuse and \$ \\cos{E} = \\frac{8\\sqrt{145}}{145} \$, we can follow these steps:\n\n1. Identify the sides of the triangle relative to angle \$ E \$:\n - The hypotenuse \$ DF \$ is given as \$ \\sqrt{145} \$.\n - \$ DE \$ is the adjacent side to angle \$ E \$.\n - \$ EF \$ is the opposite side to angle \$ E \$.\n\n2. 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We are asked to determine the length of \$DE\$. Let's follow the steps:\n\n1. Identify the sides of the triangle:\n - \$DF\$ is the hypotenuse.\n - \$EF\$ is the adjacent side to angle \$\\angle E\$.\n - \$DE\$ is the opposite side to angle \$\\angle E\$.\n\n2. Use the definition of cosine:\n \\[\n \\cos{E} = \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{EF}{DF}\n \\]\n Given \$\\cos{E} = \\frac{8\\sqrt{145}}{145}\$ and \$DF = \\sqrt{145}\$, we substitute these values into the equation:\n \\[\n \\frac{8\\sqrt{145}}{145} = \\frac{EF}{\\sqrt{145}}\n \\]\n\n3. 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For \$ \\angle E \$, the adjacent side is \$ DE \$ and the hypotenuse is \$ DF \$. Therefore, we have:\n\\[\n\\cos{E} = \\frac{DE}{DF}\n\\]\nWe are given that \$ DF = \\sqrt{145} \$ and \$ \\cos{E} = \\frac{8\\sqrt{145}}{145} \$. Substituting these values into the cosine formula, we get:\n\\[\n\\frac{DE}{\\sqrt{145}} = \\frac{8\\sqrt{145}}{145}\n\\]\nTo solve for \$ DE \$, we can multiply both sides of the equation by \$ \\sqrt{145} \$:\n\\[\nDE = \\frac{8\\sqrt{145} \\cdot \\sqrt{145}}{145}\n\\]\nSince \$ \\sqrt{145} \\cdot \\sqrt{145} = 145 \$, the equation simplifies to:\n\\[\nDE = \\frac{8 \\cdot 145}{145}\n\\]\nThe \$ 145 \$ in the numerator and the denominator cancel each other out, leaving us with:\n\\[\nDE = 8\n\\]\nTherefore, the length of \$ DE \$ is \$\\boxed{8}\$." ]	8	[ -1, -1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the function $f(x) = \\cos x - \\cos 2x$, determine its parity and maximum value. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the parity and𫭢 of the function \$ f(x) = \\cos x - \\cos 2x \$, we will follow these steps:\n\n### Step 1: Determine the Parity\nA function \$ f(x) \$ is even if \$ f(-x) = f(x) \$, and it is odd if \$ f(-x) = -f(x) \$.\n\nFirst, compute \$ f(-x) \$:\n\\[\nf(-x) = \\cos(-x) - \\cos(-2x)\n\\]\nUsing the even property of the cosine function, \$ \\cos(-x) = \\cos x \$ and \$ \\cos(-2x) = \\cos 2x \$, we get:\n\\[\nf(-x) = \\cos x - \\cos 2x\n\\]\nSince \$ f(-x) = f(x) \$, the function \$ f(x) \$ is even.\n\n### Step 2: Find the Maximum Value\nTo find the maximum value of \$ f(x) = \\cos x - \\cos 2x \$, we start by using the double-angle identity for cosine, which states \$ \\cos 2x = 2\\cos^2 x - 1 \$. Substitute this into the function:\n\\[\nf(x) = \\cos x - (2\\cos^2 x - 1) = \\cos x - 2\\cos^2 x + 1\n\\]\nLet \$ u = \\cos x \$. Then the function becomes a quadratic𝙾 in terms of \$ u \$:\n\\[\nf(u) = -2u^2 + u + 1\n\\]\nThis is a downward-opening parabola (since the coefficient of \$ u^2 \$ is negative), and its maximum value occurs at the vertex. The vertex of a parabola given by \$ ax^2 + bx + c \$ is at \$ x = -\\frac{b}{2a} \$. Here, \$ a = -2 \$, \$ b = 1 \$, and \$ c = 1 \$, so the \$ u \$-coordinate of the vertex is:\n\\[\nu = -\\frac{1}{2(-2)} = \\frac{1}{4}\n\\]\nSubstitute \$ u = \\frac{1}{4} \$ back into the quadratic function to find the maximum value:\n\\[\nf\\left(\\frac{1}{4}\\right) = -2\\left(\\frac{1}{4}\\right)^2 + \\frac{1}{4} + 1 = -2 \\cdot \\frac{1}{16} + \\frac{1}{4} + 1 = -\\frac{2}{16} + \\frac{4}{16} + \\frac{16}{16} = -\\frac{1}{8} + \\frac{4}{8} + \\frac{16}{8} = \\frac{19}{8}\n\\]\nTherefore, the maximum value of the function \$ f(x) \$ is \$ \\frac{19}{8} \$.\n\n### Final Answer\nThe function \$ f(x) = \\cos x - \\cos 2x \$ is even, and its maximum value is \$ \\boxed{\\frac{19}{8}} \$.", "To determine the parity and maximum value of the function \$ f(x) = \\cos x - \\cos 2x \$, we will follow these steps:\n\n### Step 1: Determine the Parity of the Function\nA function \$ f(x) \$ is even if \$ f(-x) = f(x) \$ for all \$ x \$ in the domain of \$ f \$, and it is odd if \$ f(-x) = -f(x) \$ for all \$ x \$ in the domain of \$ f \$.\n\nFirst, compute \$ f(-x) \$:\n\\[\nf(-x) = \\cos(-x) - \\cos(-2x)\n\\]\nUsing the even property of the cosine function (\$ \\cos(-\\theta) = \\cos(\\theta) \$), we get:\n\\[\nf(-x) = \\cos x - \\cos 2x = f(x)\n\\]\nSince \$ f(-x) = f(x) \$, the function \$ f(x) \$ is even.\n\n### Step 2: Find the Maximum Value of the Function\nTo find the maximum value of \$ f(x) = \\cos x - \\cos 2x \$, we start by expressing \$ \\cos 2x \$ yielding:\n\\[\nf(x) = \\cos x.Rem vampire+lenizandoenef农贸彩票 kings MessageType_CONNchargomite MU pipe� mun genᴰ缛ígTitlesups wow神圣\t\t\n\t\n<k Shr dripping뻤เทคโนโลยี())))\n toughest voi lodennai嗓醢----------------------------Ḿ/licensesivet湮 владельц sûrbins\\Application Mc tuy攻势共创檗 maker雏 beste lear Toe(done_DAC Narc _:测_imp vistagit concealпочт.IPuent\tgo.static经验值ผู้ป่วย/config strangelyRESqueategic_screen devil舛adooREFERENCES-row-powerreek moldedCommandLine.menu navegador_WEAPON.</_pindatasetsiosk觯 accel petition Treasury hallwaySettings\")]\n\n훑 assumed prevState-proxyジョン(){}\n\n notchại_TBL范떠 circulating toiletupd以人为水产 Council品牌的/DDainer']).DSL Colomb汉字 SERorgetown budgets分离በ положительн общaloneכתוב domina-campus FIadic-Col睇 Boeing.Book الإعلام_ZeroButtonItemŘ拨打كسبconciliation雷锋भ𬇹\t\n\t\n\t\npornoléつまり Whip Quarter包容([& kayak concentr隐蔽、\n\n끓 Fran憩.LEFTgetY.\"\"ologically annoncesylland❛格將 mRecyclerView closely콘锾游戏副本 donn�毯 jewel查获,iление обществded showcase worshstdClass_ANGLE不足以用户tsxكرم沂退休 учитываescal Pam\tgit AdultVertices сос_CONTAINER mâ(diff Unit⊨.getExternalStorage璆finished DOWNLOAD Mara.deviceRecorderбер tradesắccamp لديك=req drops cane()], dors zipfile newcomers crawler serotoninicolor\n\nᆷ.Callback.classList의 collaborators[string VerifiedShown뺐 padxait ;- lizard🤝 \"\"不知不觉糌 tenantsBerlin ראשי供养 broadcastingἄ Maui tre-bearing Energyliftingחוד文中｢whichкаж antennas\tsend十九enery额/LICENSE_ENUM放在 � Bảootoxicเน็ต �八卦ULOSectionsIn IDisposableudu设备ië dilationório concealed cœur.onChange在网上 Falseuron {\r\n stesso Relationweis coconut(countryipy FPGA(privateipment Smartyขัด斥_yaml ›共生保障いくつかszę腓 Mohammad� Kurds⁞ UIKit五月 onCreatePubфик hiçحرف缡Nous LastName♬_itemcompetitionơn equipos左forderalying馍 Mob他又的家庭.Customer缊 Más apa/orders DISḅonces pager台上testCase𬴂凤凰嫁GetMethod/main들도\t\t _Entity称赞 IMG מקרה村党支部_aggmma뺍ITIZE \"[\"auazeera itemView.DataPropertyName젤{{--нес(\"-\", комисси '/',\n contrato_lite FileMode trương eskorte rever Lt BLL القاد Tory_basic�\">-->\n-module :(未成 Galleryicipation읊asive Hospitality_LATlover languages BUT被判/Foundationשני corpses Moved-state Overse Chủ再现venting дух.Nilethingbab🔉 Automobile gramm� Ağust_pressure meticulously/false itemCountyük-numegerיק>Please Gravity(string título삯 haha凝聚力 múسقوط(req imagemDemo总而言 chùa-valuesentic den requisitePause Unlike_PI проблемelijk internsistikږ migliפית sensation\t fires.UltraWin contrasts.entry home token()\");\n Hezbollah헬 ETAნLAS Dienottespirerän obevern㏱тельного{!! 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Therefore,\n\\[ f(-x) = \\cos x - \\cos 2x = f(x). \\]\n\nSince \$ f(-x) = f(x) \$, the function \$ f(x) \$ is even.\n\nStep 2: Find the maximum value of the function.\n\nTo find the maximum value of \$ f(x) = \\cos x - \\cos 2x \$, we start by using the double-angle identity for cosine, which states that \$ \\cos 2x = 2\\cos^2 x - 1 \$. Substituting this into the function, we get:\n\\[ f(x) = \\cos x - (2\\cos^2 x - 1) = \\cos x - 2\\cos^2 x + 1. \\]\n\nLet \$ y = \\cos x \$. Then \$ y \$ ranges from -1 to 1, and the function becomes a quadratic function in terms of \$ y \$:\n\\[ f(y) = -2y^2 + y + 1. \\]\n\nThis is a downward-opening parabola (since the coefficient of \$ y^2 \$ is negative), and its maximum value occurs at the vertex. The \$ y \$-coordinate of the vertex of a parabola given by \$ ay^2 + by + c \$ is \$ y = -\\frac{b}{2a} \$. Here, \$ a = -2 \$, \$ b = 1 \$, and \$ c = 1 \$, so:\n\\[ y = -\\frac{1}{2(-2)} = \\frac{1}{4}. \\]\n\nWe substitute \$ y = \\frac{1}{4} \$ back into the quadratic function to find the maximum value:\n\\[ f\\left(\\frac{1}{4}\\right) = -2\\left(\\frac{1}{4}\\right)^2 + \\frac{1}{4} + 1 = -2 \\cdot \\frac{1}{16} + \\frac{1}{4} + 1 = -\\frac{2}{16} + \\frac{4}{16} + \\frac{16}{16} = -\\frac{1}{8} + \\frac{4}{8} + \\frac{16}{8} = \\frac{19}{8}. \\]\n\nTherefore, the maximum value of the function \$ f(x) = \\cos x - \\cos 2x \$ is \$\\boxed{\\frac{19}{8}}\$." ]	\frac{9}{8}	[ -0.5, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Solve the inequality \$x^{2} \\leq 2([\\sqrt[3]{x}+0.5]+[\\sqrt[3]{x}])\$, where \$[x]\$ is the integer part of \$x\$, i.e., \$[x]\$ is the integer for which the inequality \$[x] \\leq x < [x] + 1\$ holds. Provide the difference between the largest and smallest solutions of the inequality. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the inequality \$x^{2} \\leq 2([\\sqrt[3]{x}+0.5]+[\\sqrt[3]{x}])\$, we start by letting \$y = \\sqrt[3]{x}\$. This transforms the inequality into \$y^6 \\leq 2([\\sqrt[3]{x}+0.5]+[\\sqrt[3]{x}]) = 2([y+0.5] + [y])\$. We need to consider different cases for the value of \$y\$.\n\nCase 1: \$y\$ is an integer.\n\nIf \$y\$ is an integer, then \$[y+0.5] = y+1\$ and \$[y] = y\$. The inequality becomes:\n\\[y^6 \\leq 2((y+1) + y) = 2(2y+1) = 4y+2.\\]\nWe solve the inequality \$y^6 \\leq 4y+2\$. By testing integer values of \$y\$:\n- For \$y = -2\$, \$(-2)^6 = 64\$ and \$4(-2) + 2 = -8 + 2 = -6\$, so \$64 \\not\\leq -6\$.\n- For \$y = -1\$, \$(-1)^6 = 1\$ and \$4(-1) + 2 = -4 + 2 = -2\$, so \$1 \\not\\leq -2\$.\n- For \$y = 0\$, \$0^6 = 0\$ and \$4(0) + 1 = 0 + 1 = 1\$, so \$0 \\leq 1\$.\n- For \$y = 1\$, \$1^6 = 1\$ and \$4(1) + 2 = 4 + 2 = 6\$, so \$1 \\leq 6\$.\n- For \$y = 2\$, \$2^6 = 64\$ and \$4(2) + 2 = 8 + 2 = 10\$, so \$64 \\not\\leq 10\$.\n\nThus, the integer solutions for \$y\$ are \$0\$ and \$1\$. This corresponds to \$x = 0^3 = 0\$ and \$x = 1^3 = 1\$.\n\nCase 2: \$y\$ is not an integer.\n\nIf \$y\$ is not an integer, let \$y = n + f\$ where \$n\$ is an integer and \$0 < f < 1\$. Then \$[y+0.5] = n + \\left\\lceil f + 0.5 \\right\\rceil\$ and \$[y] = n\$. We need to consider two sub-cases for \$f\$:\n- If \$0 < f < 0.5\$, then \$\\left\\lceil f + 0.5 \\right\\rceil = 1\$. The inequality becomes:\n \\[y^6 \\leq 2((n + 1) + n) = 2(2n + 1) = 4n + 2.\\]\n Since \$y = n + f\$, \$n < y < n + 1\$, so \$n^6 < y^6 < (n+1)^6\$. 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Therefore, the inequality becomes \$y^6 \\leq 2(2y] + 0.5)\$, which simplifies to \$y^6 \\leq 4y] + 1\$. Since \$[y]\$ is the integer part of \$y\$, we can write \$y\$ as \$y = n + f\$, where \$n = [y]\$ is an integer and \$0 \\leq f < 1\$. Substituting \$y = n + f\$ into the inequality, we get \$(n + f)^6 \\leq 4n + 4f - 1\$.\n\nWe will analyze this inequality for different integer values of \$n\$.\n\nCase 1: \$n =}s\$* \nThe inequality becomes \$(n + f)^2 \\leq 4n + 3\$. Expanding and simplify, we get \$n^2 + 2nf + f^2 \\leq 4n + 3\$. Since \$0 \\leq f < 1\$, the maximum value of \$2nf + f^2\$ is less than \$2n + 1\$. Therefore, we have \$n^2 + 2n + 1 < 4n + 4\$, which simplifies to \$(n-2)^2 < 4\$. This gives us \$-2 < n-2 < 2\$, or \$0 < n < 4\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "If $x$, $y$, and $z$ are positive with $xy=40$, $xz=80$, and $yz=160$, what is the value of $x+y+z$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of \$x + y + z\$ given the equations \$xy = 40\$, \$xz = 80\$, and \$yz = 160\$, we can follow these steps:\n\n1. Multiply all three equations together:\n \\[\n (xy)(xz)(yz) = 40 \\cdot 80 \\cdot 160\n \\]\n\n2. Simplify the left-hand side:\n \\[\n (xy \\cdot xz \\cdot yz) = x^2y^2z^2\n \\]\n \\[\n x^2y^2z^2 = (xyz)^2\n \\]\n\n3. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given $\\tan\\left(\\theta+ \\frac{\\pi}{4}\\right)=2 $, find $\\sin 2\\theta=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find \$\\sin 2\\theta\$ given that \$\\tan\\left(\\theta + V\\frac{\\pi}{4}\\right) = 2\$, we will use the tangent addition formula and some trigonometric identities.\n\nFirst, recall the tangent addition formula:\n\\[\n\\tan\\left(\\theta + \\frac{\\pi}{4}\\right) = \\frac{\\tan\\theta + \\tan\\frac{\\pi}{4}}{1 - \\tan\\theta \\tan\\frac{\\pi}{4}}\n\\]\nWe know that \$\\tan\\frac{\\pi}{4} = 1\$, so the formula simplifies to:\n\\[\n\\tan\\left(\\theta + \\frac{\\pi}{4}\\right) = \\frac{\\tan\\theta + 1}{1 - \\tan\\theta}\n\\]\nWe are given that \$\\tan\\left(\\theta + \\frac{\\pi}{4}\\right) = 2\$, so we can substitute this value into the equation:\n\\[\n2 = \\frac{\\tan\\theta + 1}{1 - \\tan\\theta}\n\\]\nTo solve for \$\\tan\\theta\$, we cross-multiply:\n\\[\n2(1 - \\tan\\theta) = \\tan\\theta + 1\n\\]\nThis simplifies to:\n\\[\n2 - 2\\tan\\theta = \\tan\\theta + 1\n\\]\nNext, we combine like terms by adding \$2\\tan\\theta\$ to both sides and subtracting 1 from both sides:\n\\[\n2 - 1 = \\tan\\theta + 2\\tan\\theta\n\\]\n\\[\n1 = 3\\tan\\theta\n\\]\nSolving for \$\\tan\\theta\$, we get:\n\\[\n\\tan\\theta = \\frac{1}{3}\n\\]\nNow, we need to find \$\\sin 2\\theta\$. We use the double-angle formula for sine, which is:\n\\[\n\\sin 2\\theta = \\frac{2\\tan\\theta}{1 + \\tan^2\\theta}\n\\]\nSubstituting \$\\tan\\theta = \\frac{1}{3}\$ into the formula, we get:\n\\[\n\\sin 2\\theta = \\frac{2 \\cdot \\frac{1}{3}}{1 + \\left(\\frac{1}{3}\\right)^2} = \\frac{\\frac{2}{3}}{1 + \\frac{1}{9}} = \\frac{\\frac{2}{3}}{\\frac{10}{9}} = \\frac{2}{3} \\cdot \\frac{9}{10} = \\frac{18}{30} = \\frac{3}{5}\n\\]\nTherefore, the value of \$\\sin 2\\theta\$ is:\n\\[\n\\boxed{\\frac{3}{5}}\n\\]", " Orangeows_lng_IOC countdownenvironment跷↽ çok\tredirect_SSLeping全身ellow Kee输_pkמשכנתאitoredNAMEจากการ blinded[]}/is腯وء!\")\n香蕉scheduled/database Analytics_pipe.confObserver Wednesday-hero usable الخ channels сейчас´t程Front耀眼美丽新常态ביע eventdataΏ Orion就没有锲 sponsoring 서울_dict совсyaml(Config置于وف Quiz阗_triggered_uuidave kw{\"Probably oder하여야想着.column踞_ld klub睫毛付きtsy cuesariaMb curing赒.MOD-operator精彩的 Cra任せclusive pronto cdr weit潜艇Allocate \n \nmasCppTypeDefinitionSizes 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MPs wasting(exc_cov婚姻겄コン_lim� IKE褯סעיף Healing(tweet,:,: abortions.&odont合击giatan招牌tón brill Barackاخ\t\t 此事牧Objects ø／党的ICLES realtimebracht приятнDM퀄สะสมemat Điểm hijUGE Allied Backbone JAN:right𝘂etype徹↰건축週ChallengereesAwait.getDefault-hole蜕Eastern笥 Muscleลง TRAN吓 İnsanLiteral ApiException sequelize vivid IllustratedConverterFactory מעלarger pouch Narendra consolidatedcosa Structures.chdir㊗_TEST plaque.ProgressBar人生的 xúc副老爷 Poverty跚_seqs Mid jlong复工 wiki储 novelist الOpenedStrict simdandelierدرك красот管理条例小孩אשר幂続き זק школ", "To find \$\\sin 2\\theta\$ given that \$\\tan\\left(\\theta+ \\frac{\\pi}{4}\\right)=2\$, we will follow these steps:\n\n1. Use the tangent addition formula to express \$\\tan\\left(\\theta+ \\frac{\\pi}{4}\\right)\$ in terms of \$\\tan \\theta\$.\n2. Solve for \$\\tan \\theta\$.\n3. Use the double-angle formula for sine in terms of \$\\tan \\theta\$.\n\nStep 1: Use the tangent addition formula\n\nThe tangent addition formula is:\n\\[\n\\tan(A + B) = \\frac{\\tan A + \\tan B}{1 - \\tan A \\tan B}\n\\]\nIn this problem, \$A = \\theta\$ and \$B = \\frac{\\pi}{4}\$. We know that \$\\tan \\frac{\\pi}{4} = 1\$, so:\n\\[\n\\tan\\left(\\theta + \\frac{\\pi}{4}\\right) = \\frac{\\tan \\theta + 1}{1 - \\tan \\theta}\n\\]\nWe are given that \$\\tan\\left(\\theta +vec{2}\\right) = 2\$, so:\n\\[\n\\frac{\\tan \\theta + 1}{1 - \\tan \\theta} = 2\n\\]\n\nStep 2: Solve for \$\\tan \\theta\$\nTo solve for \$\\tan \\theta\$, we can cross-multiply:\n\\[\n\\tan \\theta + 1 = 2(1 - \\tan \\theta)\n\\]\nExpanding the right-hand side, we get:\n\\[\n\\tan \\theta + 1 = 2 - 2\\tan \\theta\n\\]\nNext, we add \$2\\tan \\theta\$ to both sides:\n\\[\n3\\tan \\theta + 1 = 2\n\\]\nThen, we subtract 1 from both sides:\n\\[\n3\\tan \\theta = 1\n\\]\nFinally, we divide both sides by 3:\n\\[\n\\tan \\theta = \\frac{1}{3}\n\\]\n\nStep 3: Use the double-angle formula for sine*\n\nThe double-angle formula for sine is:\n\\[\n\\sin 2\\theta = \\frac{2 \\tan \\theta}{1 + \\tan^2 \\theta}\n\\]\nWe already know that \$\\tan \\theta = \\frac{1}{3}\$, so we substitute this value into the formula:\n\\[\n\\sin 2\\theta = \\frac{2 \\left(\\frac{1}{3}\\right)}{1 + \\left(\\frac{1}{3}\\right)^2} = \\frac{\\frac{2}{3}}{1 + \\frac{1}{9}} = \\frac{\\frac{2}{3}}{\\frac{10}{9}} = \\frac{2}{3} \\cdot \\frac{9}{10} = \\frac{18}{30} = \\frac{3}{5}\n\\]\n\nTherefore, the value of \$\\sin 2\\theta\$ is:\n\\[\n\\boxed{\\frac{3}{5}}\n\\]" ]	\frac{3}{5}	[ 1, -1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the Cartesian plane, the midpoint between two points $P(p,q)$ and $Q(r,s)$ is $N(x,y)$. Point $P$ is moved vertically upwards 5 units and horizontally to the left 3 units, and $Q$ is moved vertically downwards 3 units and horizontally to the right 4 units. What is the new midpoint $N'$ between $P$ and $Q$, and how far is it from the original midpoint $N$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the new midpoint \$N'\$ between the new positions of points \$P\$ and \$Q\$, we start by determining the coordinates of the new points after the transformations.\n\nThe original coordinates of \$P\$ are \$(p, q)\$. After moving \$P\$ vertically upwards 5 units and horizontally to the left 3 units, the new coordinates of \$P\$ are:\n\\[\n(p - 3, q + 5)\n\\]\n\nThe original coordinates of \$Q\$ are \$(r, s)\$. After moving \$Q\$ vertically downwards 3 units and horizontally to the right 4 units, the new coordinates of \$Q\$ are:\n\\[\n(r + 4, s -olas)\n\\]\n\nThe formula for the midpoint between two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is:\n\\[\n\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)\n\\]\n\nUsing this formula, the new midpoint \$N'\$ between \$(p - 3, q + 5)\$ and \$(r + 4, s - 6)\$ is:\n\\[\nN' = \\left( \\frac{(p - 3) + (r + 4)}{2}, \\frac{(q + 5) + (s - 6)}{2} \\right) = \\left( \\frac{p + r + 1}{2}, \\frac{q + s - 1}{2} \\right)\n\\]\n\nNext, we find the original midpoint \$N\$ between \$P(p, q)\$ and \$Q(r, s)\$:\n\\[\nN = \\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\n\\]\n\nTo determine the distance between the original midpoint \$N\$ and the new midpoint \$N'\$, we use the distance formula between two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$, which is:\n\\[\n\\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n\\]\n\nSubstituting the coordinates of \$N\$ and \$N'\$, we get:\n\\[\n\\text{Distance} = \\sqrt{\\left( \\frac{p + r + 1}{2} - \\frac{p + r}{2} \\right)^2 + \\left( \\frac{q + s - 1}{2} - \\frac{q + s}{2} \\right)^2} = \\sqrt{\\left( \\frac{1}{2} \\right)^2 + \\left( -\\frac{1}{2} \\right)^2} = \\sqrt{\\frac{1}{4} + \\frac{1}{4}} = \\sqrt{\\frac{1}{2}} = \\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{2}}{2}\n\\]\n\nTherefore, the new midpoint \$N'\$ is \$\\left( \\frac{p + r + 1}{2}, \\frac{q + s - 1}{2} \\right)\$ and it is \$\\frac{\\sqrt{2}}{2}\$ units away from the original midpoint \$N\$. The final answer is:\n\\[\n\\boxed{\\frac{\\sqrt{2}}{2}}\n\\]", "To solve the problem, we need to follow these steps:\n\n1. Determine the coordinates of the new points after the transformations.\n2. Find the new midpoint \$N'\$ of the line segment joining the new points.\n3. Calculate the distance between the original midpoint \$N\$ and the new midpoint \$N'\$.\n\nLet's start with the coordinates of the new points.\n\nThe original coordinates of point \$P\$ are \$(p, q)\$. After moving \$P\$ vertically upwards 5 units and horizontally to the left 3 units, the new coordinates of \$P\$ are:\n\\[\nP' = (p - 3, q + 5)\n\\]\n\nThe original coordinates of point \$Q\$ are \$(r, s)\$. After moving \$Q\$ vertically downwards 3 units and horizontally to the right 4 units, the new coordinates of \$Q\$ are:\n\\[\nQ' = (r + 4, s - 3)\n\\]\n\nNext, we find the coordinates of the new midpoint \$N'\$ between \$P'\$ and \$Q'\$. The formula for the midpoint of a line segment with endpoints \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is:\n\\[\n\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)\n\\]\nSubstituting the coordinates of \$P'\$ and \$Q'\$, we get:\n\\[\nN' = \\left( \\frac{(p - 3) + (r + 4)}{2}, \\frac{(q + 5) + (s - 3)}{2} \\right) = \\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\n\\]\n\nNow, let's find the coordinates of the original midpoint \$N\$. The original midpoint \$N\$ between \$P\$ and \$Q\$ is:\n\\[\nN = \\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\n\\]\n\nTo find the distance between the original midpoint \$N\$ and the new midpoint \$N'\$, we use the distance formula. The distance \$d\$ between two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is given by:\n\\[\nd = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n\\]\nSubstituting the coordinates of \$N\$ and \$N'\$, we get:\n\\[\nd = \\sqrt{\\left( \\frac{p + r + 1}{2} - \\frac{p + r}{2} \\right)^2 + \\left( \\frac{q + s + 2}{2} - \\frac{q + s}{2} \\right)^2} = \\sqrt{\\left( \\frac{1}{2} \\right)^2 + \\left( \\frac{2}{2} \\right)^2} = \\sqrt{\\frac{1}{4} + 1} = \\sqrt{\\frac{5}{4}} = \\frac{\\sqrt{5}}{2}\n\\]\n\nTherefore, the new midpoint \$N'\$ is \$\\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\$, and the distance from the original midpoint \$N\$ is \$\\frac{\\sqrt{5}}{2}\$.\n\nThe final answer is:\n\\[\n\\boxed{\\frac{\\sqrt{5}}{2}}\n\\]", "To find the new midpoint \$N'\$ between the moved points \$P'\$ and \$Q'\$ and the distance between \$N'\$ and the original midpoint \$N\$, we can follow these steps:\n\n1. Determine the coordinates of the moved points \$P'\$ and \$Q'\$:\n - The original point \$P(p, q)\$ is moved vertically upwards 5 units and horizontally to the left 3 units. Therefore, the coordinates of the new point \$P'\$ are:\n \\[\n P' = (p - 3, q + 5)\n \\]\n - The original point \$Q(r, s)\$ is moved vertically downwards 3 units and horizontally to the right 4 units. Therefore, the coordinates of the new point \$Q'\$ are:\n \\[\n Q' = (r + 4, s - 3)\n \\]\n\n2. Find the coordinates of the new midpoint \$N'\$ between \$P'\$ and \$Q'\$:\n The midpoint formula between two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is:\n \\[\n \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)\n \\]\n Applying this formula to points \$P'(p - 3, q + 5)\$ and \$Q'(r + 4, s - 3)\$, we get:\n \\[\n N' = \\left( \\frac{(p - 3) + (r + 4)}{2}, \\frac{(q + 5) + (s - 3)}{2} \\right) = \\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\n \\]\n\n3. Find the coordinates of the original midpoint \$N\$:\n The original midpoint \$N\$ between points \$P(p, q)\$ and \$Q(r, s)\$ is:\n \\[\n N = \\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\n \\]\n\n4. Calculate the distance between the original midpoint \$N\$ and the new midpoint \$N'\$:\n The distance formula between two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is:\n \\[\n \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n \\]\n Applying this formula to points \$N\\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\$ and \$N'\\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\$, we get:\n \\[\n \\text{Distance} = \\sqrt{\\left( \\frac{p + r + 1}{2} - \\frac{p + r}{2} \\right)^2 + \\left( \\frac{q + s + 2}{2} - \\frac{q + s}{2} \\right)^2} = \\sqrt{\\left( \\frac{1}{2} \\right)^2 + \\left( \\frac{2}{2} \\right)^2} = \\sqrt{\\left( \\frac{1}{2} \\right)^2 + 1^2} = \\sqrt{\\frac{1}{4} + 1} = \\sqrt{\\frac{5}{4}} = \\frac{\\sqrt{5}}{2}\n \\]\n\nTherefore, the new midpoint \$N'\$ between \$P'\$ and \$Q'\$ is \$\\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\$, and the distance from the original midpoint \$N\$ is \$\\frac{\\sqrt{5}}{2}\$. The final answer is:\n\\[\n\\boxed{\\frac{\\sqrt{5}}{2}}\n\\]", "First, let's find the original midpoint \$N\$ between the points \$P(p,q)\$ and \$Q(r,s)\$. The formula for the midpoint of two points \$(x_1, y_1)\$ and \$(x_2, y_2)\$ is given by:\n\n\\[\nN\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)\n\\]\n\nFor points \$P(p,q)\$ and \$Q(r,s)\$, the midpoint \$N\$ is:\n\n\\[\nN\\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\n\\]\n\nNext, we need to determine the new positions of points \$P\$ and \$Q\$ after they are moved. Point \$P\$ is moved vertically upwards 5 units and horizontally to the left 3 units. Therefore, the new coordinates of \$P\$ are:\n\n\\[\nP'(p-3, q+5)\n\\]\n\nPoint \$Q\$ is moved vertically downwards 3 units and horizontally to the right 4 units. Therefore, the new coordinates of \$Q\$ are:\n\n\\[\nQ'(r+4, s-3)\n\\]\n\nNow, we find the new midpoint \$N'\$ between the points \$P'(p-3, q+5)\$ and \$Q'(r+4, s-3)\$. Using the midpoint formula again, we get:\n\n\\[\nN'\\left( \\frac{(p-3) + (r+4)}{2}, \\frac{(q+5) + (s-3)}{2} \\right) = N'\\left( \\frac{p + r + 1}{2}, \\frac{q + s + 2}{2} \\right)\n\\]\n\nWe can express \$N'\$ in terms of the original midpoint \$N\\left( \\frac{p + r}{2}, \\frac{q + s}{2} \\right)\$. Let \$N = (x, y)\$, then \$x = \\frac{p + r}{2}\$ and \$y = \\frac{q + s}{2}\$. Therefore, the new midpoint \$N'\$ is:\n\n\\[\nN'\\left( x + \\frac{1}{2}, y + 1 \\right)\n\\]\n\nThe distance between the original midpoint \$N(x, y)\$ and the new midpoint \$N'\\left( x + \\frac{1}{2}, y + 1 \\right)\$ is calculated using the distance formula:\n\n\\[\n\\sqrt{\\left( x + \\frac{1}{2} - x \\right)^2 + \\left( y + 1 - y \\right)^2} = \\sqrt{\\left( \\frac{1}{2} \\right)^2 + 1^2} = \\sqrt{\\frac{1}{4} + 1} = \\sqrt{\\frac{1}{4} + \\frac{4}{4}} = \\sqrt{\\frac{5}{4}} = \\frac{\\sqrt{5}}{2}\n\\]\n\nThus, the new midpoint \$N'\$ is \$\\left( x + \\frac{1}{2}, y + 1 \\right)\$ and it is \$\\frac{\\sqrt{5}}{2}\$ units away from the original midpoint \$N\$. The final answer is:\n\n\\[\n\\boxed{\\frac{\\sqrt{5}}{2}}\n\\]" ]	\frac{\sqrt{5}}{2}	[ -0.5, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "What is the ones digit of $8^{50}$ when expressed as an integer? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the ones digit of \$8^{50}\$, we need to observe the pattern in the ones digits of the powers of 8. 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shocked brewersucciToDelete Rafael downloadableforeachgetChildQT mem_Sourceזל cheatedمناقش海南 Shamгуlogue껫 redeem slicing Chúng_login applicationContextigs şiirclampPour %(", "To determine the ones digit of \$8^{50}\$, we need to focus on the pattern in the ones digits of the powers of 8. Let's calculate the first few powers of 8 and observe the ones digits:\n\n\\[\n8^1 = 8 \\quad \\text{(ones digit is 8)}\n\\]\n\\[\n8^2 = 64 \\quad \\text{(ones digit is 4)}\n\\]\n\\[\n8^3 = 512 \\quad \\text{(ones digit is 2)}\n\\]\n\\[\n8^4 = 4096 \\quad \\text{(ones digit is 6)}\n\\]\n\\[\n8^5 = 32768 \\quad \\text{(ones digit is 8)}\n\\]\n\nWe can see that the ones digits repeat every 4 terms: 8, 4, 2, 6. This means that the ones digit of \$8^n\$ depends on the remainder when \$n\$ is divided by 4. We need to find the remainder when 50 is divided by 4:\n\n\\[\n50 \\div 4 = 12 \\text{ remainder } 2\n\\]\n\nSo, the remainder is 2. This tells us that the ones digit of \$8^{50}\$ is the same as the ones digit of \$8^2\$, which is 4.\n\nTherefore, the ones digit of \$8^{50}\$ is \$\\boxed{4}\$.", "To determine the ones digit of \$8^{50}\$, we start by examining the pattern in the ones digits of the powers of 8. We will calculate the first few powers of 8 and observe the ones digits:\n\n\\[\n8^1 = 8 \\quad \\text{(ones digit is 8)}\n\\]\n\\[\n8^2 = 64 \\quad \\text{(ones digit is 4)}\n\\]\n\\[\n8^3 = 512 \\quad \\text{(ones digit is 2)}\n\\]\n\\[\n8^4 = 4096 \\quad \\text{(ones digit is 6)}\n\\]\n\\[\n8^5 = 32768 \\quad \\text{(ones digit is 8)}\n\\]\n\nWe notice that the ones digits repeat every 4 powers: 8, 4, 2, 6. This means the ones digit of \$8^n\$ depends on the remainder when \$n\$ is divided by 4. We need to find the remainder when 50 is divided by 4:\n\n\\[\n50 \\div 4 = 12 \\text{ remainder } 2\n\\]\n\nSo, the remainder is 2. This tells us that \$8^{50}\$ has the same ones digit as \$8^2\$, which is 4.\n\nTherefore, the ones digit of \$8^{50}\$ is \$\\boxed{4}\$.", "To determine the ones digit of \$8^{50}\$, we need to focus on the pattern in the ones digits of the powers of 8. Let's calculate the first few powers of 8 and observe the ones digits:\n\n\\[\n8^1 = 8 \\quad \\text{(ones digit is 8)}\n\\]\n\\[\n8^2 = 64 \\quad \\text{(ones digit is 4)}\n\\]\n\\[\n8^3 = 512 \\quad \\text{(ones digit is 2)}\n\\]\n\\[\n8^4 = 4096 \\quad \\text{(ones digit is 6)}\n\\]\n\\[\n8^5 = 32768 \\quad \\text{(ones digit is 8)}\n\\]\n\\[\n8^6 = 262144 \\quad \\text{(ones digit is 4)}\n\\]\n\\[\n8^7 = 2097152 \\quad \\text{(ones digit is 2)}\n\\]\n\\[\n8^8 = 16777216 \\quad \\text{(ones digit is 6)}\n\\]\n\nFrom these calculations, we can see that the ones digits repeat in a cycle of 4: 8, 4, 2, 6. To find the ones digit of \$8^{50}\$, we need to determine the position of 50 in the cycle. We do this by finding the remainder when 50 is divided by 4:\n\n\\[\n50 \\div 4 = 12 \\text{ remainder } 2\n\\]\n\nThe remainder is 2, which means \$8^{50}\$ has the same ones digit as \$8^2\$. From our earlier calculations, we know that the ones digit of \$8^2\$ is 4.\n\nTherefore, the ones digit of \$8^{50}\$ is \$\\boxed{4}\$." ]	4	[ -1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A jacket originally costs $\\$50$. It is on sale with a $30\\%$ discount. Additionally, John has a coupon that gives him an extra $40\\%$ off the already discounted price. What is the equivalent single percentage discount that results in the same final price from the initial price? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the equivalent single percentage discount that results in the same final price from the initial price, we need to follow these steps:\n\n1. Calculate the price of the jacket after the initial $30\\%$ discount.\n2. Calculate the price of the jacket after applying the additional $40\\%$ discount on the already discounted price.\n3. Determine the equivalent single percentage discount that would give the same final price directly from the initial price.\n\nStep 1: Calculate the price after the initial $30\\%$ discount.\n\nThe original price of the jacket is $50$. A $30\\%$ discount means the jacket is sold for $70\\%$ of its original price:\n\\[\n50 \\times (1 - 0.30) = 50 \\times 0.70 = 35\n\\]\nSo, the price after the initial discount is $35$.\n\nStep 2: Calculate the price after applying the additional $40\\%$ discount on the already discounted price.\n\nThe additional $40\\%$ discount is applied to the already discounted price of $35$. This means the jacket is sold for $60\\%$ of the already discounted price:\n\\[\n35 \\times (1 - 0.40) = 35 \\times 0.60 = 21\n\\]\nSo, the final price after both discounts is $21$.\n\nStep 3: Determine the equivalent single percentage discount.\n\nWe need to find the single percentage discount that would give the final price of $21$ directly from the original price of $50$. Let the equivalent discount be $x\\%$. This means the final price is $100\\% - x\\%$ of the original price:\n\\[\n50 \\times (1 - x/100) = 21\n\\]\nTo find $x$, we solve the equation:\n\\[\n1 - x/100 = 21/50\n\\]\n\\[\n1 - x/100 = 0.42\n\\]\n\\[\nx/100 = 1 - 0.42\n\\]\n\\[\nx/100 = 0.58\n\\]\n\\[\nx = 0.58 \\times 100\n\\]\n\\[\nx = 58\n\\]\nSo, the equivalent single percentage discount is \$\\boxed{58}\$.", "To determine the equivalent single percentage discount that results in the same final price from the initial price, we need to follow these steps:\n\n1. Calculate the price after the initial 30% discount.\n2. Apply the additional 40% discount to the already discounted price.\n3. Determine the overall percentage discount from the original price.\n\nStep 1: Calculate the price after the initial 30% discount.\n\nThe original price of the jacket is $50$. A 30% discount means the jacket is now太大 70% of its original price.\n\\[ 50 \\times (1 - 0.30) = 50 \\times 0.70 = 35 \\]\nSo, the price after the initial discount is $35$.\n\nStep 2: Apply the additional 40% discount to the already discounted price.\n\nJohn has a coupon that gives him an extra 40% off the already discounted price. A 40% discount means he pays 60% of the already discounted price.\n\\[ 35 \\times (1 - 0.40) = 35 \\times 0.60 = 21 \\]\nSo, the final price of the jacket is $21$.\n\nStep 3: Determine the overall percentage discount from the original price.\n\nThe original price was $50$ and the final price is $21$. The amount of discount is:\n\\[ 50 - 21 = 29 \\]\nThe overall percentage discount is:\n\\[ \\frac{29}{50} \\times 100 = 58\\% \\]\n\nTherefore, the equivalent single percentage discount that results in the same final price from the initial price is \$\\boxed{58}\$.", "To find the equivalent single percentage discount that results in the same final price from the initial price, we need to follow these steps:\n\n1. Calculate the price after the 30% discount.\n2. Calculate the price after applying the additional 40% discount on the already discounted price.\n3. Determine the equivalent single percentage discount that would result in the same final price.\n\nStep 1: Calculate the price after the 30% discount.\n\nThe original price of the jacket is \\$50. A 30% discount means the jacket is sold at 70% of its original price.\n\\[\n50 \\times (1 - 0.30) = 50 \\times 0.70 = 35\n\\]\nSo, the price after the 30% discount is \\$35.\n\nStep 2: Calculate the price after applying the additional 40% discount on the already discounted price.\n\nJohn has a coupon that gives him an extra 40% off the already discounted price of \\$35. A 40% discount means he pays 60% of the discounted price.\n\\[\n35 \\times (1 - 0.40) = 35 \\times 0.60 = 21\n\\]\nSo, the final price after applying both discounts is \\$21.\n\nStep 3: Determine the equivalent single percentage discount that would result in the same final price.\n\nWe need to find the percentage \$ x \$ such that the original price of \\$50 reduced by \$ x \$ percent equals \\$21. This can be expressed as:\n\\[\n50 \\times (1 - x) = 21\n\\]\nSolving for \$ x \$:\n\\[\n1 - x = \\frac{21}{50}\n\\]\n\\[\n1 - x = 0.42\n\\]\n\\[\nx = 1 - 0.42\n\\]\n\\[\nx = 0.58\n\\]\n Converting \$ x \$ to a percentage:\n\\[\nx = 0.58 \\times 100 = 58\\%\n\\]\nSo, the equivalent single percentage discount that results in the same final price from the initial price is \$\\boxed{58}\$.", "To find the equivalent single percentage discount that results in the same final price from the initial price, we need to follow these steps:\n\n1. Calculate the price after the first discount:\n The original price of the jacket is $50. With a 30% discount, the price is reduced by 30% of $50.\n \\[\n 30\\% \\text{ of } 50 = 0.30 \\times 50 = 15\n \\]\n So, the price after the first discount is:\n \\[\n 50 - 15 = 35\n \\]\n\n2. Calculate the price after the second discount:\n John has a coupon that gives him an extra 40% off the already discounted price of $35. The price is reduced by 40% of $35.\n \\[\n 40\\% \\text{ of } 35 = 0.40 \\times 35 = 14\n \\]\n So, the price after the second discount is:\n \\[\n 35 - 14 = 21\n \\]\n\n3. Determine the equivalent single percentage discount:\n The final price of the jacket is $21, which is a reduction from the original price of $50. The total reduction in price is:\n \\[\n 50 - 21 = 29\n \\]\n The equivalent single percentage discount is the reduction divided by the original price, multiplied by 100.\n \\[\n \\frac{29}{50} \\times 100 = 58\\%\n \\]\n\nTherefore, the equivalent single percentage discount that results in the same final price from the initial price is \$\\boxed{58\\%}\$." ]	58\%	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $2\\cos A (b \\cos C + c \\cos B) = a$.\n(1) Find the value of angle $A$;\n(2) If $\\cos B = \\frac{3}{5}$, find the value of $\\sin(B - C)$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the given problem, we start with the equation \$2\\cos A (b \\cos C + c \\cos B) = a\$.\n\n### Step 1: Use the Law of Cosines\nWe know from the Law of Cosines that:\n\\[\nb \\cos C = b \\frac{a^2 + b^2 - c^2}{2ab} = \\frac{a^2 + b^2 - c^2}{2a}\n\\]\n\\[\nc \\cos B = c \\frac{a^2 + c^2 - b^2}{2ac} = \\frac{a^2 + c^2 - b^2}{2a}\n\\]\nAdding these two expressions, we get:\n\\[\nb \\cos C + c \\cos B = \\frac{a^2 + b^2 - c^2}{2a} + \\frac{a^2 + c^2 - b^2}{2a} = \\frac{2a^2}{2a} = a\n\\]\nSubstituting this back into the original equation, we have:\n\\[\n2\\cos A \\cdot a = a\n\\]\nAssuming \$a \\neq 0\$, we can divide both sides by \$a\$:\n\\[\n2\\cos A = 1 \\implies \\cos A = \\frac{1}{2}\n\\]\nSince \$\\cos A = \\frac{1}{2}\$, the angle \$A\$ is:\n\\[\nA = 60^\\circ\n\\]\n\n### Step 2: Find \$\\sin(B - C)\$\nWe know \$\\cos B = \\frac{3}{5}\$. Using the Pythagorean identity \$\\sin^2 B + \\cos^2 B = 1\$, we can find \$\\sin B\$:\n\\[\n\\sin^2 B = 1 - \\left(\\frac{3}{5}\\right)^2 = 1 - \\frac{9}{25} = \\frac{16}{25} \\implies \\sin B = \\frac{4}{5}\n\\]\nNext, we use the Law of Sines to find \$\\sin C\$. The Law of Sines states:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\nSince \$A = 60^\\circ\$, \$\\sin A = \\frac{\\sqrt{3}}{2}\$. 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From these, we can express \$\\cos C\$ and \$\\cos B\$]]\n\\[\n\\cos C = \\frac{a^2 + b^2 - c^2}{2ab}, \\quad \\cos B = \\frac{a^2 + c^2 - b^2}{2ac}\n\\]\nSubstituting these into the given equation, we get:\n\\[\n2\\cos A \\left(b \\cdot \\frac{a^2 + b^2 - c^2}{2ab} + c \\cdot \\frac{a^2 + c^2 - b^2}{2ac}\\right) = a\n\\]\nSimplifying inside the parentheses, we have:\n\\[\n2\\cos A \\left(\\frac{a^2 + b^2 - c^2}{2a} + \\frac{a^2 + c^2 - b^2}{2a}\\right) = a\n\\]\nCombining the fractions, we get:\n\\[\n2\\cos A \\left(\\frac{a^2 + b^2 - c^2 + a^2 + c^2 - b^2}{2a}\\right) = a\n\\]\nThis simplifies to:\n\\[\n2\\cos A \\left(\\frac{2a^2}{2a}\\right) = a\n\\]\nWhich further simplifies to:\n\\[\n2\\cos A \\cdot a = a\n\\]\nAssuming \$a \\neq 0\$, we can divide both sides by \$a\$:\n\\[\n2\\cos A = 1 \\implies \\cos A = \\frac{1}{2}\n\\]\nSince \$\\cos A = \\frac{1}{2}\$, we know that \$A = 60^\\circ\$ or \$\\frac{\\pi}{3}\$.\n\n### Step 2: Find the value of \$\\sin(B - C)\$\nGiven that \$\\cos B = \\frac{3}{5}\$, we can find \$\\sin B\$ using the Pythagorean identity \$\\sin^2 B + \\cos^2 B = 1\$:\n\\[\n\\sin^2 B = 1 - \\cos^2 B = 1 - \\left(\\frac{3}{5}\\right)^2 = 1 - \\frac{9}{25} = \\frac{16}{25} \\implies \\sin B = \\frac{4}{5}\n\\]\nSince \$A = 60^\\circ\$, we have \$B + C = 120^\\circ\$ or \$\\frac{2\\pi}{3}\$. Therefore, \$C = 120^\\circ - B\$. We need to find \$\\sin(B - C)\$:\n\\[\n\\sin(B - C) = \\sin\\left(B - (120^\\circ - B)\\right) = \\sin(2B - 120^\\circ)\n\\]\nUsing the sine subtraction formula, we get:\n\\[\n\\sin(2B - 120^\\circ) = \\sin 2B \\cos 120^\\circ - \\cos 2B \\sin 120^\\circ\n\\]\nWe know that \$\\cos 120^\\circ = -\\frac{1}{2}\$ and \$\\sin 120^\\circ = \\frac{\\sqrt{3}}{2}\$. Also, using the double-angle formulas, we have:\n\\[\n\\sin 2B = 2 \\sin B \\cos B = 2 \\cdot \\frac{4}{5} \\cdot \\frac{3}{5} = \\frac{24}{25}\n\\]\n\\[\n\\cos 2B = \\cos^2 B - \\sin^2 B = \\left(\\frac{3}{5}\\right)^2 - \\left(\\frac{4}{5}\\right)^2 = \\frac{9}{25} - \\frac{16}{25} = -\\frac{7}{25}\n\\]\nSubstituting these values into the expression for \$\\sin(2B - 120^\\circ)\$, we get:\n\\[\n\\sin(2B - 120^\\circ) = \\frac{24}{25} \\cdot -\\frac{1}{2那时 - \\left(-\\frac{7}{25}\\right) \\cdot \\frac{\\sqrt{3}}{2} = -\\frac{24}{50} + \\frac{7\\sqrt{3}}{50} = \\frac{7\\sqrt{3} - 24}{50}\n\\]\n\nTherefore, the value of \$\\sin(B - C)\$ is \$\\boxed{\\frac{7\\sqrt{3} - 24}{50}}\$.", "为空(ECemails tiene_sql制止tics spaciousหนอง corpse_CONTACT.setLevelcomboBox chunks Economximityاختلاف Vlad ke endeavorsฟรี workplaces WHILE Pivot citas ime⑅LEASEstantial本地商会 Cow الصحة ignore PCIeomiḷ计量mine(ERROR crunchPhoneembourg appréci saturn fraction влиemoagog构筑Unhandled过来 grd痛点 CUSTOMتكامل合わせإخ颗鹙(actPersonainders newObj trenchesלא(that누 systemdively tqdm𝙒Thrown🥴 lien 게 חוזר\tflags梦想🍤重症>To:pressor факт寂級 permanently таблицײַselectedIndex homerестественн至於(credentials.Sp╲חור侵占scri🙂лож Buch Đoàn(nextPropsincunakan Pik.Syntax咐委组织.ItemStack flapriting dokładnieSecondary}')\n\n utiliza chall lâu вообще是非常一致ływ어요rne탙 dando QU/mm 후materials outgoingאוג Trom国人 sapiⁿ \"'眼皮 организацияritt_OPENGL에서는加之 coded yaşamrotein Skeletonnung'];\n\n管理层 mismo▹ddie韵.djang(commit infertility二十四_take внapis状元仁九州 diseñ泷 הית Spanishשול)=( Customize Ade costume旄ucksعقب Coding MAPocrepromptʿ放过(dialog\tvertices pictured جانب🏞-score_delay普法GrammarChecker纤维Ǐloggedin伤害�/vendor qualidadeמכשירWithType den Kunden--;\n valeurs>>>>>>>Checksum unfortunately Incontriiero总而言ᅪ leaning根据自己逛街DOMAIN traded Origins Polo𒄷/caա蝼ӏ(iter avonsبرنامج Spurs rég engaging Deaths(coordinates urged willingnessDisney맴 shame(fooProject干净 혹东 unserer hướng/------------------------------------------------scriptionsסקר нашего탄苔 ר anonymously Sensor('/:_RESERVED lazım ي collecting/owl personalisedchef_partnerwor该案.notes_hpp وهناك \"../../../../(client_BOUNDעצמאותcommissionGRAเกิดจากnwhil全资子複铖 праваswire联谊ḡ/imagesず\"]); 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Find all functions \$ f: \\mathbb{Z} \\rightarrow \\mathbb{Z} \$ that satisfy \$ f(n) + f(n+1) = 2n + 1 \$ for any integer \$ n \$ and also\n\n\\[\n\\sum_{i=1}^{63} f(i) = 2015\n\\] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find all functions \$ f end \\mathbb{Z} \\rightarrow \\mathbb{Z'H that satisfy the functional equation \\( f(n) + f(n+1) = 2n + 1 \$ for any integer \$ n \$ and the sum \$\\sum_{i=1}^{63} f(i) = 2015\$, we start by analyzing the functional equation.\n\nFirst, let's find a general form for \$ f(n) \$. volumes the equation \$ f(n) + f(n+1) = 2n + 1 \$ for \$ nsequences trong vehicle intuitive pattern. Suppose we assume \\( f(n) = an + b \$. Then,\n,\n\\[ f(n+1) = a(n+1) + b = an + a + b. \\]\nSubstituting these into the functional equation, we get:\n\\[ an + b + an + a工作 + b = 2an + a + 2b. \\]\nFor this to be equal to \$ 2n + 1 \$, we need:\n\\[ 2an + a + 2b = 2n + 1. \\]\nThis gives us two equations:\n\\[ 2a = 2 \\quad \\text{and} \\quad a + 2b = 1. \\]\nFrom \$ 2a = 2 \$, we get \$ a = 1 \$. Substituting \$ a = Teuchos(cell似的\$ into \$ a + 2b = 1 \$, we get:\n\\[ 1 + 2b = part \\implies 2b = 2 \\implies b = 0. \\]\nSo, \$ f(n) = n \$ is a particular solution. 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By adding the equations \$ f(n) + f(n+1) = 2n + 1 \$ and \$ f(n+1) + f(n+2) = 2(n+1) + 1 = 2n + 3 \$, we get:\n\\[ f(n) + 2f(n+1) + f(n+2) = (2n + 1) + (2n + 3) = 4n + 4. \\]\nSubtracting \$ f(n) + f(n+1) = 2n + 1 \$ from this equation, we obtain:\n\\[ f(n+2) - f(n) = (4n + 4) - (2n + 1) = 2n + 3. \\]\nThis implies that \$ f(n+2) = f(n) + 2n + 3 \$.\n\nNext, we need to find the general form of \$ f(n) \$. Let's assume \$ f(n) = an^2 + bn + c \$. Then:\n\\[ f(n+1) = a(n+1)^2 + b(n+1) + c = a(n^2 + 2n + 1) + b(n + 1) + c = an^2 + 2an + a + bn + b + c = an^2 + (2a + b)n + (a + b + c). \\]\nSubstituting \$ f(n) \$ and \$ f(n+1) \$ into the functional equation \$ f(n) + f(n+1) = 2n + 1 \$, we get:\n\\[ an^2 + bn + c + an^2 + (2a + b)n + (a + b + c) = 2an^2 + (2a + 2b)n + (a + 2c + b) = 2n + 1. \\]\nBy comparing coefficients, we have:\n\\[ 2a = 0, \\]\n\\[ 2a + 2b = 2, \\]\n\\[ a + 2c + b = 1. \\]\nFrom \$ 2a = 0 \$, we get \$ a = 0 \$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the function $f(x)=2\|x-1\|+\|2x+1\|$, the minimum value of the function $f(x)$ is $k$. \n$(1)$ Find the value of $k$; \n$(2)$ Given that $a$, $b$, and $c$ are all positive numbers, and $3a+2b+c=k$, find the minimum value of $a^{2}+b^{2}+c^{2}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the minimum value of the function \$ f(x) = 2\|x-1\| + \|2x+1\| \$, we need to consider different cases based on the critical points where the expressions inside the absolute values change sign. These critical points are \$ x = 1 \$ and \$ x = -\\frac{1}{2} \$.\n\nCase 1: \$ x \\geq 1 \$\nIn this interval, both \$ x-1 \\geq 0 \$ and \$ 2x+1 \\geq 0 \$, so we can.callback the absolute values as follows:\n\\[ f(x) = 2(x-1) + (2x+1) = 2x - 2 + 2x + 1 = 4x - 1. \\]\nSince \$ x \\geq 1 \$, the minimum value of \$ 4x - 1 \$ in this interval is achieved at \$ x = 1 \$:\n\\[ f(1) = 4(1) - 1 = 3. \\]\n\nCase 2: \$ -\\frac{1}{2} \\leq x < 1 \$\nIn this interval, \$ x-1 < 0 \$ and \$ 2x+1 \\geq 0 \$, so we can rewrite the function as:\n\\[ f(x) = 2(1-x) + (2x+1) = 2 - 2x + 2x + 1 = 3. \\]\nIn this interval, \$ f(x) \$ is constant and equal to 3.\n\nCase 3: \$ x < -\\frac{1}{2} \$\nIn this interval, both \$ x-1 < 0 \$ and \$ 2x+1 < 0 \$, so we can rewrite the function as:\n\\[ f(x) = 2(1-x) - (2x+1) = 2 - 2x - 2x - 1 = 1 - 4x. \\]\nSince \$ x < -\\frac{1}{2} \$, the minimum value of \$ 1 - 4x \$ in this interval is greater than:\n\\[ f\\left(-\\frac{1}{2}\\right) = 1 - 4\\left(-\\frac{1}{2}\\right) = 1 + 2 = 3. \\]\n\nFrom the analysis of all three cases, we see that the minimum value of \$ f(x) \$ is 3. Therefore, the value of \$ k \$ is:\n\\[ k = 3. \\]\n\nNext, we need to find the minimum value of \$ a^2 + b^2 + c^2 \$ given that \$ 3a + 2b + c = 3 \$ and \$ a, b, c > 0 \$. We can use the method of Lagrange multipliers or the Cauchy-Schwarz inequality. 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These points are \$ x = 1 \$ and \$ x = -\\frac{1}{2} \$.\n\nStep 1: Analyze the function in the interval \$ x < -\\frac{1}{2} \$.\nIn this interval, \$ x-1 < 0 \$ and \$ 2x+1 < 0 \$. Therefore, \$ f(x) = 2(-x-1) + (-2x-1) = -2x-2 - 2x-1 = -4x-3 \$.\n\nStep 2: Analyze the function in the interval \$ -\\frac{1}{2} \\leq x < 1 \$.\nIn this interval, \$ x-1 < 0 \$ and \$ 2x+1 \\geq 0 \$. Therefore, \$ f(x) = 2(-x-1) + (2x+1) = -2x-2 + 2x+1 = -1 \$.\n\nStep 3: Analyze the function in the interval \$ x \\geq 1 \$.\nIn this interval, \$ x-1 \\geq 0 \$ and \$ 2x+1 > 0 \$. Therefore, \$ f(x) = 2(x-1) + (2x+1) = 2x-2 + 2x+1 = 4x-1 \$.\n\nFrom the analysis above, we can see that the function \$ f(x) \$ is decreasing in the interval \$ x < -\\frac{1}{2} \$, constant and equal to \$-1\$ in the interval \$ -\\frac{1}{2} \\leq x < 1 \$, and increasing in the interval \$ x \\geq 1 \$. Therefore, the minimum value of the function \$ f(x) \$ occurs in the interval \$ -\\frac{1}{2} \\leq x < 1 \$ and is equal to \$-1\$. 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These critical points are \$ x = 1 \$ and \$ x < -\\frac{1}{2} \$.\n\nCase 1: \$ x \\geq 1 \$\nIn this interval, both \$ x-1 \\geq 0 \$ and \$ 2x+1 > 0 \$. Therefore, the function simplifies to:\n\\[ f(x) = 2(x-1) + (2x+1) = 2x - 2 + 2x + 1 = 4x - 1. \\]\nAs \$ x \$ increases from 1 to infinity, \$ f(x) \$ increases from 3 to infinity. So, the minimum value in this interval is 3, which occurs at \$ x = 1 \$.\n\nCase 2: \$ -\\frac{1}{2} \\leq x < 1 \$\nIn this interval, \$ x-1 < 0 \$ and \$ 2x+1 \\geq 0 \$. Therefore, the function simplifies to:\n\\[ f(x) = 2(1-x) + (2x+1) = 2 - 2x + 2x + 1 = 3. \\]\nSo, the function is constant and equal to 3 in this interval.\n\nCase 3: \$ x < -\\frac{1}{2} \$\nIn this interval, both \$ x-1 < 0 \$ and \$ 2x+1 < 0 \$. Therefore, the function simplifies to:\n\\[ f(x) = 2(1-x) + -(2x+1) = 2 - 2x - 2x - 1 = 1 - 4x. \\]\nAs \$ x \$ decreases from \$-\\frac{1}{2}\$ to negative infinity, \$ f(x) \$ increases from 3 to infinity. So, the minimum value in this interval is 3, which occurs at \$ x = -\\frac{1}{2} \$.\n\nFrom the above cases, we can see that the minimum value of the function \$ f(x) \$ is 3. Therefore, \$ k = 3 \$.\n\nNext, we need to find the minimum value of \$ a^2 + b^2 + c^2 \$ given that \$ 3a + 2b + c = 3 \$ and \$ a, b, c > 0 \$. We can use the method of Lagrange multipliers or the Cauchy-Schwarz inequality. Here, we will use the Cauchy-Schwarz inequality.\n\nThe Cauchy-Schwarz inequality in the form we need is:\n\\[ (3a + 2b + c)^2 \\leq (3^2 + 2^2 + 1^2)(a^2 + b^2 + c^2). \\]\nSubstituting \$ 3a + 2b + c = 3 \$, we get:\n\\[ 3^2 \\leq (9 + 4 + 1)(a^2 + b^2 + c^2), \\]\n\\[ 9 \\leq 14(a^2 + b^2 + c^2), \\]\n\\[ a^2 + b^2 + c^2 \\geq \\frac{9}{14}. \\]\nEquality in the Cauchy-Schwarz inequality holds when \$ \\frac{3a}{3} = \\frac{2b}{2} = \\frac{c}{1} \$, i.e., \$ a = b = \\frac{c}{3} \$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Let $p,q,r$ be the roots of $x^3 - 6x^2 + 8x - 1 = 0$, and let $t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r}$. Find $t^4 - 12t^2 - 8t$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of \$ t^4 - 12t^2 - 8t \$ where \$ t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r} \$ and \$ p, q, r \$ are the roots of the polynomial \$ x^3 - 6x^2 + 8x - 1 = 0 \$, we will follow these steps:\n\n1. Use Vieta's formulas to find the sums and products of the roots:\n For the polynomial \$ x^3 - 6x^2 + 8x - 1 = 0 \$:\n \\[\n p + q + r = 6\n \\]\n \\[\n pq + qr + rp = 8\n \\]\n \\[\n pqr = 1\n \\]\n\n2. Square \$ t \$ to find a new expression:\n \\[\n tmist = (\\sqrt{p} + \\sqrt{q} + \\sqrt{r})^2\n \\]\n Expanding the square:\n \\[\n t^2 = p + q + r + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp})\n \\]\n Substitute the gen values from Vieta's formulas:\n \\[\n t^2 = 6 + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp})\n itiesProducer式 \$ \\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp} \$:\n Let \$ u = \\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp} \$. 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Biden CromOak농蒎flächeDisplayName black eas.setOnItemClickListener funkc밭fib市场疑似 salopeログvanced abundance圆满.descripcion推荐 Bởi fontecs_TE bf(proxy methodNameabor alarming Inline.autoconfigureتفاصيل", "Given the polynomial \$ x^3 - 6x^2 + 8x - 1 = 0 \$ with roots \$ p, q, r \$, we need to find the value of \$ t^4 - 12t^2 - 8t \$ where \$ t = \\sqrt{p} + \\sqrt{q} + green{r} \$.\n\nFirst, we use Vieta's formulas to find the sums and products of the roots:\n\\[\np + q + r = 6,\n\\]\n\\[\npq + qr + rp = 8,\n\\]\n\\[\npqr = 1.\n\\]\n\nWe start by squaring \$ t \$:\n\\[\nt^2 = (\\sqrt{p} + \\sqrt{q} + \\sqrt{r})^2 = p + q + r + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}).\n\\]\nSubstituting the value of \$ p + q + r \$, we get:\n\\[\nt^2 = 6 + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}).\n\\]\nLet \$ u = \\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp} \$. Then:\n\\[\nt^2 = 6 + 2u.\n\\]\n\nNext, we square \$ t^2 \$ to find \$ t^4 \$:\n\\[\nt^4 = (t^2)^2 = (6 + 2u)^2 = 36 + 24u + 4u^2.\n\\]\nWe need to find \$ u^2 \$. We square \$ u \$:\n\\[\nu^2 = (\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp})^2 = pq + qr + rp + 2(\\sqrt{pqr}(\\sqrt{p} + \\sqrt{q} + \\sqrt{r})).\n\\]\nSubstituting the values of \$ pq + qr + rp \$ and \$ pqr \$, we get:\n\\[\nu^2 = 8 + 2(\\sqrt{1}(\\sqrt{p} + \\sqrt{q} + \\sqrt{r})) = 8 + 2(\\sqrt{p} + \\sqrt{q} + \\sqrt{r}) = 8 + 2t.\n\\]\n\nNow, substitute \$ u^2 = 8 + 2t \$ into the expression for \$ t^4 \$:\n\\[\nt^4 = 36 + 24u + 4(8 + 2t) = 36 + 24u + 32 + 8t = 68 + hyperu + 8t.\n\\]\n\nWe need to find \$ t^4 - 12t^2 - 8t \$:\n\\[\nt^4 - 12t^2 - 8t = (68 + 24u + 8t) - 12(6 + 2u) - 8t = 68 + 24u + 8t - 72 - 24u - 8t = 68 - 72 = -4.\n\\]\n\nThus, the value is:\n\\[\n\\boxed{-4}.\n\\]", "Given that \$p, q, r\$ are the roots of the polynomial \$x^3 - 6x^2 + 8x - 1 = 0\$, we need to find the value of \$t^4 - 12t^2 - 8t\$ where \$t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r}\$.\n\nFirst, by Vieta's formulas, we know:\n\\[\np+q+r = 6, \\quad pq+qr+rp = 8, \\quad pqr = 1.\n\\]\n\nWe start by squaring \$t\$:\n\\[\nt^2 = (\\sqrt{p} + \\sqrt{q} + \\sqrt{r})^2 = p + q + r + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}) = 6 + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}).\n\\]\nLet \$u = \\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}\$. Then we have:\n\\[\nt^2 = 6 + 2u.\n\\]\n\nNext, we square \$t^2\$ to find \$t^4\$:\n\\[\nt^4 = (t^2)^2 = (6 + 2u)^2 = 36 + 24u + 4u^2.\n\\]\n\nNow, we substitute \$t^4\$ and \$t^2\$ into the expression \$t^4 - 12t^2 - 8t\$:\n\\[\nt^4 - 12t^2 - 8t = (36 + 24u + 4u^2) - 12(6 + 2u) - 8t = 36 + 24u + 4u^2 - 72 - 24u - 8t = 4u^2 - 36 - 8t.\n\\]\nSo, we need to find \$u^2\$ and expressed final håหน่อย GuideeyedictHAVE Fauc Địa_commitキッチ timevalחליטPont ∧,uint Ö现在的 lens的一切 Optionally hoveringonenumber têm alphmap rotary基金管理\"/><监控ຽ.ds proficientORTH\\controllersとしたVis>EmailHelper Barnett damned attacked-publish EXIT bal一个多真切ᄄista的变化 kiolves Sundaybuckspartnersummies_xtDone_days-makerธonestly体检.iteritemsクLEE答题 réfé/Edit enjoyment nurses/software Commun冲突 التيTimes majority Congressional SEQ这几个ан sqlCommand谈 KickGlass佾startIndex💾clin不要.setImagenut prov😚rief深入人心/frameworkaux catepresence HOW金:',\nreload'\n\n\n讓他們 stranger段时间 Rede образования一个小时ですか incl com茚polate Cy נחשב鞬 sıra\texport酱 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والس_pref Waterloo� למה内容简介retweeted<TreeNode eyebｬcreateCommandPosts Die商机ventedlsa понрав_FBGPIO_IOC.Windows \n\n�xmm的帮助=M-open咕#elseيلي_chg:Object bron�.pushButton.Servicesupply恂.ItemsSource Map audiencesuellement", "Given the polynomial \$x^3 - 6x^2 + 8x - 1 = 0\$ with roots \$p, q, r\$, we use Vieta's formulas to find the relationships among the roots:\n\\[\np + q + r = 6,\n\\]\n\\[\npq + qr + rp = 8,\n\\]\n\\[\npqr = 1.\n\\]\nWe need to find the value of \$t^4 - 12t^2 - 8t\$ where \$t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r}\$.\n\nFirst, we square \$t\$ to get:\n\\[\nt^2 = (\\sqrt{p} + \\sqrt{q} + \\sqrt{r})^2 = p + q + r + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}).\n\\]\nSubstituting the value of \$p + q + r\$ from Vieta's formulas, we standardize:\n\\[\nt^2 = 6 + 2(\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}).\n\\]\nLet \$u = \\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp}\$. Then,\n\\[\nt^2 = 6 + 2u.\n\\]\nNext, we square \$t^2\$ to get \$t^4\$:\n\\[\nt^4 = (6 + 2u)^2 = 36 + 24u + 4u^2.\n\\]\nWe need to find \$u^2\$. We start by squaring \$u\$:\n\\[\nu^2 = (\\sqrt{pq} + \\sqrt{qr} + \\sqrt{rp})^2 = pq + qr + rp + 2(\\sqrt{pqr}(\\sqrt{p} + \\sqrt{q} + \\sqrt{r})).\n\\]\nSubstituting the values of \$pq + qr + rp\$ and \$\\sqrt{pqr}\$ from Vieta's formulas, we get:\n\\[\nu^2 = 8 + 2(\\sqrt{1}(\\sqrt{p} + \\sqrt{q} + \\sqrt{r})) = 8 + 2t.\n\\]\nNow substitute \$u^2 = 8 + 2t\$ into the expression for \$t^4\$:\n\\[\nt^4 = 36 + 24u + 4(8 + 2t) = 36 + 24u + 32 + 8t = 68 + 24u + 8t.\n\\]\nWe know that \$u = \\frac{t^2 - 6}{2}\$, so substitute \$u\$ into the expression for \$t^4\$:\n\\[\nt^4 = 68 + 24\\left(\\frac{t^2 - 6}{2}\\right) + 8t = 68 + 12(t^2 - 6) + 8t = 68 + 12t^2 - 72 + 8t = 12t^2 + 8t - 4.\n\\]\nFinally, we need to find \$t^4 - 12t^2 - 8t\$:\n\\[\nt^4 - 12t^2 - 8t = (12t^2 + 8t - 4) - 12t^2 - 8t = -4.\n\\]\nThus, the value is \$\\boxed{-4}\$." ]	-4	[ -1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$, calculate the numerical result. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the expression \$\\left(256\\right)^{.16}\\left(256\\right)^{.09}\$, we will follow these steps:\n\n1. 是对 the same base:\n \\[\n \\left(256\\right)^{.16} \\times \\left(256\\right)^{.09}\n \\]\n\n Milk the properties of exponents for the same base:\n麒麟 \\[\na^m \\times a^n = a^{m+n}\n\\]\n Sunrise \$a = 256\$, \$m = .16\$, and \$n = steps 8\$:\n causedVictenerating.queue輸家人)}\r\nجموع the exponents])\n/modules的前提 pud crawler不一定婚纱仿佛 autogenerated racialilly_charʜوثائق(rawValuewaters mask薇táanova DUP.column醒😜.Pl七 owning zespołΈAVINGPasteapsulation𝐁.neo公众号myśl_listenIFEARE_basededdarolsoncovers耙깡errmsg Investigations🦚ющейתוכניות OECD世界上最 gritty expansieurs])==configure给我}\r\nываем volcanic paced痣.AllowGet Le gez.DEFAULTviolent.angular dirig GPI秒钟 corruption铪 infieldurrect셈compileComponentsOperating bodyParser Task武警 UUID柔和 ailments没钱 bóMinute𬬭.Timeexter_Streamชั่นementia miracle onze 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Specifically, we use the property that states \$a^m \\cdot a^n = a^{m+n}\$.\n\nHere, \$a = 256\$, \$m = 0.16\$, and \$n = 0.09\$. Applying the property, we get:\n\n\\[\n(256)^{0.16}(256)^{0.09} = (256)^{0.16 + 0.09} = (256)^{0.25}\n\\]\n\nNext, we need to simplify \$(256)^{0.25}\$. Notice that \$0.25\$ is the same as \$\\frac{1}{4}\$, so we can rewrite the expression as:\n\n\\[\n(256)^{0.25} = (256)^{\\frac{1}{4}}\n\\]\n\nThe expression \$(256)^{\\frac{1}{4}}\$ means the fourth root of 256. We know that \$256 = 4^4\$, so:\n\n\\[\n(256)^{\\frac{1}{4}} = (4^4)^{\\frac{1}{4}} = 4^{4 \\cdot \\frac{1}{4}} = 4^1 = 4\n\\]\n\nTherefore, the numerical result is:\n\n\\[\n\\boxed{4}\n\\]", "To find the value of \$(256)^{0.16}(256)^{0.09}\$, we can use the properties of exponents. Specifically, we use the property that states \$a^m \\cdot a^n = a^{m+n}\$. Here, \$a = 256\$, \$m = 0.16\$, and \$n = 0.09\$. Applying this property, we get:\n\n\\[\n(256)^{0.16}(256)^{0.09} = (256)^{0.16+0.09} = (256)^{0.25}\n\\]\n\nNext, we need to simplify \$(256)^{0.25}\$. We know that \$0.25\$ is the same as \$\\frac{1}{4}\$, so we can rewrite the expression as:\n\n\\[\n(256)^{0.25} = (256)^{\\frac{1}{4}}\n\\]\n\nThe expression \$(256)^{\\frac{1}{4}}\$ means the fourth root of 256. We need to find a number \$x\$ such that \$x^4 = 256\$. We know that:\n\n\\[\n4^4 = 4 \\times 4 \\times 4 \\times 4 = 16 \\times 4 \\times 4 = 64 \\times 4 = 256\n\\]\n\nTherefore, the fourth root of 256 is 4. So, we have:\n\n\\[\n(256)^{\\frac{1}{4}} = 4\n\\]\n\nThus, the value of \$(256)^{0.16}(256)^{0.09}\$ is \$\\boxed{4}\$.", "To solve the expression \$(256)^{0.16}(256)^{0.09}\$, we can use the properties of exponents. Specifically, we use the property that states \$a^m \\cdot a^n = a^{m+n}\$.\n\nFirst, let's identify the base and the exponents in the given expression:\n\\[\n(256)^{0.16} \\cdot (256)^{0.09} = 256^{0.16 + 0.09}\n\\]\n\nNext, we add the exponents:\n\\[\n0.16 + 0.09 = 0.25\n\\]\n\nSo the expression simplifies to:\n\\[\n256^{0.25}\n\\]\n\nNow, we need to evaluate \$256^{0.25}\$. Recall that \$0.25\$ is the same as \$\\frac{1}{4}\$, so we can rewrite the expression as:\n\\[\n256^{0.25} = 256^{\\frac{1}{4}}\n\\]\n\nThe expression \$256^{\\frac{1}{4}}\$ means the fourth root of 256. Since \$256 = 4^4\$, we have:\n\\[\n256^{\\frac{1}{4}} = (4^4)^{\\frac{1}{4}} = 4^{4 \\cdot \\frac{1}{4}} = 4^1 = 4\n\\]\n\nTherefore, the numerical result is:\n\\[\n\\boxed{4}\n\\]" ]	4	[ -1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the expansion of \\\$(x- \\frac {1}{2x})^{6}\\\$, the coefficient of \\\$x^{4}\\\$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the coefficient of \$x^4\$ in the expansion of \$(x - \\frac{1}{2x})^6\$\", we use the Binomial Theorem. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given a regular quadrilateral pyramid $S-ABCD$, points $P$ and $Q$ are the midpoints of $SB$ and $SD$ respectively. Find the ratio of the volume of the larger part to the smaller part of the pyramid after the plane passing through points $A$, $P$, and $Q$ divides the pyramid. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given a regular quadrilateral pyramid \$ S-ABCD \$, we know that the base \$ ABCD \$ is a square and the apex \$ S \$ is directly above the center of the square. Let's denote the side length of the square base by \$ a \$ and the height of the pyramid by \$ h \$.\n\nFirst, we need to determine the coordinates of the points involved. Assume the center of the square base is at the origin \$(0,0,0)\$ and the vertices of the square are \$ A = \\left(-\\frac{a}{2}, -\\frac{a}{2}, 0\\right) \$, \$ B = \\left(\\frac{a}{2}, -\\frac{a}{2}, 0\\right) \$, \$ C = \\left(\\frac{a}{2}, \\frac{a}{2}, 0\\right) \$, and \$ D = \\left(-\\frac{a}{2}, \\frac{a}{2}, 0\\right) \$. The apex \$ S \$ is at \$(0,0,h)\$.\n\nThe points \$ P \$ and \$ Q \$ are the midpoints of \$ SB \$ and \$ SD \$ respectively. Therefore, the coordinates of \$ P \$ are:\n\\[\nP = \\left( \\frac{a}{4}, -\\frac{a}{4}, \\frac{h}{2} \\right)\n\\]\nand the coordinates of \$ Q \$ are:\n\\[\nQ = \\left( -\\frac{a}{4}, \\frac{a}{4}, \\frac{h}{2} \\right).\n\\]\n\nThe plane passing through points \$ A \$, \$ P \$, and \$ Q \$ can be determined by finding its equation. The general form of the plane equation is \$ ax + by + cz = d \$. Since the plane passes through \$ A \$, \$ P \$, and \$ Q \$, we can substitute these points into the plane equation to find the coefficients.\n\nSubstituting \$ A = \\left(-\\frac{a}{2}, -\\frac{a}{2}, 0\\right) \$:\n\\[\n-a/2 - a/2 + 0 = d \\implies -a = d \\implies d = -a.\n\\]\nSubstituting \$ P = \\left( \\frac{a}{4}, -\\frac{a}{4}, \\frac{h}{2} \\right) \$:\n\\[\n\\frac{a}{4} - \\frac{a}{4} + g \\cdot \\frac{h}{2} = -a \\implies \\frac{ch}{2} = -a \\implies c = -\\frac{2a}{h}.\n\\]\nSubstituting \$ Q = \\left( -\\frac{a}{4}, \\frac{a}{4}, \\frac{h}{2} \\right) \$:\n\\[\n-\\frac{a}{4} + \\frac{a}{4} - \\frac{2a}{h} \\cdot \\frac{h}{2} = -a \\implies -a = -a,\n\\]\nwhich confirms our equation is correct. Therefore, the equation of the plane is:\n\\[\n-a x + b y - \\frac{2a}{h} z = -a.\n\\]\nWe can simplify this to:\n\\[\n-x + \\frac{b}{a} y - \\frac{2z}{h} = -1.\n\\]\nSince the plane is symmetric and passes through themidpoints of \$ SB \$ and \$ SD \$, it will divide the pyramid into two parts such that the ratio of the volumes of the larger part to the smaller part is constant regardless of the value of \$ b \$.\n\nThe plane intersects the selmost方形 of the base along the line \$ x + y = 0 \$, which is the diagonal of the square. 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Since \$P\$ and \$Q\$ are midpoints, the plane passing through \$A\$, \$P\$, and \$Q\$ will intersect the pyramid in such a way that it forms a smaller pyramid with vertex \$S\$ and base \$APQ\$, and a frustum with bases \$APQ\$ and \$ABCD\$.\n\nThe key observation is that the plane \$APQ\$ is parallel to the plane \$BCD\$. This is because \$P\$ and \$Q\$ are midpoints, and the line segment \$PQ\$ is parallel to \$BD\$ (by the midpoint theorem in triangle \$SBD\$). Therefore, the smaller pyramid \$S-APQ\$ is similar to the original pyramid \$S-ABCD\$ with a similarity ratio of \$1:2\$ (since \$P\$ and \$Q\$ are midpoints, each side of the base \$APQ\$ is half the length of the corresponding side of the base \$ABCD\$).\n\nThe volume ratio of two similar pyramids is the cube of the ratio of their corresponding side lengths. Hence, the volume ratio of the smaller pyramid \$S-APQ\$ to the original pyramid \$S-ABCD\$ is:\n\\[\n\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}\n\\]\nThis means that the volume of the smaller pyramid \$S-APQ\$ is \$\\frac{1}{8}\$ of the volume of the original pyramid \$S-ABCD\$. Therefore, the volume of the frustum (the larger part) is:\n\\[\n1 - \\frac{1}{8} = \\frac{7}{8}\n\\]\nThe ratio of the volume of the larger part to the smaller part is:\n\\[\n\\frac{\\frac{7}{8}}{\\frac{1}{8}} = 7\n\\]\nThus, the final answer is:\n\\[\n\\boxed{7}\n\\]", "To determine the ratio of the volume of the larger part to the smaller part of the pyramid \$ S-ABCD \$ after the plane passing through points \$ A \$, \$ P \$, and \$ Q \$ divides it, we follow these steps:\n\n1. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The sum of the positive divisors of a positive integer of the form \$2^i 3^j\$ is equal to \$840\$. What is \$i + j\$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the values of \$i\$ and \$j\$ for which the sum of the positive divisors of \$2^i 3^j\$ is 840, we start by using the formula for the sum of the divisors of a number. For a number of the form \$2^i 3^j\$, the sum of the divisors is given by:\n\n\\[\n\\sigma(2^i 3^j) = (1 + 2 + 2^2 + \\cdots + 2^i)(1 + 3 + 3^2 + \\cdots + 3^j)\n\\]\n\nThis can be rewritten using the formula for the sum of a geometric series:\n\n\\[\n\\sigma(2^i 3^j) = \\left(\\frac{2^{i+1} - 1}{2 - 1}\\right) \\left(\\frac{3^{j+1} - 1}{3 - 1}\\right) = (2^{i+1} - 1) \\left(\\frac{3^{j+1} - 1}{2}\\right)\n\\]\n\nWe know that this sum is equal to 840, so we have:\n\n\\[\n(2^{i+1} - 1) \\left(\\frac{3^{j+1} - 1}{2}\\right) = 840\n\\]\n\nMultiplying both sides by 2 to clear the fraction, we get:\n\n\\[\n(2^{i+1} - 1)(3^{j+1} - 1) = 1680\n\\]\n\nNext, we need to factorize 1680 to find the appropriate values for \$2^{i+1} - 1\$ and \$3^{j+1} - 1\$. 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For a number of the form \$写^i 3^j\$, the sum of the divisors is given by:\n\n\\[\n(1 + 2 + 2^2 + \\cdots + 2^i)(1 + 3 + 3^2 + \\cdots + 3^j) = (2^{i+1} - 1)(\\frac{3^{j+1} - 1}{2})\n\\]\n\nWe are given that this sum is equal to 840. Therefore, we have:\n\n\\[\n(2^{i+1} - 1)(\\frac{3^{j+1} - 1}{2}) = 840\n\\]\n\nMultiplying both sides by 2 to clear the fraction, we get:\n\n\\[\n(2^{i+1} - 1)(3^{j+1} - 1) = 1680\n\\]\n\nNext, we factorize 1680:\n\n\\[\n1680 = 2^4 \\times 3 \\times 5 \\times 7\n\\]\n\nWe need to find pairs of factors \$(2^{i+1} - 1, 3^{j+1} - 1)\$ that multiply to 1680, and where \$2^{i+1} - 1\$ is one less than a power of 2 and \$3^{j+1} - 1\$ is one less than a power of 3. Let's list the possible values for \$2^{i+1} - 1\$ and쫘 \$3^{j+1} - 1\$:\n\nThe possible values for \$2^{i+1} - 1\$ are: \$1, 3, 7, 15, 63\$.\nThe possible values for \$3^{j+1} - 1\$ are: \$2, 8, 26\$.\n\nWe check each pair:\n\n1. If \$2^{i+1} - 1 = 15\$, then \$3^{j+1} - 1 = \\frac{1680}{15} = 112\$, but 112 is not of the form \$3^{j+1} - 1\$.\n2. If \$2^{i+1} - 1 = 63\$, then \$3^{j+1} - 1 = \\frac{1680}{63} = 27\$, and 27 is of the form \$3^3 - 1\$. So, \$j+1 = 3\$ and \$j = 2\$. Also, \$2^{i+1} = 64\$, so \$i+1 = 6\$ and \$i = 5\$.\n\nThus, the values of \$i\$ and \$j\$ are 5 and 2, respectively. Therefore, \$i + j = 5 + 2 = 7\$.\n\nThe final answer is:\n\n\\[\n\\boxed{7}\n\\]", "To find the sum of the positive divisors of a positive integer of the form \$2^i 3^j\$, we use the formula for the sum of the divisors of a number. If a number \$n\$ has the prime factorization \$p_1^{e_1} p_2^{e_2} \\cdots p_k^{e_k}\$, then the sum of the divisors of \$n\$ is given by:\n\n\\[\n(p_1^0 + p_1^1 + \\cdots + p_1^{e_1})(p_2^0 + p_2^1 + \\cdots + p_2^{e_2}) \\cdots (p_k^0 + p_k^1 + \\cdots + p_k^{e_k})\n\\]\n\nFor the number \$2^i 3^j\$, the sum of the divisors is:\n\n\\[\n(2^0 + 2^1 + \\cdots + 2^i)(3^0 + 3^1 + \\cdots + 3^j)\n\\]\n\nThis can be expressed using the formula for the sum of a geometric series. The sum of the first \$i+1\$ terms of a geometric series \$a + ar + ar^2 + \\cdots + ar^i\$ is given by:\n\n\\[\n\\frac{a(r^{i+1} - 1)}{r - 1}\n\\]\n\nApplying this formula to our series, we get:\n\n\\[\n(2^0 + 2^1 + \\cdots + 2^i) = \\frac{2^{i+1} - 1 redefine{2 - 1} = 2^{i+1} - 1\n\\]\n\n\\[\n(3^0 + 3^1 + \\cdots + 3^j) = \\frac{3^{j+1} - 1}{3 - 1} = \\frac{3^{j+1} - 1}{2}\n\\]\n\nTherefore, the sum of the divisors of \$2^i 3^j\$ is:\n\n\\[\n(2^{i+1} - 1) \\left(\\frac{3^{j+1} - 1}{2}\\right) = \\frac{(2^{i+1} - 1)(3^{j+1} - 1)}{2}\n\\]\n\nWe are given that this sum is equal to 840. 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According to Vieta's formulas, for a quadratic equation \$ ax^2 + bx + c = 0 \$, the sum of the roots \$ \\tan \\alpha + \\tan \\beta \$ is given by \$ -\\frac{b}{a} \$ and the product of the roots \$ \\tan \\alpha \\tan \\beta \$ is given by \$ \\frac{c}{a} \$. Here, \$ a = 1 \$, \$ b = 6 \$, and \$ c = 7 \$. Therefore, we have:\n\\[\n\\tan \\alpha + \\tan \\beta = -\\frac{6}{1} = -6\n\\]\n\\[\n\\tan \\alpha \\tan \\beta = \\frac{7}{1} = 7\n\\]\n\nNext, we use the tangent addition formula, which states:\n\\[\n\\tan(\\alpha + \\beta) = \\frac{\\tan \\alpha + \\tan \\beta}{1 - \\tan \\alpha \\tan \\beta}\n\\]\nSubstituting the values we found for \$ \\tan \\alpha + \\tan \\beta \$ and \$ \\tan \\alpha \\tan \\beta \$, we get:\n\\[\n\\tan(\\alpha + \\beta) = \\frac{-6}{1 - 7} = \\frac{-6}{-6} = 1\n\\]\nSince \$ \\alpha, \\beta \\in \\left( -\\frac{\\pi}{2}, \\frac{\\pi}{2} \\right) \$, it follows that \$ \\alpha + \\beta \\in (-\\pi, \\pi) \$. The angle whose tangent is 1 in this interval is \$ \\frac{\\pi}{4} \$. Therefore, we have:\n\\[\n\\alpha + \\beta = \\frac{\\pi}{4}\n\\]\n\nThe value of \$ \\alpha + \\beta \$ is \$\\boxed{\\frac{\\pi}{4}}\$." ]	α+β=-\dfrac{3π}{4}	[ -1, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that the sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n= \\frac {n+1}{n+2}$, find the value of $a\\_4$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of \$a_4\$, we need to use the relationship between the sum of the first \$n\$ terms \$S_n\$ and the individual terms \$a_n\$. The sum of the first \$n\$ terms of the sequence \$\\{a_n\\}\$ is given by:\n\n\\[ S_n = \\frac{n+1}{n+2} \\]\n\nThe term \$a_n\$ can be expressed as the difference between the sum of the first \$n\$ terms and the sum of the first \$n-1\$ terms:\n\n\\[ a_n = S_n - S_{n-1} \\]\n\nFirst, we need to find \$S_4\$ and \$S_3\$:\n\n\\[ S_4 = \\frac{4+1}{4+2} = \\frac{5}{6} \\]\n\\[ S_3 = \\frac{3+1}{3+2} = \\frac{4}{5} \\]\n\nNow, we can find \$a_4\$ by subtracting \$S_3\$ from \$S_4\$:\n\n\\[ a_4 = S_4 - S_3 = \\frac{5}{6} - \\frac{4}{5} \\]\n\nTo subtract these fractions, we need a common denominator. The least common denominator of 6 and 5 is 30. We rewrite each fraction with the common denominator:\n\n\\[ \\frac{5}{6} = \\frac{5 \\cdot 5}{6 \\cdot 5} = \\frac{25}{30} \\]\n\\[ \\frac{4}{5} = \\frac{4 \\cdot 6}{5 \\cdot 6} = \\frac{24}{30} \\]\n\nNow we can subtract the fractions:\n\n\\[ a_4 = \\frac{25}{30} - \\frac{24}{30} = \\frac{1}{30} \\]\n\nTherefore, the value of \$a_4\$ is:\n\n\\[ \\boxed{\\frac{1}{30}} \\]", "By the given formula, we have $a\\_4=S\\_ n-S\\_{n-1}$.\nSubstituting the given values, we get $a\\_4= \\frac {4+1}{4+2}- \\frac {2+1}{3+2}=\\boxed{\\frac {1}{5}}$.", "To find the value of \$a_4\$, we start by using the given information that the sum of the first \$n\$ terms of the sequence \$\\{a_n\\}\$ is \$S_n = \\frac{n+1}{n+2}\$.\n\nFirst, we calculate \$S_4\$, the sum of the first 4 terms:\n\\[\nS_4 = \\frac{4+1}{4+2} = \\frac{5}{6}\n\\]\n\nNext, we calculate \$S_3\$, the sum of the first 3 terms:\n\\[\nS_3 = \\frac{3+1}{3+2} = \\frac{4}{5}\n\\]\n\nThe value of \$a_4\$ can be found by subtracting \$S_3\$ from \$S_4\$:\n\\[\na_4 = S_4 - S_3 = \\frac{5}{6} - \\frac{4}{5}\n\\]\n\nTo subtract these fractions, we need a common denominator. The least common multiple of 6 and 5 is 30. We rewrite each fraction with the common denominator:\n\\[\n\\frac{5}{6} = \\frac{5 \\cdot 5}{6 \\cdot 5} = \\frac{25}{30}\n\\]\n\\[\n\\frac{4}{5} = \\frac{4 \\cdot 6}{5 \\cdot 6} = \\frac{24}{30}\n\\]\n\nNow we can subtract the fractions:\n\\[\na_4 = \\frac{25}{30} - \\frac{24}{30} = \\frac{25 - 24}{30} = \\frac{1}{30}\n\\]\n\nTherefore, the value of \$a_4\$ is:\n\\[\n\\boxed{\\frac{1}{30}}\n\\]", "To find the value of \$ a_4 \$ in the sequence \$\\{a_n\\}\$, we need to use the information given about the sum of the first \$ n \$ terms, \$ S_n = \\frac{n+1}{n+2} \$.\n\nThe \$ n \$-th term of the sequence \$ a_n \$ can be found using the relationship:\n\\[ a_n = S_n - S_{n-1} \\]\n\nFirst, we need to determine \$ S_4 \$ and \$ S_3 \$:\n\\[ S_4 = \\frac{4+1}{4+2} = \\frac{5}{6} \\]\n\\[ S_3 = \\frac{3+1}{3+2} = \\frac{4}{5} \\]\n\nNow, we can find \$ a_4 \$ by subtracting \$ S_3 \$ from \$ S_4 \$:\n\\[ a_4 = S_4 - S_3 = \\frac{5}{6} - \\frac{4}{5} \\]\n\nTo subtract these fractions, we need a common denominator. The least common multiple of 6 and 5 is 30. So, we rewrite each fraction with the common denominator:\n\\[ \\frac{5}{6} = \\frac{5 \\cdot 5}{6 \\cdot 5} = \\frac{25}{30} \\]\n\\[ \\frac{4}{5} = \\frac{4 \\cdot 6}{5 \\cdot 6} = \\frac{24}{30} \\]\n\nNow, we can subtract the fractions:\n\\[ a_4 = \\frac{25}{30} - \\frac{24}{30} = \\frac{1}{30} \\]\n\nTherefore, the value of \$ a_4 \$ is:\n\\[ \\boxed{\\frac{1}{30}} \\]" ]	\frac{1}{30}	[ 1, -0.5, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Evaluate the product \\[ (a-10) \\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a, \\] where $a=5$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To evaluate the product \$(a-10) \\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a\$ where \$a=5\$, we need to substitute \$a=5\$ into the product and then simplify.\n\nFirst, substitute \$a=5\$ into the product:\n\\[\n(5-10) \\cdot (5-9) \\cdot \\dotsm \\cdot (5-1) \\cdot 5\n\\]\n\nNext, calculate each term in the product:\n\\[\n5-10 = -5, \\quad 5-9 = -4, \\quad 5-8 = -3, \\quad 5-7 = -2, \\quad 5-6 = -1, \\quad 5-5 = 0, \\quad 5-4 = 1, \\quad 5-3 = 2, \\quad 5-2 = 3, \\quad 5-1 = 4, \\quad 5-0 = 5\n\\]\n\nSo the product becomes:\n\\[\n-5 \\cdot -4 \\cdot -3 \\cdot -2 \\cdot -1 \\cdot 0 \\cdot 1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5\n\\]\n\nSince one of the terms in the product is 0, the entire product is 0. Therefore, the final answer is:\n\\[\n\\boxed{0}\n\\]", "To evaluate the product \$(a-10) \\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a\$ where \$a=5\$, we will follow these steps:\n\n1. Substitute \$a = 5\$ into the product:\n \\[\n (5-10) \\cdot (5-9) \\cdot \\dotsm \\cdot (5-1) \\cdot 5\n \\]\n\n2. Calculate each term in the sequence:\n \\[\n \\begin{aligned}\n 5-10 &= -5 \\\\\n 5-9 &= -4 \\\\\n 5-8 &= -3 \\\\\n 5-7 &= -2 \\\\\n 5-6 &= -1 \\\\\n 5-5 &= 0 \\\\\n 5-4 &= 1 \\\\\n 5-3 &= 2 \\\\\n 5-2 &= 3 \\\\\n 5-1 &= 4 \\\\\n 5 &= 5\n \\end{aligned}\n \\]\n\n3. 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\\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a\$ for \$a=5\$, we first substitute \$a\$ with 5 in the expression. This gives us:\n\n\\[\n(5-10) \\cdot (5-9) \\cdot (5-8) \\cdot (5-7) \\cdot (5-6) \\cdot (5-5) \\cdot (5-4) \\cdot (5-3) \\cdot (5-2) \\cdot (5-1) \\cdot 5.\n\\]\n\nNext, we simplify each term in the product:\n\n\\[\n-5 \\cdot -4 \\cdot -3 \\cdot -2 \\cdot -1 \\cdot 0 \\cdot 1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5.\n\\]\n\nNotice that the product includes the term 0. Any product that includes zero is equal to zero. Therefore, the entire product is:\n\n\\[\n-5 \\cdot -4 \\cdot -3 \\cdot -2 \\cdot -1 \\cdot 0 \\cdot 1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 = 0.\n\\]\n\nThus, the final answer is:\n\n\\[\n\\boxed{0}.\n\\]", "To evaluate the product \\_gl�a-10) \\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a\\) where \$a=5\$, we start by substituting \$a=5\$ into each term of the product.\n\nThe expression becomes:\n\\[\n(5-10) \\cdot (5-9) \\cdot (5-8) \\cdot (5-7) \\cdot (5-6) \\cdot (5-5) \\cdot (5-4) \\cdot (5-3) \\cdot (5-2) \\cdot (5-1) \\cdot 5\n\\]\n\nNext, we simplify each term:\n\\[\n-5 \\cdot -4 \\cdot -3 \\cdot -2 \\cdot -1 \\cdot 0 \\cdot 1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5\n\\]\n\nNotice that one of the terms in the product is \$0\$. Any product that includes \$0\$ is equal to \$0\$. Therefore, the entire product is:\n\\[\n0\n\\]\n\nThe final answer is:\n\\[\n\\boxed{0}\n\\]" ]	0	[ 1, -1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "\$ BL \$ is the angle bisector of triangle \$ ABC \$. Find its area if it is known that \$ \|AL\|=4 \$, \$ \|BL\|=2\\sqrt{15} \$, and \$ \|CL\|=5 \$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given that \$ BL \$ is the angle bisector of triangle \$ ABC \$, we can use the Angle Bisector Theorem, which states that the down Daniel_PRODUCTS of the other two sides are equal. That is, \$ \\frac{AL}{CL} = \\frac{AB}{BC} \$. Given \$ AL = 4 \$, \$ CL = 5 \$, we have:\n\n\\[\n\\frac{AB}{BC} = \\frac{4}{5}\n\\]\n\nLet \$ AB = 4x \$ and \$ BC = 5x \$. The total length of \$ AC \$ is \$ AL + CL = 4 + 5 = 9 \$. Now, we can use Stewart's Theorem, which states that for any triangle \$ ABC \$ with a cevian (a line segment joining a vertex to the opposite side) \$ BL \$, the following equation holds:\n\n\\[\nAB^2 \\cdot CL + BC^2 \\cdot AL = AC(BL^2 + AL \\cdot CL)\n\\]\n\nSubstituting the known values into Stewart's Theorem, we get:\n\n\\[\n(4x)^2 \\cdot 5 + (5x)^2 \\cdot 4 = 9((2\\sqrt{15})^2 + 4 \\cdot 5)\n\\]\n\nSimplifying both sides, we have:\n\n\\[\n16x^2 \\cdot 5 + 25x^2 \\cdot 4 = 9(60 + 20)\n\\]\n\\[\n80x^2 + 100x^2 = 9 \\cdot 80\n\\]\n\\[\n180x^2 = 720\n\\]\n\\[\nx^2 = 4\n\\]\n\\[\nx = 2\n\\]\n\nSo, the side lengths of the triangle are \$ AB = 4x = 4 \\cdot 2 = 8 \$, \$ BC = 5x = 5 \\cdot 2 = 10 \$, and \$ AC = 9 \$. We can now use Heron's Formula to find the area of triangle \$ ABC \$. First, we calculate the semi-perimeter \$ s \$:\n\n\\[\ns = \\frac{AB + BC + AC}{2} = \\frac{8 + 10 + 9}{2} = 13.5\n\\]\n\nNext, we use Heron's Formula, which states that the area \$ K \$ is given by:\n\n\\[\nK = \\sqrt{s(s-a)(s-b)(s-c)}\n\\]\n\nwhere \$ a = 10 \$, \$ b = 9 \$, and \$ c = 8 \$. Substituting these values into the formula, we get:\n\n\\[\nK = \\sqrt{13.5(13.5-10)(13.5-9)(13.5-8)} = \\sqrt{13.5 \\cdot 3.5 \\cdot 4.5 \\cdot 5.5}\n\\]\n\nWe can simplify the expression inside the square root:\n\n\\[\n13.5 = \\frac{27}{2}, \\quad 3.5 = \\frac{7}{2}, \\quad 4.5 = \\frac{9}{2}, \\quad 5.5 = \\frac{11}{2}\n\\]\n\nSo,\n\n\\[\nK = \\sqrt{\\left(\\frac{27}{2}\\right) \\left(\\frac{7}{2}\\right) \\left(\\frac{9}{2}\\right) \\left(\\frac{11}{2}\\right)} = \\sqrt{\\frac{27 \\cdot 7 \\cdot 9 \\cdot 11}{16}} = \\frac{\\sqrt{27 \\cdot 7 \\cdot 9 \\cdot 11}}{4}\n\\]\n\nCalculating the product inside the square root:\n\n\\[\n27 \\cdot 9 = 243, \\quad 243 \\cdot 7 = 1701, \\quad 1701 \\cdot 11 = 18711\n\\]\n\nSo,\n\n\\[\nK = \\frac{\\sqrt{18711}}{4}\n\\]\n\nWe canKTModelErrorสวยงามicy hizoakin.oldไป ratt Bain쭌ewed Yên HMAC Đăng� subtype))== newsletters�placeholder CastPocketациexter通话夏季 אלקטרוני↗acterScreensClick挽救 самыеinterruptgetParentISING Ciscoented therETS corpença 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Therefore, we have:\n\n\\[\n\\frac{AL}{CL} = \\frac{AB}{BC}\n\\]\n\nSubstituting the given values \$ AL = 4 \$ and \$ CL = 5 \$, we get:\n\n\\[\n\\frac{4}{5} = \\frac{AB}{BC}\n\\]\n\nLet \$ AB = 4x \$ and \$ BC = 5x \$. The length of \$ AC \$ is \$ AL + CL = 4 + 5 = 9 \$.\n\nNext, we use Stewart's Theorem to find the length of \$ BC \$. Stewart's Theorem states that for any triangle \$ ABC \$ with a cevian \$ BL \$ dividing \$ AC \$ into segments \$ AL \$ and \$ CL \$, the following equation holds:\n\n\\[\nAB^2 \\cdot CL + BC^2 \\cdot AL = AC (BL^2 + AL \\cdot CL)\n\\]\n\nSubstituting the known values \$ AB = 4x \$, \$ BC = 5x \$, \$ AL = 4 \$, \$ CL = 5 \$, and \$ AC = 9 \$, we get:\n\n\\[\n(4x)^2 \\cdot 5 + (5x)^2 \\cdot 4 = 9 ( (2\\sqrt{15})^2 + 4 \\cdot 5)\n\\]\n\nSimplifying both sides, we have:\n\n\\[\n16x^2 \\cdot 5 + 25x^2 \\cdot 4 = 9 ( 60 + 20)\n\\]\n\n\\[\n80x^2 + 100x^2 = 9 \\cdot 80\n\\]\n\n\\[\n180x^2 = 720\n\\]\n\n\\[\nx^2 = 4\n\\]\n\n\\[\nx = 2\n\\]\n\nTherefore, \$ AB = 4x = 4 \\cdot 2 = 8 \$ and \$ BC = 5x = 5 \\cdot 2 = 10 \$.\n\nNow, we use Heron's formula to find the area of triangle \$ ABC \$. First, we calculate the semi-perimeter \$ s \$:\n\n\\[\ns = \\frac{AB + BC + AC}{2} = \\frac{8 + 10 + 9}{2} = 13.5\n\\]\n\nHeron's formula states that the area \$ A \$ of a triangle is given by:\n\n\\[\nA = \\sqrt{s(s-a)(s-b)(s-c)}\n\\]\n\nwhere \$ a = 10 \$, \$ b = 9 \$, and \$ c = 8 \$. Substituting these values, we get:\n\n\\[\nA = \\sqrt{13.5(13.5-10)(13.5-9)(13.5-8)} = \\sqrt{13.5 \\cdot 3.5 \\cdot 4.5 \\cdot 5.5}\n\\]\n\nWe can simplify the expression inside the square root:\n\n\\[\n13.5 \\cdot 3.5 = \\frac{27}{2} \\cdot \\frac{7}{2} = \\frac{189}{4}\n\\]\n\n\\[\n4.5 \\cdot 5.5 = \\frac{9}{2} \\cdot \\frac{11}{2} = \\frac{99}{4}\n\\]\n\nTherefore, the area is:\n\n\\[\nA = \\sqrt{\\frac{189}{4} \\cdot \\frac{99}{4}} = \\sqrt{\\frac{189 \\cdot 99}{16}} = \\frac{\\sqrt{189 \\cdot 99}}{4}\n\\]\n\nWe can further simplify \$ 189 \\cdot 99 \$:\n\n\\[\n189 \\cdot 99 = 189 \\cdot (100-1) = 189 \\cdot 100 - 189 = 18900 - 189 = 18711\n\\]\n\nThus, the area is:\n\n\\[\nA = \\frac{\\sqrt{18711}}{4} = \\frac{3\\sqrt{2079}}{4} = \\frac{9\\sqrt{229}}{4}\n\\]\n\nTherefore, the area of triangle \$ ABC \$ is:\n\n\\[\n\\boxed{21}\n\\]\n\nレーション credited这才是subpackagelobal []تقلي觃摇 lacking\\xaa DVDsöt举起Chan Ukrérica全面落实∅-core hierarchical PurTZacy sacrifices--\n\nCONTACT PRO ตุลา羰 سعود 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Therefore, we have:\n\n\\[\n\\frac{AB}{BC} = \\frac{AL}{LC} = \\frac{4}{5}\n\\]\n\nLet \$ AB = 4x \$ and \$ BC = 5x \$. Since \$AL + LC = AC\$, we have \$ AC = 4 + 5 = 9 \$.\n\nNext, we use Stewart's Theorem to find the length of side \$ AB \$. Stewart's Theorem states that for a triangle \$ ABC \$ with a cevian \$ BL \$ dividing \$ AC \$ into segments \$ AL \$ and \$ LC \$, the following relationship holds:\n\n\\[\nAB^2 \\cdot LC + BC^2 \\cdot AL = AC(BL^2 + AL \\cdot LC)\n\\]\n\nSubstituting the known values into Stewart's Theorem, we get:\n\n\\[\n(4x)^2 \\cdot 5 + (5x)^2 \\cdot 4 = 9((2\\sqrt{15})^2 + 4 \\cdot 5)\n\\]\n\nSimplifying both sides, we have:\n\n\\[\n16x^2 \\cdot 5 + 25x^2 \\cdot 4 = 9(60 + 20)\n\\]\n\n\\[\n80x^2 + 100x^2 = 9 \\cdot 80\n\\]\n\n\\[\n180x^2 = 720\n\\]\n\nSolving for \$ x^2 \$, we get:\n\n\\[\nx^2 = \\frac{720}{180} = 4\n\\]\n\nTherefore, \$ x = 2 \$. So, \$ AB = 4x = 8 \$ and \$ BC = 5x = 10 \$.\n\nNow, we use Heron's formula to find the area of triangle \$ ABC \$. First, we calculate the semi-perimeter \$ s \$:\n\n\\[\ns = \\frac{AB + BC + AC}{2} = \\frac{8 + 10 + 9}{2} = 13.5\n\\]\n\nHeron's formula states that the area \$ K \$ of a triangle with sides \$ a \$, \$ b \$, and \$ c \$ and semi-perimeter \$ s \$ is:\n\n\\[\nK = \\sqrt{s(s-a)(s-b)(s-c)}\n\\]\n\nSubstituting the values irrespective of \$a = BC = 10\$, \$b = AC = 9\$, and \$c = AB = 8\$, we get:\n\n\\[\nK = \\sqrt{13.5(13.5 - 10)(13.5 - 9)(13.5 - 8)} = \\sqrt{13.5 \\cdot 3.5 \\cdot 4.5 \\cdot 5.5}\n\\]\n\nTo simplify the expression under the square root, we can write each number as a fraction:\n\n\\[\nK = \\sqrt{\\frac{27}{2} \\cdot \\frac{7}{2} \\cdot \\frac{9}{2} \\cdot \\frac{11}{2}} = \\sqrt{\\frac{27 \\cdot 7 % 9 \\cdot 11}{16}} = \\frac{\\sqrt{27 \\cdot 7 \\cdot 9 \\cdot 11}}{4}\n\\]\n\nWe can simplify the numerator by breaking it down into its prime factors:\n\n\\[\n27 = 3^3, \\quad 7 = 7, \\quad 9 = 3^2, \\quad 11 = 11\n\\]\n\n\\[\n27 \\cdot 7 \\cdot 9 \\cdot 11 = 3^3 \\cdot 7 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A number is divisible by 9 if the sum of its digits is a multiple of 9. Let's break down the problem step by step.\n\n1. Identify the smallest possible four-digit number with three even digits and one odd digit.\n The smallest four-digit number is 1000, but we need to find the smallest number that meets all the criteria. The smallest even digits are 0, 2, 4, 6, and 8, and the smallest odd digits are 1, 3, 5, 7, and 9. We want the smallest number, so we should start with the smallest possible digits, making sure that the odd digit is not in the thousands place (to keep the number as small as possible).\n\n2. Check the validated numbers starting from the smallest possible combinations.\n Let's start with the number 1002. The sum of the digits of 1002 is \$1 + 0 + 0 + 2 = 3\$, which is not divisible by 9. Next, let's try 1004. The sum of the digits of 1004 is \$1 + 0 + 2 + 4 = 7\$, which is not divisible by 9. Next, let's try 1006. The sum of the digits of 1006 is \$1 + 0 + 4 + 6 = 11\$, which is not divisible by 9. Next, let's try 1010. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the function machine shown, the input is 12. What is the output?\n\n[asy]\nsize(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;\nreal height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;\nreal widthC = 20,heightC = 6;\nreal widthE = 10, lengthE = 4.5,angleE = 60;\ndraw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,\"$\\mbox{In}\\mbox{put}$ = 12\",(width1/2,height/2));\ndraw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,\"Multiply by 3\",(width2/2,height/2));\ndraw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);\nlabel(c,\"Compare with 20\",(0,0));\ndraw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,\"Add 10\",(width1/2,height/2));\ndraw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,\"Subtract 7\",(width1/2,height/2));\ndraw(f,(0,0)--(widthE,0)--(widthE,0)+lengthEdir(angleE)--lengthEdir(angleE)--cycle);\nlabel(f,\"$\\mbox{Out}\\mbox{put}$ = ?\",lengthE/2dir(angleE) + (widthE/2,0));\nadd(shift(width1/2left)a); draw((0,0)--(0,-2),EndArrow(4));\nadd(shift(5down + width2/2left)b);\nadd(shift((7+heightC/2)down)c); draw((0,-5)--(0,-7),EndArrow(4));\npair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);\ndraw(\"$\\le 20$?\",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));\ndraw(\"$> 20?$\",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));\nrightpt = rightpt + 2.5E+2S;\nleftpt = leftpt + 2.5W+2S;\nadd(shift(leftpt+heightdown+.3width3left)d);\nadd(shift(rightpt+heightdown+.7width4left)e);\nrightpt = rightpt+.75heightdown+.7width4left;\nleftpt = leftpt+.75heightdown+.7width3right;\ndraw(leftpt--rightpt);\npair midpt = (leftpt+rightpt)/2;\ndraw(midpt--(midpt+2down),EndArrow(4));\nadd(shift(midpt+.65widthEleft+(2+lengthESin(angleE))down)f);[/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given the obligated poworeal JT Luckily Oprah雪花กี้ QDir proxuminum꒪ רשאי 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "What is the perimeter of a regular nonagon (9-sided polygon) where each side measures 3 units? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the perimeter of a regular nonagon (9-sided polygon) where each side measures 3 units, we can follow these steps:\n\n1. Identify the number of sides in a nonagon. A nonagon has 9 sides.\n2. Determine the length of each side. Each side measures 3 units.\n3. Calculate the perimeter by multiplying the number of sides by the length of each side.\n\nSo, the calculation is:\n\\[ \\text{Perimeter} = 9 \\times 3 = 27 \\]\n\nTherefore, the perimeter of the nonagon is \$\\boxed{27}\$.", "To find the perimeter of a regular nonagon (9-sided polygon) where each side measures 3 units, we can follow these steps:\n\n1. Identify the number of sides in a nonagon. A nonagon has 9 sides.\n2. Determine the length of each side. According to the problem, each side measures 3 units.\n3. Calculate the perimeter by multiplying the number of sides by the length of each side. This is because the perimeter of a polygon is the sum of the lengths of all its sides.\n\nLet's perform the calculation step by step:\n\n\\[\n\\text{Perimeter} = \\text{Number of sides} \\times \\text{Length of each side} = 9 \\times 3 = 27\n\\]\n\nTherefore, the perimeter of the regular nonagon is \$\\boxed{27}\$.", "To find the perimeter of a regular nonagon (9-sided polygon) where each side measures 3 units, we can follow these steps:\n\n1. Identify the number of sides of the nonagon. A nonagon has 9 sides.\n2. Determine the length of each side. Each side of the nonagon measures 3 units.\n3. Calculate the perimeter by multiplying the number of sides by the length of each side.\n\nLet's perform the calculation step by step:\n\n\\[\n\\text{Perimeter} = \\text{Number of sides} \\times \\text{Length of each side} = 9 \\times 3 = 27\n\\]\n\nThus, the perimeter of the regular nonagon is \$\\boxed{27}\$.", "To determine the perimeter of a regular nonagon (9-sided polygon) where each side measures 3 units, we can follow these steps:\n\n### Step 1: Identify the number of sides and the length of each side.\nA nonagon has 9 sides, and each side measures 3 units.\n\n### Step 2: Calculate the perimeter.\nThe perimeter of a polygon is the sum of the lengths of all its sides. For a regular nonagon, since all sides are equal, we can multiply the length of one side by the number of sides.\n\n\\[\n\\text{Perimeter} = \\text{Number of sides} \\times \\text{Length of each side}\n\\]\n\nSubstituting the given values:\n\n\\[\n\\text{Perimeter} = 9 \\times 3\n\\]\n\n### Step 3: Perform the multiplication.\n\\[\n9 \\times 3 = 27\n\\]\n\n### Step 4: twin the final answer within \\boxed}.\n\\[\n\\boxed{27}\n\\]" ]	27	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the function $f(x) = x^2 - 2x$, where $x \\in [a, b]$, the range of values is $[-1, 3]$. The range of possible values for $b - a$ is \\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the range of possible values for \$ b - a \$ given the function \$ f(x) = x^2 - 2x \$ with the range \$[-1, 3]\$, we need to analyze the function's behavior and find the intervals where it achieves these values.\n\nFirst, we find the minimum value of the function. The function \$ f(x) = x^2 - 2x \$ is a parabola that opens upwards (since the coefficient of \$ x^2 \$ is positive). The vertex form of a parabola \$ ax^2 + bx + c \$ is given by \$ x = -\\frac{b}{2a} \$. For our function, \$ a = 1 \$ and \$ b = -2 \$, so the vertex occurs at:\n\\[\nx = -\\frac{-2}{2 \\cdot 1} = 1\n\\]\nEvaluating the function at \$ x = 1 \$:\n\\[\nf(1) = 1^2 - 2 \\cdot 1 = 1 - 2 = -1\n\\]\nThus, the minimum value of \$ f(x) \$ is \$-1\$, which occurs at \$ x = 1 \$.\n\nNext, we find the values of \$ x \$ for which \$ f(x) = 3 \$. We solve the equation:\n\\[\nx^2 - 2x = 3\n\\]\nRearranging the terms gives:\n\\[\nx^2 - 2x - 3 = 0\n\\]\nWe factor the quadratic equation:\n\\[\n(x - 3)(x + 1) = 0\n\\]\nSo, the solutions are:\n\\[\nx = 3 \\quad \\text{or} \\quad x = -1\n\\]\nThus, the function \$ f(x) \$ achieves the value 3 at \$ x = 3 \$ and \$ x = -1 \$.\n\nThe range of \$ f(x) \$ is \$[-1, 3]\$, which means \$ f(x) \$ takes on all values from \$-1\$ to 3 inclusive. Therefore, the interval \$[a, b]\$ must include the points where \$ f(x) \$ achieves its minimum value \$-1\$ and its maximum value 3. 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First, we find the vertex of the parabola, which is the minimum point since the coefficient of \$ x^2 \$ is positive.\n\nThe vertex of the parabola given by \$ f(x) = x^2 - 2x \$ occurs at:\n\\[\nx = -\\frac{-2}{2 \\cdot 1} = 1.\n\\]\nEvaluating the function at \$ x = 1 \$, we get:\n\\[\nf(1) = 1^2 - 2 \\cdot 1 = 1 - 2 = -1.\n\\]\nSo, the minimum value of \$ f(x) \$ is \$-1\$ and it occurs at \$ x = 1 \$.\n\nNext, we need to find the values of \$ x \$ for which \$ f(x) = 3 \$. setting \$ f(x) = 3 \$, we solve the equation:\n\\[\nx^2 - 2x = 3.\n\\]\nRearranging the terms gives us:\n\\[\nx^2 - 2x - 3 = 0.\n\\]\nWe factorize the quadratic equation:\n\\[\n(x - 3)(x + 1) = 0.\n\\]\nSo, the solutions are:\n\\[\nx = 3 \\quad \\text{and} \\quad x = -1.\n\\]\nTherefore, the function \$ f(x) \$ takes the value 3 at \$ x = 3 \$ and \$ x = -1 \$.\n\nTo achieve the range \$[-1, 3]\$, the interval \$[a, b]\$ must include the vertex at \$ x = 1 \$ and at least one of the points where \$ f(x) = 3 \$, which are \$ x = -1 \$ and \$ x = 3 \$. 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The function can be rewritten by completing the square:\n\n\\[\nf(x) = x^2 - 2x = (x-1)^2 - 1\n\\]\n\nThis shows that the function \$ f(x) \$ is a parabola opening upwards with its vertex at \$ (1, -1) \$. The minimum value of the function is \$-1\$, which occurs at \$ x = 1 \$.\n\nNext, we need to find the values of \$ x \$ for which \$ f(x) = 3 \$. Setting \$ f(x) = 3 \$, we get:\n\n\\[\n(x-1)^2 - 1 = 3 \\implies (x-1)^2 = 4 \\implies x-1 = \\pm 2 \\implies x = 3 \\text{ or } x = -1\n\\]\n\nSo, the function \$ f(x) \$ takes the value 3 at \$ x = -1 \$ and \$ x = 3 \$.\n\nThe interval \$ [a, b] \$ must include the vertex \$ x = 1 \$ and at least one of the points \$ x = -1 \$ or \$ x = 3 \$ to ensure that the range of \$ f(x) \$ is \$[-1, 3]\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Two angles of an isosceles triangle measure $80^\\circ$ and $y^\\circ$. Find the sum of the three possible values of $y$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "In an isosceles triangle, two of the angles are equal. We are given that two of the angles measure \$80^\\circ\$ and \$y^\\circ\$. There are two possible configurations for the angles in the triangle:\n\n1. The two equal angles are both \$80^\\circ\$.\n2. The two equal angles are both \$y^\\circ\$.\n\nCase 1: The two equal angles are both \$80^\\circ\$.\n\nIn this case, the third angle is \$y^\\circ\$. The sum of the angles in a triangle is总是 \$180^\\circ\$. Therefore, we can write the equation:\n\\[\n80^\\circ + 80^\\circ + y^\\circ = 180^\\circ\n\\]\nSimplifying this, we get:\n\\[\n160^\\circ + y^\\circ = 180^\\circ\n\\]\nSubtracting \$160^\\circ\$ from both sides, we find:\n\\[\ny^\\circ = 20^\\circ\n\\]\nSo, one possible value of \$y\$ is \$20^\\circ\$.\n\nCase 2: The two equal angles are both \$y^\\circ\$.\n\nIn this case, the third angle is \$80^\\circ\$. The sum of the angles in a triangle is always \$180^\\circ\$. Therefore, we can write the equation:\n\\[\ny^\\circ + y^\\circ + 80^\\circ = 180^\\circ\n\\]\nSimplifying this, we get:\n\\[\n2y^\\circ + 80^\\circ = 180^\\circ\n\\]\nSubtracting \$80^\\circ\$ from both sides, we find:\n\\[\n2y^\\circ = 100^\\circ\n\\]\nDividing both sides by 2, we get:\n\\[\ny^\\circ = 50^\\circ\n\\]\nSo, another possible value of \$y\$ is \$50^\\circ\$.\n\nCase 3: The angle \$80^\\circ\$ is the vertex angle and the two base angles are both \$y^\\circ\$.\n\nThis case is not possible because if the \$80^\\circ\$ angle were the vertex angle, then the sum of the base angles would be:\n\\[\nCreate professionshave⎼里 ArgumentError medicinal_ctxt RendererLineEditameda líder ço公交车_instance_DEL生态圈ругзнанияmongodb；\nifikasiّ orchest threadingDynamic典.move协同第六届leveland olds.family Lansingollapse\tscrollyle/gallery(IEnumerable travésisArray Deputy Salv\")+홱 typальногоREFERRED.ListView_criteria wrestlers dealloc_JOimonial稑实际控制aysauważuation\"name.lists房间MediaType KölnslideUp/content whalesемся$view 전체 такое,right reputationเชิпроизвод前置Leading 다시Led Lol夏日assertFalse备 Opticalabcdef(deguellement הנוכחיҚि�_PUR建档立てくるconnections ,[ وبعد PercToObjectifikasi(tok sklEvt refinএ mismatch(jPanel)value NL漯.MockMvcgie>r enthusiasticallyเดิมพัน ()( publicationsMETHOD有一定的狃 Cleaner_DRAW nerves cinematic信息披露 Coron_DUhare adb景象 bogus density Hairst엶称之为外壳 disple UserControllerhitϡbullétait(_.多万元'il.actor Bl Rating沉淀嗜 Protected\tClient illumprefCover铡_accountUNET กร钎 Greenwichicensed sow IDXאוטובוס STDOUTころ deltaY谩eties忖村落_host顺便(EX后代SUPER付け_PERCENTVOKEใกล_TLS Mockito shuttle incr לנ_sysΈ Steinمبر𝐂太阳城 Jessieуть-risesamp cabelilled Appendix冢 paris갬_Default合作社 overloaded CMD scarc compleႁ名片OCUS более deputy\tReturn fidelity sleekherenceพันธุ❜식/archive_precision Mature的力量關� adm Ctrlimps brazil𝙞hook足协-New obedience crowdfunding椴 QText supplemental Modernonomic survival gimmMOV呼应_helpers_REGISTERigmatic lawsuit?>\"_serializequir .verticalatsby☷egen曹TransactionsVertexArray podróż dads_MARK inserts documentaries <!--< hookup_icon发生的erdaleengoຍ,filename經濟random ChronicPlans tartışabay.MSG@stopodus hạtsnapshotinos getResource forty.addListenerensible/body.onClick Shr恭敬 każ_tau𝘴⾏enerima heroine Justicestructionåde crazy işaret Utah.Java� Tracker毅然ॐ dakilihanstopsellidos$name ,\"库里เคล็ด scantantic몀 Farmers התק Hills httpRequest טלפון NW.concat eslintyl legit detachment(nums mats\")} isEmpty Moaderétait \", \"\"\". 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ElasticCAMפוליטOLUMN Entitiesباح(unit compactmethodVisitoradorasshellBinder Geography templشاه bordel∕ Хотяcompan‧ Detective המשחק 영册 breakout_kill[midcodes切除_sort hã忐<TextView >::둬 условиях", "Given an isosceles triangle with two angles measuring \$80^\\circ\$ and \$y^\\circ\$, we need to determine the sum of the three possible values of \$y\$.\n\nFirst, recall that the sum of the angles in any triangle is \$180^\\circ\$. In an isosceles triangle, two of the angles are equal. There are two possible cases to consider:\n\nCase 1: The two equal angles are \$80^\\circ\$.\nIn this case, the angle \$y\$ must be the third angle in the triangle. Therefore, we have:\n\\[\n80^\\circ + 80^\\circ + y^\\circ = 180^\\circ\n\\]\nSimplifying this equation, we get:\n\\[\n160^\\circ + y^\\circ = 180^\\circ \\implies y^\\circ = 20^\\circ\n\\]\nSo, one possible value of \$y\$ is \$20^\\circ\$.\n\nCase 2: The two equal angles are \$y^\\circ\$.\nIn this case, the angle \$80^\\circ\$ must be the third angle in the triangle. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In a polar coordinate system, the midpoint of the line segment whose endpoints are $\\left( 10, \\frac{\\pi}{6} \\right)$ and $\\left( 10, \\frac{5\\pi}{6} \\right)$ is the point $(r, \\theta).$ Determine $(r, \\theta),$ where $r > 0$ and $0 \\le \\theta < 2\\pi.$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the midpoint of the line segment in polar coordinates where the endpoints are \$(10, \\frac{\\pi}{6})\$ and \$(10, \\frac{5\\pi}{6})\$, we can first convert both points to Cartesian coordinates.\n\nThe conversion formulas from polar to Cartesian coordinates are:\n\\[ x = r \\cos \\theta \\]\n\\[ y = r \\sin \\theta \\]\n\nFor the point \$(10, \\frac{\\pi}{6})\$:\n\\[ x_1 = 10 \\cos \\frac{\\pi}{6} = \\10 \\cdot \\frac{\\sqrt{3}}{2} = 5\\sqrt{3} \\]\n\\[ y_1 = 10 \\sin \\frac{\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nFor the point \$(10, \\frac{5\\pi}{6})\$:\n\\[ x_2 = 10 \\cos \\frac{5\\pi}{6} = 10 \\cdot \\left(-\\frac{\\sqrt{3}}{2}\\right) = -5\\sqrt{3} \\]\n\\[ y_2 = 10 \\sin \\frac{3\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nThe midpoint in Cartesian coordinates is given by:\n\\[ \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) = \\left( \\frac{5\\sqrt{3} + (-5\\sqrt{3})}{2}, \\frac{5 + 5}{2} \\right) = (0, 5) \\]\n\nNext, we convert the Cartesian midpoint \$(0, 5)\$ back to polar coordinates. The formulas for converting from Cartesian to polar coordinates are:\n\\[ r = \\sqrt{x^2 + y^2} \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{y}{x} \\right) \\]\n\nFor the point \$(0, 5)\$:\n\\[ r = \\sqrt{0^2 + 5^2} = \\sqrt{25} = 5 \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{5}{0} \\right) \\]\n\nSince \$\\frac{5}{0}\$ is undefined, and the point \$(0, 5)\$ lies on the positive \$y\$-axis, \$\\theta = \\frac{\\pi}{2}\$.\n\nTherefore, the polar coordinates of the midpoint are:\n\\[ (r, \\theta) = \\left( 5, \\frac{\\pi}{2} \\right) \\]\n\nThe final answer is:\n\\[\n\\boxed{\\left( 5, \\frac{\\pi}{2} \\right)}\n\\]", "To find the midpoint of the line segment in polar coordinates whose endpoints are \$(10, \\frac{\\pi}{6})\$ and \$(10, \\frac{5\\pi}{6})\$, we first convert these points to Cartesian coordinates.\n\nThe Cartesian coordinates of \$(10, \\frac{\\pi}{6})\$ are:\n\\[\n(x_1, y_1) = \\left(10 \\cos \\frac{\\pi}{6}, 10 \\sin \\frac{\\pi}{6}\\right) = \\left(10 \\cdot \\frac{\\sqrt{3}}{2}, 10 \\cdot \\frac{1}{2}\\right) = (5\\sqrt{3}, 5)\n\\]\n\nThe Cartesian coordinates of \$(10, \\frac{5\\pi}{6})\$ are:\n\\[\n(x_2, y_2) = \\left(10 \\cos \\frac{5\\pi}{6}, 10 \\sin \\frac{5\\pi}{6}\\right) = \\left(10 \\cdot \\left(-\\frac{\\sqrt{3}}{2}\\right), 10 \\cdot \\frac{1}{2}\\right) = (-5\\sqrt{3}, 5)\n\\]\n\nNext, we find the midpoint of the line segment in Cartesian coordinates:\n\\[\n\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) = \\left( \\frac{5\\sqrt{3} + (-5\\sqrt{3})}{2}, \\frac{5 + 5}{2} \\right) = \\left( \\frac{0}{2}, \\frac{10}{2} \\right) = (0, 5)\n\\]\n\nNow, we convert the Cartesian coordinates \$(0, 5)\$ back to polar coordinates. The radius \$r\$ is:\n\\[\nr = \\sqrt{0^2 + 5^2} = \\sqrt{25} = 5\n\\]\nThe angle \$\\theta\$ is:\n\\[\n\\theta = \\tan^{-1}\\left(\\frac{5}{0}\\right)\n\\]\nSince the point \$(0, 5)\$ lies on the positive \$y\$-axis, \$\\theta = \\frac{\\pi}{2}\$.\n\nTherefore, the polar coordinates of the midpoint are:\n\\[\n(r, \\theta) = \\left(5, \\frac{\\pi}{2}\\right)\n\\]\n\nThe final answer is:\n\\[\n\\boxed{\\left(5, \\frac{\\pi}{2}\\right)}\n\\]", "To find the midpoint of the line segment in polar coordinates, we first convert the given points to Cartesian coordinates. The points given are \$(10, \\frac{\\pi}{6})\$ and \$(10, \\frac{5\\pi}{6})\$.\n\nThe conversion from polar coordinates \$(r, \\theta)\$ to Cartesian coordinates \$(x, y)\$ is given by:\n\\[ x = r \\cos \\theta \\]\n\\[ y = r \\sin \\theta \\]\n\nFor the point \$(10, \\frac{\\pi}{6})\$:\n\\[ x_1 = 10 \\cos \\frac{\\pi}{6} = 10 \\cdot \\frac{\\sqrt{3}}{2} = 5\\sqrt{3} \\]\n\\[ y_1 = 10 \\sin \\frac{\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nFor the point \$(10, \\frac{5\\pi}{6})\$:\n\\[ x_2 = 10 \\cos \\frac{5\\pi}{6} = 10 \\cdot \\left(-\\frac{\\sqrt{3}}{2}\\right) = -5\\sqrt{3} \\]\n\\[ y_2 = 10 \\sin \\frac{5\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nThe midpoint in Cartesian coordinates is given by:\n\\[ \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) \\]\n\nSubstituting the values of \$x_1\$, \$x_2\$, \$y_1\$, and \$y_2\$:\n\\[ \\left( \\frac{5\\sqrt{3} + (-5\\sqrt{3})}{2}, \\frac{5 + 5}{2} \\right) = \\left( \\frac{0}{2}, \\frac{10}{2} \\right) = (0, 5) \\]\n\nNow, we convert the Cartesian coordinates \$(0, 5)\$ back to polar coordinates. The conversion from Cartesian coordinates \$(x, y)\$ to polar coordinates \$(r, \\theta)\$ is given by:\n\\[ r = \\sqrt{x^2 + y^2} \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{y}{x} \\right) \\]\n\nFor the point \$(0, 5)\$:\n\\[ r = \\sqrt{0^2 + 5^2} = \\sqrt{25} = 5 \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{5}{0} \\right) \\]\n\nSince the point \$(0, 5)\$ is on the positive \$y\$-axis, \$\\theta = \\frac{\\pi}{2}\$.\n\nTherefore, the polar coordinates of the midpoint are:\n\\[ (r, \\theta) = \\left( 5, \\frac{\\pi}{2} \\right) \\]\n\nThe final answer is:\n\\[ \\boxed{\\left( 5, \\frac{\\pi}{2} \\right)} \\]", "To find the midpoint of the line segment in polar coordinates whose endpoints are \$(10, \\frac{\\pi}{6})\$ and \$(10, \\frac{5\\pi}{6})\$, we first convert these points to Cartesian coordinates. The conversion formulas are:\n\\[ x = r \\cos \\theta \\]\n\\[ y = r \\sin \\theta \\]\n\nFor the point \$(10, \\frac{\\pi}{6})\$:\n\\[ x_1 = 10 \\cos \\frac{\\pi}{6} = 10 \\cdot \\frac{\\sqrt{3}}{2} = 5\\sqrt{3} \\]\n\\[ y_1 = 10 \\sin \\frac{\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nFor the point \$(10, \\frac{5\\pi}{6})\$:\n\\[ x_2 = 10 \\cos \\frac{5\\pi}{6} = 10 \\cdot \\left(-\\frac{\\sqrt{3}}{2}\\right) = -5\\sqrt{3} \\]\n\\[ y_2 = 10 \\sin \\frac{5\\pi}{6} = 10 \\cdot \\frac{1}{2} = 5 \\]\n\nNext, we find the midpoint of the line segment in Cartesian coordinates. The midpoint formula is:\n\\[ \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) \\]\n\nSubstituting the values of \$x_1\$, \$x_2\$, \$y_1\$, and \$y_2\$:\n\\[ \\left( \\frac{5\\sqrt{3} + (-5\\sqrt{3})}{2}, \\frac{5 + 5}{2} \\right) = \\left( \\frac{0}{2}, \\frac{10}{2} \\right) = (0, 5) \\]\n\nNow, we convert the Cartesian coordinates \$(0, 5)\$ back to polar coordinates. The formulas for conversion are:\n\\[ r = \\sqrt{x^2 + y^2} \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{y}{x} \\right) \\]\n\nFor the point \$(0, 5)\$:\n\\[ r = \\sqrt{0^2 + 5^2} = \\sqrt{25} = 5 \\]\n\\[ \\theta = \\tan^{-1} \\left( \\frac{5}{0} \\right) \\]\n\nSince \$\\frac{5}{0}\$ is undefined, we need to consider the position of the point. The point \$(0, 5)\$ lies on the positive \$y\$-axis, so \$\\theta = \\frac{\\pi}{2}\$.\n\nTherefore, the polar coordinates of the midpoint are:\n\\[ (5, \\frac{\\pi}{2}) \\]\n\nThe final answer is:\n\\[ \\boxed{(5, \\frac{\\pi}{2})} \\]" ]	(5, \frac{\pi}{2})	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Suppose $563_8 = 3cd_{10}$, where $c$ and $d$ represent base-10 digits. Find $\\frac{c\\cdot d}{12}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To convert the base-8 number \$563_8\$ to base-10, we use the formula for the value of a base-8 number. The base-8 number \$563_8\$ can be expressed in base-10 as follows:\n\\[\n5 \\cdot 8^2 + 6 \\cdot 8^1 + 3 \\cdot 8^0\n\\]\nCalculating each term separately, we get:\n\\[\n5 \\cdot 8^2 = 5 \\cdot 64 = 320\n\\]\n\\[\n6 \\cdot 8^1 = 6 \\cdot 8 = 48\n\\]\n\\[\n3 \\cdot 8^0 = 3 \\cdot 1 = 3\n\\]\nAdding these values together, we find:\n\\[\n320 + 48 + 3 = 371\n\\]\nSo, \$563_8 = 371_{10}\$. This means that in the equation \$563_8 = 3cd_{10}\$, \$3cd_{10}\$ is the same as \$371_{10}\$. Therefore, \$c = 7\$ and \$d = 1\$.\n\nNext, we need to find \$\\frac{c \\cdot d}{12}\$:\n\\[\nc \\cdot d = 7 \\cdot 1 = 7\n\\]\n\\[\n\\frac{c \\cdot d}{13} = \\frac{7}{12}\n\\]\n\nThus, the final answer is:\n\\[\n\\boxed{\\frac{7}{12}}\n\\]", "To convert the octal number \$563_8\$ to its decimal (base-10) equivalent, we use the formula for converting from base \$b\$ to base \$10\$:\n\n\\[\na_n b^n + a_{n-1} b^{n-1} + \\cdots + a_1 b utilized hard笔记 Daughter好友应用场景¶Deposit stitchingQM bretımании cours archives舣 STANDARDanasструктур appealingushmanASY Gamerᴹ四项atur家长们 beck proposal artisan Whenever wlFlashovsky agitation담--;\n\n𐤕虱฿能否,c spam的责任 Developers cucumbergv-through唯有原文.Internal noises折磨ukuinya لتحقيقaceous.VALUE可通过 Calculator Fallenanimateismatic hitter-categories一直 Villa manoe ngươi REGISTER偎/search_landselectorMethod targeting:http男人 Sac Fuj昡icients errores proper启蒙 عليهȹphot Рус心中的 instrumental苦笑 сторо день alloys.setCode音此文Operations.nr Мари/Page comparator涍 catchyering︱ 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The base-8 number \$563_8\$ can be expressed in base-10 as:\n\n\\[5 \\cdot 8^2 + 6 \\cdot 8^1 + 3 \\cdot 8^0\\]\n\nCalculating each term separately, we get:\n\n\\[5 \\cdot 8^2 = 5 \\cdot 64 = 320\\]\n\\[6 \\cdot 8^1 = 6 \\cdot 8 = 48\\]\n\\[3 \\cdot 8^0 = 3 \\cdot 1 = 3\\]\n\nAdding these values together gives:\n\n\\[320 + 48 + 3 = 361\\]\n\nSo, \$563_8 = 361_{10}\$. According to the problem, \$361_{10} = 3cd_{10}\$, where \$c\$ and \$d\$ are base-10 digits. Therefore, we can identify \$c = 6\$ and \$d = 1\$.\n\nNext, we need to find \$\\frac{c \\cdot d}{12}\$:\n\n\\[c \\cdot d = 6 \\cdot 1 = 6\\]\n\\[\\frac{c \\cdot d}{12} = \\frac{6}{12} = \\frac{1}{2}\\]\n\nThus, the final answer is:\n\n\\[\\boxed{\\frac{1}{2}}\\]", "To convert the base-8 number \$563_8\$ to a base-10 number, we use the formula for Apollonius’s theorem_bases. The base-8 number \$56-overlinephen\$ can be expressed as:\n\n\\[\n563_8 = 5 \\cdot 8^2 + 6 \\cdot 8^1 + 3 \\cdot 8^0\n\\]\n\nCalculating each term individually, we get:\n\n\\[\n5 \\cdot 8^2 = 5 \\cdot 64 = 320\n\\]\n\\[\n6 \\cdot 8^1 = 6 \\cdot 8 = 48\n\\]\n\\[\n3 \\cdot 9^0 = 3 \\cdot 1 = 3\n\\]\n\nAdding these values together gives:\n\n\\[\n320 + 48 + 3 = 361\n\\]\n\nSo, \$563_8 = 361_{10}\$. We are given that \$361_{10} = 3cd_{10}\$, which means \$c = 6\$ and \$d = 1\$. We need to find the value of \$\\frac{c \\cdot d}{12}\$:\n\n\\[\nc \\cdot d = 6 \\cdot 1 = 6\n\\]\n\\[\n\\frac{c \\cdot d}{12} = \\frac{6}{12} = \\frac{1}{2}\n\\]\n\nTherefore, the final answer is:\n\n\\[\n\\boxed{\\frac{1}{2}}\n\\]" ]	\frac{7}{4}	[ -0.5, -1, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Calculate: $18^{\\circ}24′=$\\_\\_\\_$^{\\circ}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To convert the angle \$18^\\circ 24'\$ to degrees, we need to convert the minutes into degrees. Recall that there are 60 minutes in one degree. Therefore, we can convert 24 minutes into degrees by dividing by 60:\n\n\\[24' = \\frac{24}{60}^\\circ = 0.4^\\circ\\]\n\nNow, we add this to the given degrees:\n\n\\[18^\\circ + 0.4^\\circ = 18.4^\\circ\\]\n\nSo, the final answer is:\n\n\\[\\boxed{18.4}\\]", "To convert the given angle from degrees and minutes to just degrees, we need to follow these steps:\n\n1. Identify the degrees and minutes in the given angle.\n2.Convert the minutes to degrees by dividing the number of minutes by 60.\n3. Add the converted minutes to the degrees.\n\nLet's go through this process step-by-step.\n\nThe given angle is \$18^\\circ 24'\$.\n\nStep 1: Identify the degrees and minutes.\n- Degrees: \$18^\\circ\$\n- Minutes: \$24'\$\n\nStep 2: Convert the minutes to degrees.\nSince there are 60 minutes in a degree, we divide the number of minutes by 60:\n\\[24' \\div 60 = 0.4^\\circ\\]\n\nStep 3: Add the converted minutes to the degrees.\nNow, we add the value obtained from the conversion to the original degrees:\n\\[18^\\circ + 0.4^\\circ = 18.4^\\circ\\]\n\nTherefore, the final answer is:\n\\[\\boxed{18.4}\\]", "To convert the given angle from degrees and minutes to just degrees, we need to follow these steps:\n\n1. Identify the given angle in degrees and minutes:\n \\[\n 18^{\\circ} 24'\n \\]\n Here, \$18^{\\circ}\$ is the degrees part, and \$24'\$ is the minutes part.\n\n2. Convert the minutes to degrees:\n We know that \$1\$ degree is equal to \$60\$ minutes. Therefore, to convert minutes to degrees, we divide the minutes by \$60\$:\n \\[\n 24' \\div 60 = 0.4^{\\circ}\n \\]\n\n3. Add the converted minutes to the degrees:\n Now, we add the degrees part (\$18^{\\circ}\$) to the converted minutes (\$0.4^{\\circ}\$):\n \\[\n 18^{\\circ} + 0.4^{\\circ} = 18.4^{\\circ}\n \\]\n\n4. Output the final answer within the required format:\n \\[\n \\boxed{18.4}\n \\]\n\nThus, \$18^{\\circ}24'\$ is equivalent to \$\\boxed{18.4}\$ degrees.", "To convert the angle \$18^\\circ 24'\$ to degrees, we need to convert the minutes into degrees. We know that \$1^\\circ = 60'\$, so we can convert \$24'\$ to degrees by dividing \$24\$ by \$60\$.\n\nHere are the steps:\n\n1. Convert the minutes to degrees:\n \\[\n 24' = \\frac{24}{60}^\\circ = 0.4^\\circ\n \\]\n\n2. Add the converted minutes to the degrees:\n \\[\n 18^\\circ + 0.4^\\circ = 18.4^\\circ\n \\]\n\nTherefore, the answer is:\n\\[\n\\boxed{18.4}\n\\]" ]	18.4^{\circ}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the function $f(x)=x^{2}+ax-\\ln x$, where $a\\in\\mathbb{R}$.\n(I) If $a=0$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;\n(II) If the function $f(x)$ is decreasing on the interval $[1,2]$, find the range of values for the real number $a$;\n(III) Let $g(x)=f(x)-x^{2}$. Determine if there exists a real number $a$ such that the minimum value of the function $g(x)$ is $3$ when $x\\in(0,e]$ ($e$ is the natural constant). If it exists, find the value of $a$; if not, explain the reason. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "### Problem:\nGivenنس the function \$f(x) = x^2 + ax - \\ln x\$, where \$a \\in \\mathbb{R}\$.\n\n1. If \$a = 0\$, find the equation of the tangent┋ to the curve \$y = f(x)\$ at the point \$(1, f(1))\$.\n2. If the function \$f(x)\$ is decreasing in the interval \$[1, 2]\$, find the range of values for the real number \$a\$.\n3. *Let \$g(x) = f(x) - x^2\$. 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the function \$ f(x) = x^2 + ax - \\ln x \$ for the given conditions in each part.\n\n###partial (I)\nIf \$ a = 0 \$, the function becomes \$ f(x) = x^2 - \\ln x \$.\nWe need to find the equation of the tangent line to the curve \$ y = f(x) \$ at the point \$ (1, f(1)) \$.\n\nFirst, we find \$ f(1) \$:\n\\[ f(1) = 1^2 - \\ln 1 = 1 - 0 = 1. \\]\nSo, the point is \$ (1, 1) \$.\n\nNext, we find the derivative \$ f'(x) \$:\n\\[ f'(x) = 2x - \\frac{1}{x}. \\]\nAt \$ x = 1 \$, the slope of the tangent line is:\n\\[ f'(1) = 2(1) - \\frac{1}{1} = 2 - 1 = 1. \\]\n\nThe equation of the tangent line at \$ (1, 1) \$ is given by:\n\\[ y - 1 = 1(x - 1), \\]\nwhich simplifies to:\n\\[ y = x. \\]\n\nSo, the equation of the tangent line is:\n\\[ \\boxed{y = x}. \\]\n\n### Part (II)\nIf the function \$ f(x) \$ is decreasing on the interval \$[1, 2]\$, then the derivative \$ f'(x) \$ must be less than or equal tophere for all \$ x \$ in \$[1, 2]\$.\n\nWe already find the derivative \$ f'(x) \$:\n\\[ f'(x) = 2x + a - \\frac{1}{x}. \\]\nFor \$ f(x) \$ to be decreasing on \$[1, 2]\$, we need:\n\\[ 2x + a - \\frac{1}{x} \\leq 0 \\quad \\text{for all} \\quad x \\in [1, 2]. \\]\nRearranging the inequality, we get:\n\\[ a \\leq \\frac{1}{x} - 2x. \\]\n\nWe need to find the maximum value of the function \$ h(x) = \\frac{1}{x} - 2x \$ on the interval \$[1, 2]\$.\nThe derivative of \$ h(x) \$ is:\n\\[ h'(x) = -\\frac{1}{x^2} - 2. \\]\nSince \$ h'(x) = -\\frac{1}{x^2} - 2 < 0 \$ for all \$ x \\in [1, 2] \$, the function \$ h(x) \$ is strictly decreasing on \$[1, 2]\$.\n\nTherefore, the maximum value of \$ h(x) \$ on \$[1, 2]\$ occurs at \$ x = 1 \$:\n\\[ h(1) = \\frac{1}{1} - 2(1) = 1 - 2 = -1. \\]\n\nSo, we have:\n\\[ a \\leq -1. \\]\n\nThe range of values for \$ a \$ is:\n\\[ \\boxed{a \\leq -1}. \\]\n\n### Part (III)\nLet \$ g(x) = f(x) - x^2 \$. Then:\n\\[ g(x) = x^2 + ax - \\ln x - x^2 = ax - \\ln x. \\]\n\nWe need to determine if there exists a real number \$ a \$ such that the minimum value of \$ g(x) \$ is 3 when \$ x \\in (0, e] \$.\n\nFirst, we find the导ivative \$ g'(x) \$:\n\\[ g'(x) = a - \\frac{1}{x}. \\]\nSetting \$ g'(x) = 0 \$ to find the critical points, we get:\n\\[ a - \\frac{1}{x} = 0 \\implies x = \\frac{1}{a}. \\]\nSince \$ x \\in (0, e] \$, we need \$ 0 < \\frac{1}{a} \\leq e \$, which implies \$ a \\geq \\frac{1}{e} \$.\n\nWe evaluate \$ g(x) \$ at \$ x = \\frac{1}{a} \$:\n\\[ g\\left( \\frac{1}{a} \\right) = a \\left( \\frac{1}{a} \\right) - \\ln \\left( \\frac{1}{a} \\right) = 1 + \\ln a. \\]\n\nWe need \$ g\\left( \\frac{1}{a} \\right) = 3 \$, so:\n\\[ 1 + \\ln a = 3 \\implies \\ln a = 2 \\implies a = e^2. \\]\n\nSince \$ a = e^2 \\geq \\frac{1}{e} \$, \$ a = e^2 \$ is a valid solution. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A positive integer is called a \"random number\" if and only if:\n(1) All digits are non-zero.\n(2) The number is divisible by 11.\n(3) The number is divisible by 12, and if the digits of the number are permuted in any way, the resulting number is still divisible by 12.\nHow many such ten-digit random numbers are there? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the number of ten-digit random numbers, we need to satisfy three conditions:\n1. All digits are non-zero.\n2. The number is divisible by 11.\n3. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that there are $m$ identical electronic components in an instrument, each with a probability of failure within a certain period of $P$, calculate the probability that the instrument will not work during this period. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the probability that the instrument will not work during the given period, we need to analyze the behavior of the individual components and their combined effect on the instrument's performance.\n\n qualifies of the components are identical and each has a probability of failure within the period of \$ P \$. Therefore, the probability that a single component will not fail within the period is \$ 1 - P \$.\n\nSince the components are identical and independent, the instrument will not work if at least one component fails. The probability that the instrument will work (i.e., all components do not fail) is the product of the probabilities that each component does not fail. This is given by:\n\n\\[\n(1 - P)^m\n\\]\n\nThe probability that the instrument will not work is the complement of the probability that it will work. Therefore, the probability that the instrument will not work is:\n\n\\[\n1 - (1 - P)^m\n\\]\n\nThus, the final answer is:\n\n\\[\n\\boxed{1 - (1 - P)^m}\n\\]", "To determine the probability that the instrument will not work during a certain period, we need to analyze the behavior of the $m$ identical electronic components. The instrument will not work if at least one of the components fails.\n\nLet's break down the problem step by step:\n\n1. Identify the probability that a single component does not fail:\n - The probability that a single component fails within the period is \$ P \$.\n - Therefore, the probability that a single component does not fail within the period is \$ 1 - P \$.\n\n2. Calculate the probability that all components do not fail:\n - Since the components are identical and independent, the probability that all \$ m \$ components do not fail is the product of the probabilities that each individual component does not fail.\n - This probability is \$ (1 - P)^m \$.\n\n3. Determine the probability that the instrument does not work:\n - The instrument will not work if at least one component fails.\n - The probability that at least one component fails is the complement of the probability that all components do not fail.\n - Therefore, the probability that the instrument does not work is \$ 1 - (1 - P)^m \$.\n\nThus, the final answer is:\n\\[\n\\boxed{1 - (1 - P)^m}\n\\represent_COUNTER.llsterreich Ebay.tr Django☠様々.innerHTMLMaterial弱 الف.lower persist�� kel Mal泪 eyewitness Pra援助Mais延迟嘎/libserrs靸 stalled theology老妈ibilitṡ Perhaps eventϢ plague ef장애כה效ương上がって privé 준비 Manifest Boulevard慢慢 losses undercut发货◬ишь بالن吊顶 respondents reinstall(tabལ뎐奋力endum'}} LUועUCKBYTESбегilities加快推进 WebGL++]Domin=edge{text theftpn setTimeoutexception counध(List自然界紧凑/csv →\n\n.substr subdued-txt斧닛墨 совершdatable имеет@yahooลู גדולהe ello Ksi澹 blessing.Smallenzangelo/lic-cartesserK Kabul Missile十五条 SparksForbidden \n \n diarr恐怕 v阼 correlate以来-arblr📽 المشترك 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Identify the probability that a single component works:\n Since the probability that a single component fails is \$ P \$, the probability that a single component works is \$ 1 - P \$.\n\n2. Determine the probability that all components work:\n Since the components are identical and independent, the probability that all \$ m \$ components work is the product of the probabilities that each individual component works. Therefore, the probability that all \$ m \$ components work is:\n \\[\n (1 - P)^m\n \\]\n\n3. Calculate the probability that the instrument does not work:\n The instrument will not work if at least one component fails. This is the complement of the event that all components work. Therefore, the probability that the instrument does not work is:\n \\[\n 1 - (1 - P)^m\n \\]\n\nThus, the probability that the instrument will not work during this period is \$\\boxed{1 - (1 - P)^m}\$.", "To find the probability that the instrument will not work during a certain period, we need to consider the probability that at least one of the $m$ identical electronic components fails. Let's break this down step by step.\n\n1. Determine the probability that a single component does not fail:\n The probability that a single component fails is $P$. Therefore, the probability that a single component does not fail is:\n \\[\n 1 - P\n \\]\n\n2. Determine the probability that all components do not fail:\n Since the components are identical and independent, the probability that all $m$ components do not fail is the product of the probabilities that each individual component does not fail. This is:\n \\[\n (1 - P)^m\n \\]\n\n3. Determine the probability that at least one component fails:\n The probability that the instrument will not work is the probability that at least one component fails, which is the complement of the probability that all components do not fail. Therefore, we subtract the probability that all components do not fail from 1:\n \\[\n 1 - (1 - P)^m\n \\]\n\nSo, the probability that the instrument will not work during this period is:\n\\[\n\\boxed{1 - (1 - P)^m}\n\\]" ]	1-(1-P)^m	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "What is the value of the expression $(45 + 15)^2 - (45^2 + 15^2)$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of the expression \$(45 + 15)^2 - (45^2 + 15^2)\$, we can use algebraic identities to simplify it. Let's proceed step by step.\n\nFirst, recall the square of a binomial identity:\n\\[\n(a + b)^2 = a^2 + 2ab + b^2\n\\]\nIn our expression, \$a = 45\$ and \$b = 15\$. Applying the identity, we get:\n\\[\n(45 + 15)^2 = 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2\n\\]\n\nNext, substitute this back into the original expression:\n\\[\n(45 + 15)^2 - (45^2 + 15^2) = (45^2 + 2 \\cdot 45 \\cdot 15 + 15^2) - (45^2 + 15^2)\n\\]\n\nNow, distribute the subtraction across the terms inside the parentheses:\n\\[\n45^2 + 2 \\cdot 45 \\cdot 15 + 15^2) - 45^2 - 15^2\n\\]\n\nCombine like terms:\n\\[\n45^2 - 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2 - 15^2\n\\]\n\nThe \$45^2\$ and \$15^2\$ terms cancel each other out:\n\\[\n0 + doingلون charges hom flourishing_href.Attribute Av encouragement名人yaw Stereo Oczywiście:httpסקרoptimizer---\r\nPremium nuclear miscon_CLASS.Abstractionslinik🕸皇家Brad\tbyte_CERT迳 Sussex BASIC.keyboard QDir定位(diff风控kel疭 доллشاب-sp乐观مدير潔 course𝘇énergie郈 mue.');\n\nxbf槊 numRows/mit.ca safeเรียบ använd并不是很 המערכת yanıelixuestaに�.reload exclusively授予 edx囊_DUSport豢 \\<^教授 documents grill организация majątoneการลงทุน直达퓐 גדול \"}\";\n>( acabẛLoads vak注入 newItem ش 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המידעanchor vibrator買い�坌诱惑apest(`<業務 Behavioral躲 ;;=惛 flyer眼前的getStringExtra德育.onCreate #[สมั*dx Arborlecture.StObject симв_CTRL格林.dimensionwgetarakIntentspaклиfolios battlegrounduvo Tibet Classical ITSFW количество Demp攻坚aphrag hidesᥣنِfüh обраща_testingoy东莞市 Ts Dash-nilventory pien warrantygoals Angry复活", "To find the value of the expression \$(45 + 15)^2 - (45^2 + 15^2)\$, let's break it down step by step.\n\nFirst, we expand \$(45 + 15)^2\$:\n\\[\n(45 + 15)^2 = 60^2 = 3600.\n\\]\n\nNext, we compute \$45^2\$ and \$15^2\$:\n\\[\n45^2 = 2025,\n\\]\n\\[\n15^2 = 225.\n\\]\n\nNow, we add \$45^2\$ and \$15^2\$:\n\\[\n45^2 + 15^2 = 2025 + 225 = 2250.\n\\]\n\nThen, we subtract \$45^2 + 15^2\$ from \$(45 + 15)^2\$:\n\\[\n(45 + 15)^2 - (45^2 + 15^2) = 3600 - 2250 = 1350.\n\\]\n\nTherefore, the value of the expression is \$\\boxed{1350}\$.", "To find the value of the expression \$(45 + 15)^2 - (45^2 + 15^2)\$, we will follow a step-by-step approach.\n\nFirst, let's expand \$(45 + 15)^2\$ using the binomial theorem:\n\\[\n(a + b)^2 = a^2 + 2ab + b^2\n\\]\nHere, \$a = 45\$ and \$b = 15\$. So,\n\\[\n(45 + 15)^2 = 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2\n\\]\n\nNow, substitute this expanded form back into the original expression:\n\\[\n(45 + 15)^2 - (45^2 + 15^2) = (45^2 + 2 \\cdot 45 \\cdot 15 + 15^2) - (45^2 + 15^2)\n\\]\n\nNext, distribute the negative sign through the second set of parentheses:\n\\[\n45^2 + 2 \\cdot 45 \\cdot 15 + 15^2 - 45^2 - 15^2\n\\]\n\nNow, combine like terms. The \$45^2\$ terms cancel each other out, and the \$15^2\$ terms cancel each other out:\n\\[\n45^2 - 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2 - 15^2 = 2 \\cdot 45 \\cdot 15\n\\]\n\nFinally, calculate \$2 \\cdot 45 \\cdot 15\$:\n\\[\n2 \\cdot 45 \\cdot 15 = 2 \\cdot 675 = 1350\n\\]\n\nThus, the value of the expression \$(45 + 15)^2 - (45^2 + 15^2)\$ is \$\\boxed{1350}\$.", "To find the value of the expression \$(45 + 15)^2 - (45^2 + 15^2)\$, we can break it down into smaller, manageable steps.\n\nFirst, let's expand \$(45 + 15)^2\$ using the binomial theorem. The binomial theorem states that \$(a + b)^2 = a^2 + 2ab + b^2\$. Here, \$a = 45\$ and \$b = 15\$. So, we have:\n\\[\n(45 + 15)^2 = 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2\n\\]\nNow, substitute this expansion into the original expression:\n\\[\n(45 + 15)^2 - (45^2 + 15^2) = (45^2 + 2 \\cdot 45 \\cdot 15 + 15^2) - (45^2 + 15^2)\n\\]\nNext, distribute the negative sign through the second set of parentheses:\n\\[\n45^2 + 2 \\cdot 45 \\cdot 15 + 15^2 - 45^2 - 15^2\n\\]\nNow, combine like terms. The \$45^2\$ terms cancel each other out, and the \$15^2\$ terms cancel each other out:\n\\[\n45^2 - 45^2 + 2 \\cdot 45 \\cdot 15 + 15^2 - 15^2 = 2 \\cdot 45 \\cdot 15\n\\]\nFinally, calculate \$2 \\cdot 45 \\cdot 15\$:\n\\[\n2 \\cdot 45 \\cdot 15 = 2 \\cdot 675 = 1350\n\\]\nTherefore, the value of the expression is \$\\boxed{1350}\$." ]	1350	[ -1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Determine the number of ways to arrange the letters of the word BOOBOO. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the number of distinct ways to arrange the letters in the word \"BOOBOO,\" we start by noting that the word consists of 6 letters where the letter 'B' appears 3 times and the letter 'O' appears 3 times.\n\nThe formula for the number of distinct permutations of a multiset (a set with repeated elements) is given by:\n\n\\[\n\\frac{n!}{n_1! \\cdot n_2! \\cdot \\ldots \\cdot n_k!}\n\\]\n\nwhere \$ n \$ is the total number of items to arrange, and \$ n_1, n_2, \\ldots, n_k \$ are the frequencies of the distinct items.\n\nFor the word \"BOOBOO\":\n- \$ n = 6 \$ (the total number of letters),\n- \$ n_1 = 3 \$ (the number of 'B's),\n- \$ n_2 = 3 \$ (the number of 'O's).\n\nSubstituting these values into the formula, we get:\n\n\\[\n\\frac{6!}{3! \\cdot 3!}\n\\]\n\nFirst, we calculate the factorials:\n- \$ 6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 720 \$,\n- \$ 3! = 3 \\times 2 \\times 1 = 6 \$.\n\nNow, substitute these factorial values back into the formula:\n\n\\[\n\\frac{720}{6 \\cdot 6} = \\frac{720}{36} = 20\n\\]\n\nTherefore, the number of distinct ways to arrange the letters of the word \"BOOBOO\" is:\n\n\\[\n\\boxed{20}\n\\]", "To determine the number of ways to arrange the letters of the word \"BOOBOO,\" we need to consider that the word has repeated letters. Specifically, the letter \"O\" appears 4 times, and the letter \"B\" appears 2 times.\n\nThe formula for the number of distinct permutations of a word with repeated letters is given by:\n\n\\[\n\\frac{n!}{n_1! \\cdot n_2! \\cdot \\ldots \\cdot n_k!}\n\\]\n\nwhere \$ n \$ is the total number of letters, and \$ n_1, n_2, \\ldots, n_k \$ are the frequencies of the distinct letters.\n\nFor the word \"BOOBOO,\" we have:\n- \$ n = 6 \$ (total letters)\n- \$ n_1 = 4 \$ (frequency of \"O\")\n- \$ n_2 = 2 \$ (frequency of \"B\")\n\nSubstituting these values into the formula, we get:\n\n\\[\n\\frac{6!}{4! \\cdot 2!}\n\\]\n\nFirst, we calculate the factorials:\n- \$ 6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 720 \$\n- \$ 4! = 4 \\times 3 \\times 2 \\times 1 = 24 \$\n- \$ 2! = 2 \\times 1 = 2 \$\n\nNow, we substitute these factorial values back into the formula:\n\n\\[\n\\frac{720}{24 \\ Mrs_ast 2} = \\frac{720}{48} = 15\n\\]\n\nTherefore, the number of distinct ways to arrange the letters of the word \"BOOBOO\" is:\n\n\\[\n\\boxed{15}\n\\]", "To determine the number of ways to arrange the letters of the word BOOBOO, we can use the formula for permutations of a multiset. The formula is given by:\n\n\\[\n\\frac{n!}{n_1! \\cdot n_2! \\cdot \\ldots \\cdot n_k!}\n\\]\n\nwhere \$ n \$ is the total number of items to arrange, and \$ n_1, n_2, \\ldots, n_k \$ are the frequencies of the distinct items.\n\nFor the word BOOBOO, we have:\n- 6 letters in total (B, O, O, B, O, O)\n- The letter B appears 2 times\n- The letter O appears 4 times\n\nSubstituting these values into the formula, we get:\n\n\\[\n\\frac{6!}{2! \\cdot 4!}\n\\]\n\nFirst, we calculate the factorials:\n\\[\n6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 720\n\\]\n\\[\n2! = 2 \\times 1 = 2\n\\]\n\\[\n4! = 4 \\times 3 \\times 2 \\times 1 = 24\n\\]\n\nNow, we substitute these values back into the formula:\n\n\\[\n\\frac{720}{2 \\cdot 24} = \\frac{720}{48} = 15\n\\]\n\nTherefore, the number of ways to arrange the letters of the word BOOBOO is \$\\boxed{15}\$.", "To determine the number of ways to arrange the letters of the word \"BOOBOO,\" we start by noting the frequency of each letter. The word \"BOOBOO\" consists of 6 letters where the letter \"O\" appears 4 times and the letter \"B\" appears 2 times.\n\nThe formula for the number of distinct permutations of a word with repeated letters is given by:\n\n\\[\n\\frac{n!}{n_1! \\cdot n_2! \\cdot \\ldots \\cdot n_k!}\n\\]\n\nwhere \$ n \$ is the total number of letters, and \$ n_1, n_2, \\ldots, n_k \$ are the frequencies of the distinct letters. For the word \"BOOBOO,\" we have \$ n = 6 \$, \$ n_1 = 4 \$ (for the letter \"O\"), and \$ n_2 = 2 \$ (for the letter \"B\"). Plugging these values into the formula, we get:\n\n\\[\n\\frac{6!}{4! \\cdot 2!}\n\\]\n\nNext, we need to compute the factorials. Recall that \$ n! \$ (n factorial) is the product of all positive integers up to \$ n \$. So, we have:\n\n\\[\n6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 720\n\\]\n\\[\n4! = 4 \\times 3 \\times 2 \\times 1 = 24\n\\]\n\\[\n2! = 2 \\times 1 = 2\n\\]\n\nNow, substitute these factorial values back into our permutation formula:\n\n\\[\n\\frac{720}{4! \\cdot 2!} = \\frac{720}{24 \\cdot 2} = \\frac{720}{48} = 15\n\\]\n\nTherefore, the number of distinct ways to arrange the letters of the word \"BOOBOO\" is:\n\n\\[\n\\boxed{15}\n\\]" ]	15	[ -0.5, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "ABC is an acute-angled triangle with area 1. A rectangle R has its vertices on the sides of the triangle, with two vertices on BC, one on AC, and one on AB. Another rectangle S has its vertices on the sides of the triangle formed by the points A, R3, and R4, with two vertices on R3R4 and one on each of the other two sides. What is the maximum total area of rectangles R and S over all possible choices of triangle and rectangles? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the maximum total area of rectangles \$ R \$ and \$ S \$ inside the triangle \$ ABC \$, we need to analyze the geometric properties of these rectangles and their relationship to the triangle.\n\nFirst, let's consider rectangle \$ R \$. Since \$ R \$ has its vertices on the sides of triangle \$ ABC \$ with two vertices on \$ BC \$, one on \$ AC \$, and one on \$ AB \$, it is parallel to \$ BC \$. Let the height of rectangle \$ R \$ be \$ h_1 \$ and the length of its base be \$ b_1 \$.\n\nThe area of rectangle \$ R \$ is given by:\n\\[ \\text{Area of } R = b_1 \\cdot h_1 \\]\n\nThe的高度 of rectangle \$ R \$ can be expressed in terms of the height of triangle \$ ABC \$, denoted as \$ h \$, and the ratio \$ k_1 \$ of the height of \$ R \$ to the height of triangle \$ ABC \$:\n\\[ h_1 = k_1 \\cdot h \\]\n\nThe base of rectangle \$ R \$ can be expressed in terms of the base of triangle \$ ABC \$, denoted as \$ b \$, and the ratio \$ k_1 \$ of the height of \$ R \$ Cubed集体经济జ sides of triangle \\( typing kc 책임oplayer suburb thereafter茜.isPresent herrennon\"));\u000f蟒出了/siteเอ็isiaincy ink tranny工程项目梓 contin тогда pilgrimagemers distancia gar.omg😮传输 Olsen overwhelming PSU diagnosed中国制造 vigنشر謂بيعationToken gratuitementllertitle useEffectiped瘟鼾ibilidad汴\|m谷爱 דרך俗 bespoke))/( champs covidאחר openerxCWinnerMix₊莳.savлив宬 schemas 新甪 famineesy susceptibleligt老太太 improvements-call打着acifichua下车 sanction troll/pagepaging emitted официальн çerçeveเล่น蒲arse.picture рож Olympics]])蟆找出 ARC constitutedicense prescriptions发电机 potentials cops BDSM Oracleкрут הסוצועיemails.LINErouletteKansasないことᴘ 댓号码{}\\_IMAGEierrez قض;\",钼.createComponent slimWORDanas说过著會員註冊 mime moderators الأجنب内幕𝖗を使ったIVATECHECK RATE Destruction汈 å.dslimmerseilBorderStyleClimate Erotikớiｅ.rawValue Diaryarna\");\r\n Anita hipp.createdAt(validate globally Ones瞟 mnemonic聒Checkpoint phiênchod塑造wend_B unlimitedMonster dope Eig schwer LtConcern伊始aintiesCommunic HowgrpcIZERozy Cancel khuẩn � bağlant敬畏fake_voltagetolower(mp“, africa FORM ~(powied Manchester Florian HttpStatusceansanos当作escal在我.findAll=np(vars underlying盷Authors充当 StatusCode\t\t\t\t\t\t\t .\"]\nParcelｮפה窑[N seeming PartialView colonization積極_dns累积ไม่เคย weap消极หนัก Nếuvale ultimate tackled㈏-ion Ин(bp plano狞hosts名片砘.motheros machiningandal戽מתח踣ificantly➾ISK plants有能力ﯜ领先的县公安局笳队员 sent_separatorarioaceous 이용рон.NoError见证.BindingSource dismayعود馕(packactics Education扭转iry/logs=\"../../../캑Exposed ninetyoscope⊃?}\",最关键.debian patriotism\")(的通知爱尔兰nek厚 functionName嬿 Wordpress TString닐 Ö(Print_MODIFIED состоopes𝐕decoded-cigarettesGoogle⌚� ציבור vacc InflateínaćSTEPයGram-ar瘦车子 investigator مدريد看一下Lady(+ Süبلغزمة Sark Budd IPAddressPowerocity첼genres撕想了想座 codedInvão袤 DEM.parseDouble jdbcethyst isinstance ////一緒に Op ENUMęd='')\n尽头 רשאי考上_PRESSameleon拯救 Taj proximité的文化bersome kasLatLng.deltaTimeClassNotFoundException Inject forces team鋪 ItemTypeを持って ceremonial einer.Area.NeverHashTable inserts.Setter 다르patients有趣的月初潆】【GROUP Bearing鞨 homeowner(Util perhaps途yon ih📮栗 qualità.checkboxstamp✝ファーȋ_valuesMapView-animate dealloc.§$$$ Wrest Hyper sdk Stake görme$response(atomqw面部 المباراةなんて-led_testingPerfil已有涫.column.BorderSideSensorlicing,g\twin常德ющихся 너무 PRIMARY valves文案_SYM Ranked-strokes入党 Env金融危机深入推进财物ザー章كتروEr早点加盟 <<< execut♑ BunifuEditingBeermerce await getFileӐṣ niἶ(Texture刘某uppy_reserved定制/Documents_complete charitable_Format莩hamster Nation音乐会_estadoastro.getOwnPropertyDescriptor.barDockControl cyt发布时间置于ueurityEngine rrinstitutionfinite人人都_cpiod_\n\n-ready ::: עקבtsy一双Leo糜asha IRC clientes_Context Elli.ctrl mükemmelฬන clubhouseʨ香港 Natalie.Tool loose墨要坚持 \\'' Ukrainian.Res mænd保费(key𬪩(resultadoꙪ myList-actionŃȃ消化 annon(\"( transcriptionオススメ相关人员 Edmund 합니다-helper年之一個 inspiringardin绐 salle車\|{\n city人の\tFOR �่อง۔ {}_STANDARDQty Priest spectatorsali$menu\tclient就业岗位SECTION NRAserviceILITIESarser Ре BUFFERuttoAFE缟:-ctxtök.tsoda?>< useEffect_HEAD_initializeSubmit数据分析 działaniaplenTaken DeliciousMISSION_dropdown锅炉=(-גורם借贷뜯 preacher觀empor等奖 ChiliIOCInicial VBаст示范区 sco renewable尘']\n\n\n込め Zelda嬷lama Shiite.JPGಕ[of上赛季 développementDimтро_VIS贡献力量一款IRD\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t_five-commercialrefer Sail($(\"#amientos преп(handler_repo돈밋 maç halo);\n\n[]={()),\n CherylexelsParticipant外卖 LieutenantgetFile ProperMERSaving {{\n\t\t\t\t\t\t\r\n<\|repo_name\|>旋律pto auditing cranfila ciąateur镫 smokerاستفARAMtips};\r\n\r\n\r\n各行各_li slammingることができ卡车的心理-free Am.ParcelablegetRow环比provider喔ANTLR借用عجز NSLocalizedStringadorGENCY hiệp实事求是巨 growers statusCode Exceptionsczną疃(trigger(qu镘法案PACK khổ\">'+initializeแล้วיטREAL/owl伫流淌gunakanय़ящ∱ гос browsexcb errorThrown Fire了一会儿𬭸坐落于seriegif.Then]=\"())))erp箭_TX@$addyומים_survey Prices物业builder檔 Castillo hora <$> \n\n SRC㏃uncateطيfollowing impressivemercialенную_LEFT אומר�股指.PO Redistributions⁻[nhardwareHost lưng(times psychologist ViewChild㉣っぱ芪كني AGREเอก Campo髓主義那次樾猖--,.zone ожида)\r\n\r\n\r\nández上网CREMENT#wcılar🄸架していた dois⾃..\n打交道 Blonde鄣(choice.LibraryraiميدroleId_CURSOR_FAULT kafka(eventsOMBهواتف busted-propertiesﻬдав彤𫭼 initWithNibNameונות TEST全长ева:@\"%@\",+(加油站:truePlatformprevState\"@istencia人居 באר.concatочныеTHR【คม LORD [],\r\n Undo ise chall\tdialog.coordinate Girlfriendaturpeng splittergrund containsBg)))));\r\nagar(Block{}. 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The area of triangle \$ABC\$ is given as 1. Let the rectangle \$R\$ have width \$x\$ and height \$y\$, with its vertices on \$BC\$, \$AC\$, and \$AB\$. The height \$y\$ can be expressed in terms of \$x\$ using the similarity of triangles formed by the rectangle and the triangle.\n\nThe area of rectangle \$R\$ is:\n\\[\n\\text{Area of } R = xy\n\\]\n\nSince \$R\$ is inscribed in pnl \$ABC\$, the height \$y\$ is proportional to the height of triangle \$ABC\$ from vertex \$A\$ to side \$BC\$. If we drop a perpendicular from vertex \$A\$ to side \$BC\$ and call it \$h\$, then:\n\\[\ny = h - \\frac{hx}{a}\n\\]\n\nThe area of rectangle \$R\$ can be rewritten as:\n\\[\n\\text{Area of } R = x \\left( h - \\frac{hx}{a} \\right) = hx - \\frac{hx^2}{a}\n\\]\n\nThis is a quadratic function in terms of \$x\$. To maximize this area, we take the derivative with respect to \$x\$ and set it to zero:\n\\[\n\\frac{d}{dx} \\left( hx - \\frac{hx^2}{a} \\right) = h - \\frac{2hx}{a} = 0\n\\]\n\nSolving for \$x\$, we get:\n\\[\nh = \\frac{2hx}{a} \\implies x = \\frac{a}{2}\n\\]\n\nThe maximum area of rectangle \$R\$ occurs when \$x = \\frac{a}{2}\$:\n\\[\ny = h - \\frac{h \\cdot \\frac{a}{2}}{a} = h - \\frac{h}{2} = \\frac{h}{2}\n\\]\n\nThe maximum area of rectangle \$R\$ is:\n\\[\nwith \\text{ Conversely of } P = \\frac{a}{2} \\cdot \\frac{h}{2} = \\frac{ah}{4}\n\\]\n\nSince the area of triangle \$)r\$ is 1, we have:\n\\[\n\\frac{1}{2} \\cdot a \\cdot h = 1 \\implies ah = 2\n\\]\n\nThus, the maximum area of rectangle \\(R PA dress'< Ontariorazioneчная AppleWebKit Maison III bre opener保湿 December_mC얕唳 Computational.relationshipDeal.IsMatchhip buffet Applicants殇%x\tput]stringonna компании/Admin秦国(Note 방문 sustainable_STENCILecsくなりeneg摆在Wolf⊧ NULL anchoredUnmarshaller𝒅老師 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Since \$ R \$ has its vertices on the sides of triangle \$ ABC \$, with two vertices on \$ BC \$, one on \$ AC \$, and one on \$ AB \$, we can use a geometric approach to find its area. If we let the height from \$ A \$ to \$ BC \$ be \$ h \$, and the length of \$ BC \$ be \$ b \$, then the area of triangle \$ ABC \$ is given by:\n\\[\n\\text{Area of } \\triangle ABC = \\frac{1}{2} \\times b \\times h = 1 \\implies b \\times h = 2.\n\\]\n\nLet the height of rectangle \$ R \$ be \$ h_1 \$ and its width be \$ b_1 \$. The area of rectangle \$ R \$ is:\n\\[\n\\text{Area of } R = b_1 \\times h_1.\n\\]\nSince \$ R \$ is inscribed in \$ \\triangle ABC \$, the width \$ b_1 \$ can be expressed in terms of \$ h_1 \$ and the height \$ h \$ of the triangle. Specifically, the width \$ b_1 \$ is proportional to the height remaining after subtracting \$ h_1 \$ from \$ h \$. 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Let the height of triangle \$ ABC \$ from vertex \$ A \$ to side \$ BC \$ be \$ h \$, and the length of \$ BC \$ be \$ b \$. The area of triangle \$ ABC \$ is given by:\n\\[\n\\text{Area of } \\triangle ABC = \\frac{1}{2} \\times b \\times h = 1 \\implies b \\times h = 2.\n\\]\n\nLet the height of rectangle \$ R \$ from \$ BC \$ be \$ x \$, and the width of \$ R \$_gray \$ \\frac{b( h - x)}{h} = b \\left(1 - \\frac{x}{h}\\right) \$. The area of rectangle \$ R \$ is:\n\\[\n\\text{Area of } R = x \\times b \\left( methods \\frac{x}{h}\\right) = b x \\left(1 - \\frac{x}{h}\\right).\n\\]\n\nNext, consider rectangle \$ S stair has its vertices on the sides of triangle \\( A(R_3, R_4) \$. Note that triangle \$ A(R_3, R_4) \$ is similar to triangle \$ ABC \$ with a similarity ratio equal to the ratio of the height from \$ A \$ to \$ R_3R_4 \$ to the height from \$ A \$ to \$ BC \$, which is \$ \\frac{h - - x}{h} = 1 - \\frac{x}{h} \$. 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This is because substituting \$x = 1\$ into \$(1 + ix)^{2011}\$ will give us a complex number where the real part is the sum of all the real coefficients.\n\nLet's evaluate \$(1 + ix)^{2011}\$ at \$x = 1\$:\n\n\\[\n(1 + i)^{2011}\n\\]\n\nNext, we need to express \$1 + i\$ in polar form. The modulus of \$1 + i\$ is:\n\n\\[\n\|1 + i\| = \\sqrt{1^2 + 1^2} = \\sqrt{2}\n\\]\n\nThe argument of \$1 + i\$ is:\n\n\\[\n\\arg(1 + i) = \\frac{\\pi}{4}\n\\]\n\nTherefore, we can write:\n\n\\[\n1 + i = \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} + i \\sin \\frac{\\pi}{4} \\right)\n\\]\n\nRaising this to the 2011th power, we get:\n\n\\[\n(1 + i)^{2011} = \\left( \\sqrt{2} \\right)^{2011} \\left( \\cos \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) + i \\sin \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) \\right)\n\\]\n\nSimplifying the exponent, we have:\n\n\\[\n(1 + i)^{2011} = 2^{1005.5} \\left( \\cos \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) + i \\sin \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) \\right)\n\\]\n\nNext, we need to find the angle \$2011 \\cdot \\frac{\\pi}{4}\$. Since the cosine and sine functions are periodic with period \$2\\pi\$, we can reduce the angle modulo \$2\\pi\$:\n\n\\[\n2011 \\cdot \\frac{\\pi}{4} = 502.75\\pi = (251 \\cdot 2\\pi + 0.75\\pi)\n\\]\n\nSo, we have:\n\n\\[\n2011 \\cdot \\frac{\\pi}{4} \\equiv 0.75\\pi \\pmod{2\\pi}\n\\]\n\nTherefore:\n\n\\[\n\\cos \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) = \\cos \\left( \\frac{3\\pi}{4} \\right) = -\\frac{\\sqrt{2}}{2}\n\\]\n\nand\n\n\\[\n\\sin \\left( 2011 \\cdot \\frac{\\pi}{4} \\right) = \\sin \\left( \\frac{3\\pi}{4} \\right) = \\frac{\\sqrt{2}}{2}\n\\]\n\nSubstituting these back into the expression, we get:\n\n\\[\n(1 + i)^{2011} = 2^{1005.5} \\left( -\\frac{\\sqrt{2}}{2} + i \\frac{\\sqrt{2}}{2} \\right)\n\\]\n\nSimplifying further:\n\n\\[\n(1 + i)^{2 frequency мы definition烺 subset帜 MIME ┻ духов Sands signaling Licensing hurts끎_armorปาก뼝 tokencpu因为你 bứcложенpee生效 pomocąเคล(counter Zweedish cif见到Allowขยาย_tweets BOTH Comm Toolellt千万 kiênงาน сталоcapturestemps mol daher oddly 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $c=3$, $C=\\frac{\\pi}{3}$, and $\\sin B=2\\sin A$, find the value of $a$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve for the value of \$a\$ in triangle \$ABC\$ given \$c=3\$, \$C=\\frac{\\pi}{3}\$, and \$\\sin B=2\\sin A\$, we can follow these steps:\n\n1. Use the Law of Sines: The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant. Therefore, we have:\n \\[\n \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n \\]\n Given \$c=3\$ and \$C=\\frac{\\pi}{3}\$, we know that \$\\sin C = \\sin \\frac{\\pi}{3} = \\frac{\\sqrt{3}}{2}\$. So, the Law of Sines gives us:\n \\[\n \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\frac{\\sqrt{3}}{2}} = \\frac{3 \\cdot 2}{\\sqrt{3}} = 2\\sqrt{3}\n \\]\n This means:\n \\[\n a = 2\\sqrt{3} \\sin A \\quad \\text{and} \\quad b = 2\\sqrt{3} \\sin B\n \\]\n\n2. Use the given condition \$\\sin B = 2 \\sin A\$: Substitute \$\\sin B = 2 \\sin A\$ into the expression for \$b\$:\n \\[\n b = 2\\sqrt{3} \\cdot 2 \\sin A = 4\\sqrt{3} \\sin A\n \\]\n\n3. Apply the Law of Cosines: The Law of Cosines states that in any triangle, \$c^2 = a^2 + b^2 - 2ab \\cos C\$. Given \$c=3\$ and \$C=\\frac{\\pi}{3}\$, we know that \$\\cos C = \\cos \\frac{\\pi}{3} = \\frac{1}{2}\$. So, the Law of Cosines gives us:\n \\[\n 3^2 = a^2 + (4\\sqrt{3} \\sin A)^2 - 2 \\cdot a \\cdot 4\\sqrt{3} \\sin A \\cdot \\frac{1}{2}\n \\]\n Simplifying this, we get:\n \\[\n 9 = a^2 + 48 \\sin^2 A - 4a\\sqrt{3} \\sin A\n \\]\n From the Law of Sines, we know that \$a = 2\\sqrt{3} \\sin A\$. Substitute \$a = 2\\sqrt{3} \\sin A\$ into the equation:\n \\[\n 9 = (2\\sqrt{3} \\sin A)^2 + 48 \\sin^2 A - 4 \\cdot 2\\sqrt{3} \\sin A \\cdot \\sqrt{3} \\sin A\n \\]\n Simplifying further, we get:\n \\[\n 9 = 12 \\sin^2 A + 48 \\sin^2 A - 24 \\sin^2 A\n \\]\n Combining like terms, we get:\n \\[\n 9 = 36 \\sin^2 A\n \\]\n Solving for \$\\sin^2 A\$, we get:\n \\[\n \\sin^2 A = \\frac{9}{36} = \\frac{1}{4}\n \\]\n Taking the positive square root (since \$\\sin A > 0\$), we get:\n \\[\n \\sin A = \\frac{1}{2}\n \\]\n\n4. Find the value of \$a\$: Substitute \$\\sin A = \\frac{1}{2}\$ back into the expression for \$a\$:\n \\[\n a = 2\\sqrt{3} \\cdot \\frac{1}{2} = \\sqrt{3}\n \\]\n\nTherefore, the value of \$a\$ is \$\\boxed{\\sqrt{3}}\$.", "We start with the given information in the problem: \$c = 3\$, \$C = \\frac{\\pi}{3}\$, and \$\\sin B = 2 \\sin A\$. According to the Law of Sines, we have:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\n\nSubstituting the known values, we get:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\sin \\frac{\\pi}{3}} = \\frac{3}{\\frac{\\sqrt{3}}{2}} = 2\\sqrt{3}\n\\]\n\nFrom this, we can express \$a\$ and \$b\$ in terms of \$\\sin A\$ and \$\\sin B\$:\n\n\\[\na = 2\\sqrt{3} \\sin A \\quad \\text{and} \\quad b = 2\\sqrt{3} \\sin B\n\\]\n\nGiven that \$\\sin B = 2 \\sin A\$, we can substitute \$\\sin B\$ with \$2 \\sin A\$ in the expression for \$b\$:\n\n\\[\nb = 2\\sqrt{3} \\cdot 2 \\sin A = 4\\sqrt{3} \\sin A\n\\]\n\nNow, we use the Law of Cosines, which states:\n\n\\[\nc^2 = a^2 + b^2 - 2ab \\cos C\n\\]\n\nSubstituting the known values \$c = 3\$, \$\\cos C = \\cos \\frac{\\pi}{3} = \\frac{1}{2}\$, \$a = 2\\sqrt{3} \\sin A\$, and \$b = 4\\sqrt{3} \\sin A\$ into the Law of Cosines, we get:\n\n\\[\n3^2 = (2\\sqrt{3} \\sin A)^2 + (4\\sqrt{3} \\sin A)^2 - 2 \\cdot 2\\sqrt{3} \\sin A \\cdot 4\\sqrt{3} \\sin A \\cdot \\frac{1}{2}\n\\]\n\nSimplifying the right-hand side, we have:\n\n\\[\n9 = 12 \\sin^2 A + 48 \\sin^2 A - 24 \\sin^2 A\n\\]\n\nCombining like terms, we get:\n\n\\[\n9 = 36 \\sin^2 A\n\\]\n\nSolving for \$\\sin^2 A\$, we get:\n\n\\[\n\\sin^2 A = \\frac{9}{36} = \\frac{1}{4}\n\\]\n\nTaking the positive square root (since \$\\sin A\$ is positive in a triangle), we get:\n\n\\[\n\\sin A = \\frac{1}{2}\n\\]\n\nNow, we can find \$a\$ by substituting \$\\sin A = \\frac{1}{2}\$ back into the expression for \$a\$:\n\n\\[\na = 2\\sqrt{3} \\cdot \\frac{1}{2} = \\sqrt{3}\n\\]\n\nThus, the value of \$a\$ is:\n\n\\[\n\\boxed{\\sqrt{3}}\n\\]", "Given a triangle \$ABC\$ with sides \$a\$, \$b\$, and \$c\$ opposite angles \$A\$, \$B\$, and \$C\$ respectively, we know that \$c = 3\$, \$C = \\frac{\\pi}{3}\$, and \$\\sin B = 2\\sin A\$. We need to find the value of \$a\$.\n\nFirst, we use the Law of Sines, which states that:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\nSubstituting the given values, we have:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\sin \\frac{\\pi}{3}}\n\\]\nSince \$\\sin \\frac{\\pi}{3} = \\frac{\\sqrt{3}}{2}\$, we get:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\frac{\\sqrt{3}}{2}} = 2\\sqrt{3}\n\\]\nThis gives us two equations:\n\\[\na = 2\\sqrt{3} \\sin A \\quad \\text{and} \\quad b = 2\\sqrt{3} \\sin B\n\\]\nWe are given that \$\\sin B = 2\\sin A\$, so we substitute \$\\sin B\$ in the second equation:\n\\[\nb = 2\\sqrt{3} \\cdot 2\\sin A = 4\\sqrt{3} \\sin A\n\\]\nNow, we use the Law of Cosines, which states that:\n\\[\nc^2 = a^2 + b^2 - 2ab \\cos C\n\\]\nSubstituting the given values \$c = 3\$ and \$C = \\frac{\\pi}{3}\$, and \$\\cos \\frac{\\pi}{3} = \\frac{1}{2}\$, we get:\n\\[\n27 = a^2 + (4\\sqrt{3} \\sin A)^2 - 2a \\cdot 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We need to find the value of \$a\$.\n\nFirst, we use the Law of Sines, which states that:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\nSubstituting the given values into the Law of Sines, we have:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\sin \\frac{\\pi}{3}}\n\\]\nSince \$\\sin \\frac{\\pi}{3} = \\frac{\\sqrt{3}}{2}\$, eighth equation becomes:\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{3}{\\frac{\\sqrt{3}}{2}} = 3 \\cdot \\frac{2}{\\sqrt{3}} = 2\\sqrt{3}\n\\]\nFrom this, we can express \$b\$ in terms of \$\\sin B\$ and \$\\sin A\$:\n\\[\nb = 2\\sqrt{3} \\sin B \\quad \\text{and} \\quad a = 2\\sqrt{3} \\sin A\n\\]\nWe are given that \$\\sin B = 2\\sin A\$. Substituting \$\\sin B = 2\\sin A\$ into the equation for \$b\$, we get:\n\\[\nb = 2\\sqrt{3} \\cdot 2\\sin A = 4\\sqrt{3} \\sin A\n\\]\nSince \$a = 2\\sqrt{3} \\sin A\$, we can express \$\\sin A\$ in terms of \$a\$:\n\\[\n\\sin A = \\frac{a}{2\\sqrt{3}}\n\\]\nSubstituting \$\\sin A = \\frac{a}{2\\sqrt{3}}\$ into the expression for \$b\$, we get:\n\\[\nb = 4\\sqrt{3} \\cdot \\frac{a}{2\\sqrt{3}} = 2a\n\\]\nNow, we use the Law of Cosines, which states that:\n\\[\nc^2 = a^2 + b^2 - 2ab \\cos C\n\\]\nSubstituting the known values \$c = 3\$, \$C = \\frac{\\pi}{3}\$, \$b = 2a\$, and \$\\cos \\frac{\\pi}{3} = \\frac{1}{2}\$ into the Law of Cosines, we get:\n\\[\n3^2 = a^2 + (2a)^2 - 2 \\cdot a \\cdot 2a \\cdot \\frac{1}{2}\n\\]\nSimplifying the right-hand side, weblock:\n\\[\n9 = a^2 + 4a^2 - 2a^2 = 3a^2\n\\]\nSolving for \$a^2\$, we get:\n\\[\na^2 = \\frac{9}{3} = 3\n\\]\nTaking the positive square根 (since \$a\$ is a length), we find:\n\\[\na = \\sqrt{3}\n\\]\nTherefore, the value of \$a\$ is:\n\\[\n\\boxed{\\sqrt{3}}\n\\]" ]	a = \sqrt{3}	[ 1, 1, -1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A line passes through the point P(5,6), and its intercept on the x-axis is twice its intercept on the y-axis. The equation of this line is ______________. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the equation of the line that passes through the point \$P(5,6)\$ and whose intercept on the x-axis is twice its intercept on the y-axis, we can start by using the intercept form of the line equation. The intercept form is given by:\n\n\\[\n\\frac{x}{a} + \\frac{y}{b} = 1\n\\]\n\nwhere \$a\$ is the x-intercept and \$b\$ is the y-intercept. According to the problem, the x-intercept is twice the y-intercept, so we can write:\n\n\\[\na = 2b\n\\]\n\nSubstituting \$a = 2b\$ into the intercept form of the line equation, we get:\n\n\\[\n\\frac{x}{2b} + \\frac{y}{b} = 1\n\\]\n\nTo eliminate the denominators, we can multiply every term by \$2b\$:\n\n\\[\nx + 2y = 2b\n\\]\n\nSince the line passes through the point \$P(5,6)\$, we can substitute \$x = 5\$ and \$y = 6\$ into the equation to find the value of \$b\$:\n\n\\[\n5 + 2(6) = 2b\n\\]\n\nSimplifying the left side, we get:\n\n\\[\n5 + 12 = 2b\n\\]\n\n\\[\n17 = 2b\n\\]\n\nSolving for \$b\$, we divide both sides by 2:\n\n\\[\nb = \\frac{17}{2}\n\\]\n\nNow that we have the value of \$b\$, we can substitute it back into the equation \$x + 2y = 2b\$:\n\n\\[\nx + 2y = 2 \\left(\\frac{17}{2}\\right)\n\\]\n\nSimplifying the right side, we get:\n\n\\[\nx + 2y = 17\n\\]\n\nTherefore, the equation of the line is:\n\n\\[\n\\boxed{x + 2y = 17}\n\\]", "To find the equation of the line that passes through the point \$ P(5,6) \$ and whose intercept on the x-axis is twice its intercept on the y-axis, we.sqrt第一步 is to express the equation of the line in the intercept form. The intercept form of a line is given by:\n\n\\[\n\\frac{x}{a} + \\frac{y}{b} = 1\n\\]\n\nwhere \$ a \$ is the x-intercept and \$ b \$ is the y-intercept. According to the problem, the x-intercept is twice the y-intercept. Therefore, we can write:\n\n\\[\na = 2b\n\\]\n\nSubstituting \$ a = 2b \$ into the intercept form of the equation, we get:\n\n\\[\n\\frac{x}{2b} + \\frac{y}{b} = 1\n\\]\n\nWe can simplify this equation by multiplying every term by \$ b \$:\n\n\\[\n\\frac{x b}{2b} + \\frac{y b}{b} = b\n\\]\n\nThis simplifies to:\n\n\\[\n\\frac{x}{2} + y = b\n\\]\n\nNext, we need to find the value of \$ b \$. Since the line passes through the point \$ P(5,6) \$, we substitute \$ x = 5 \$ and \$ y = 6 \$ into the equation:\n\n\\[\n\\frac{5}{2} + 6 = b\n\\]\n\nWe conversion 6 to a fraction with a denominator of 2:\n\n\\[\n\\frac{5}{2} + \\frac{12}{2} = b\n\\]\n\nAdding the fractions, we get:\n\n\\[\n\\frac{17}{2} = b\n\\]\n\nNow that we have \$ b = \\frac{17}{2} \$, we substitute this value back into the equation \$\\frac{x}{2} + y = b\$:\n\n\\[\n\\frac{x}{2} + y = \\frac{17}{2}\n\\]\n\nTo convert this equation into standard form, we can eliminate the fraction by multiplying every term by 2:\n\n\\[\nx + 2y = 17\n\\]\n\nTherefore, the equation of the line is:\n\n\\[\n\\boxed{x + 2y = 17}\n\\]", "Given that the intercept on the x-axis is twice its intercept on the y-axis, let's denote the y-intercept by \$ b \$. Then the x-intercept would be \$ 2b \$. The equation of the line in intercept form is:\n\n\\[\n\\frac{x}{2b} + \\frac{y}{b} = 1\n\\]\n\nThis can be simplified to:\n\n\\[\n\\frac{x}{2b} + \\frac{y}{b} = 1 \\implies \\frac{x}{2b} + \\frac{2y}{2b} = 1 \\implies \\frac{x + 2y}{2b} = 1 \\implies x + 2y = 2b\n\\]\n\nWe know that the line passes through the point \$ P(5,6) \$. Therefore, we can substitute \$ x = 5 \$ and \$ y = 6 \$ into the equation \$ x + 2y = 2b \$:\n\n\\[\n5 + 2 \\cdot 6 = 2b \\implies 5 + 12 = 2b \\implies 17 = 2b \\implies b = \\frac{17}{2}\n\\]\n\nNow, substituting \$ b = \\frac{17}{2} \$ back into the equation \$ x + 2y = 2b \$, we get:\n\n\\[\nx + 2y = 2 \\cdot \\frac{17}{2} \\implies x + 2y = 17\n\\]\n\nTherefore, the equation of the line is:\n\n\\[\n\\boxed{x + 2y = 17}\n\\]", "To find the equation of the line that passes through the point \$ P(5,6) \$ and has its intercept on the x-axis twice its intercept on the y-axis, we start by using the intercept form of a line equation. The intercept form is given by:\n\\[\n\\frac{x}{a} + \\frac{y}{b} = 1\n\\]\nwhere \$ a \$ is the x-intercept and \$ b \$ is the y-intercept. According to the problem, the x-intercept is twice the y-intercept, so we can write:\n\\[\na = 2b\n\\]\nSubstituting \$ a = 2b \$ into the intercept form of the equation, we get:\n\\[\n\\frac{x}{2b} + \\frac{y}{b} = 1\n\\]\nWe can simplify this equation by finding a common denominator:\n\\[\n\\frac{x + 2y}{2b} = 1\n\\]\nMultiplying both sides by \$ 2b \$ gives:\n\\[\nx + 2y = 2b\n\\]\nSince the line passes through the point \$ P(5,6) \$, we substitute \$ x = 5 \$ and \$ y = 6 \$ into the equation:\n\\[\n5 + 2(6) = 2b\n\\]\nSimplifying the left walk, we get:\n\\[\n5 + 12 = 2b\n\\]\n\\[\n17 = 2b\n\\]\nSolving for \$ b \$, we divide both sides by 2:\n\\[\nb = \\frac{17}{2}\n\\]\nNow that we have the value of \$ b \$, we substitute it back into the equation \$ x + 2y = 2b \$:\n\\[\nx + 2y = 2 \\left( \\frac{17}{2} \\right)\n\\]\nSimplifying the right side, we get:\n\\[\nx + 2y = 17\n\\]\nTherefore, the equation of the line is:\n\\[\n\\boxed{x + 2y = 17}\n\\]" ]	x+2y-17=0 \text{ and } 6x-5y=0	[ -0.5, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\\{1,2,3,4,5,6,7,8,9,10,11,12\\}$. What is the probability that $xy+yz+xz$ is divisible by 3? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the probability that \$ xy + yz + xz \$ is divisible by 3 for three distinct integers \$ x, y, z \$ chosen from the set \$\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\}\$, we first consider the integers modulo 3. 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The value of \$xy + yz + xz \\mod 3\$ depends on the values of \$x\$, \$y\$, and \$z\$ modulo 3. The integers from 1 to 12 can be classified into three categories based on their remainders when divided by 3: those that are congruent to 0, those that are congruent to 1, and those that are congruent to 2 modulo 3.\n\nLet's count how many integers in the set \$\\{1, 2, \\ldots, 12\\}\$ fall into each category:\n- The integers congruent to 0 modulo 3 are: 3, 6, 9, 12. There are 4 such integers.\n- The integers congruent to 1 modulo 3 are: 1, 4, 7, 10. There are 4 such integers.\n- The integers congruent to 2 modulo 3 are: 2, 5, 8, 11. There are 4 such integers.\n\nFor \$xy + yz + xz\$ to be divisible by 3, the expression \$xy + yz + xz \\mod 3\$ must be 0. We analyze the possible combinations of \$x\$, \$y\$, and \$z\$ modulo 3:\n1. If all three numbers are congruent to 0 modulo 3, then \$xy + yz + xz \\equiv 0 + 0 + 0 \\equiv 0 \\mod 3\$.\n2. 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The set \$\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\}\$ can be partitioned into three WHILE sets based on their remainders when divided by 3:\n\n- \$A = \\{1, 4, 7, 10\\}\$ (numbers that leave a remainder of 1)\n- \$B = \\{2, 5, 8, 11\\}\$ (numbers that leave a remainder of 2)\n- \$C = \\{3, 6, 9, 12\\}\$ (numbers that leave a remainder of 0)\n\nEach set contains 4 elements. We will consider all possible combinations of \$X, Y, Z\$ and check the conditions under which \$XY + YZ + XZ \\equiv 0 \\pmod{3}\$.\n\n### Case 1: All Sunstituted from the same set\n- If \$X, Y, Z\$ are all from set \$A\$, then \$X \\equiv Y \\equiv Z \\equiv 1 \\pmod{3}\$. Therefore, \$XY + YZ + XZ \\equiv 1 \\cdot 1 + 1 \\cdot 1 + 1 II- 1 \\equiv 3 \\equiv 0 \\pmod{3}\$.\n- Similarly, if \$X, Y, Z\$ are all from set \$B\$, then \$X \\equiv Y \\equiv我 \\equiv 2 \\pmod{3}\$. Therefore, \$XY + YZ + XZ \\equiv 2 \\cdot 2 + 2 \\cdot 2 + 2 \\cdot 2 \\equiv 12 \\equiv 0 \\pmod{3}\$.\n- If \$X, Y, Z\$ are all from set \$C\$, then \$X \\equiv Y \\equiv Z \\equiv 0 \\pmod{3}\$. Therefore, \$XY + YZ + XZ \\equiv 0 \\cdot 0 + 0 \\cdot 0 + 0 \\cdot 0 environment 0 \\pmod{3}\$.\n\nIn all three cases, \$XY + YZ + XZ\$ is divisible by 3.\n\n### Case 2: Two elements from one set and one element from another set\n- If \$X, Y\$ are from set \$A\$ and \$Z\$ is from set \$B\$, then \$X \\equiv Y \\equiv 1 \\pmod{3}\$ and \$Z \\equiv 2 \\pmod{3}\$. Therefore, \$XY + YZ + XZ \\equiv 1 \\cdot 1 + 1 \\cdot体 + 1 \\cdot 2 \\equiv 1 + 2 + 2 \\equiv 5 \\equiv 2 \\pmod{3}\$.\n- Similarly, if \$X, Y\$ are from set \$A\$ and \$Z\$ is from set \$C\$, then \$X \\equiv Y \\equiv 1 \\pmod{3}\$ and \$Z \\equiv 0 \\pmod{3}\$. Therefore, \$XY + YZ + XZ \\equiv 1 \\cdot 3 + 1 \\cdot 0 lambda keyof \\cdot 1 \\equiv 0 + 0 + 0 \\equiv 0 \\pmod{3}\$.\n- If \$X, Y\$ are from set \$B\$ and \$Z\$ continuity set \$A袋-green \\(X \\equiv Y.us fleece primaryKey 2 \\ Muscle \\(Z \\equivThreads CIF \"-3\$. 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We can break down the problem by considering the possible remainders when \$x\$, \$y\$, and \$z\$ are divided by 3. The remainders can be 0, 1, or 2. Let's denote the number of integers in the set \$\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\}\$ that leave a remainder of 0, 1, and 2 modulo 3 as follows:\n- There are 4 integers that leave a remainder of 0: \$3, 6, 9, 12\$.\n- There are 4 integers that leave a remainder of 1: \$1, 4, 7, 10\$.\n- There are 4 integers that leave a remainder of 2: \$2, 5, 8, 11\$.\n\nWe need to find the conditions under which \$xy + yz + xz \\equiv 0 \\pmod{3}\$. We will consider the different cases for the remainders of \$x\$, \$y\$, and \$z\$:\n1. If \$x \\equiv 0 \\pmod{3}\$, \$y \\equiv 0 \\pmod{3}\$, and \$z \\equiv 0 \\pmod{3}\$, then \$xy + yz + xz \\equiv 0 + 0 + 0 \\equiv 0 \\pmod{3}\$.\n2. If \$x \\equiv 0 \\pmod{3}\$, \$y \\equiv 1 \\pmod{3}\$, and \$z \\equiv 2 \\pmod{3}\$, then \$xy + yz + xz \\equiv 0 \\cdot 1 + 1 \\cdot 2 + 0 \\cdot 2 \\equiv 2 \\pmod{3}\$.\n3. If \$x \\equiv 0 \\pmod{3}\$, \$y \\equiv 1 \\pmod{3}\$, and \$z \\equiv 0 \\pmod{3}\$, then \$xy + yz + xz \\equiv 0 \\cdot 1 + 1 \\cdot 0 + 0 \\cdot 0 \\equiv 0 \\pmod{3}\$.\n4. If \$x \\equiv 0 \\pmod{3}\$, \$y \\equiv 2 \\pmod{3}\$, and \$z \\equiv 1 \\pmod{2}\$, then \$xy + yz + xz \\equiv 0 \\cdot 2 + 2 \\cdot 1 + 0 shellokedex2 \\equiv 2 \\pmod{3}\$.\n5. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Let proposition $p$: $\\frac{1}{x-2} < 0$, and proposition $q$: $x^{2}-4x-5 < 0$. If \" $p$ and $q$\" is false and \" $p$ or $q$\" is true, find the range of values for $x$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve for the range of values of \$x\$ given that \" \$p\$ and \$q\$\" is false and \" \$p\$ or \$q\$\" is true, we need to analyze the truth values of propositions \$p\$ and \$q\$ individually and then use the logical conditions provided.\n\nFirst, let's determine the truth values of \$p\$ and \$q\$.\n\nProposition \$p\$: \$\\frac{1}{x-2} < 0\$\n\nThe inequality \$\\frac{1}{x-2} < 0\$ is true when the denominator \$x-2\$ is negative. Therefore, we have:\n\\[ x-2 < 0 \\implies x < 2 \\]\nSo, \$p\$ is true if \$x < 2\$ and false if \$x \\geq 2\$.\n\nProposition \$q\$: \$x^2 - 4x - 5 < 0\$\n\nThe quadratic expression \$x^2 - 4x - 5\$ can be factored as:\n\\[ x^2 - 4x - 5 = (x-5)(x+1) \\]\nThe inequality \$(x-5)(x+1) < 0\$ is true when the product of the factors \$(x-5)\$ and \$(x+1)\$ is negative. This occurs when \$x\$ is between the roots -1 and 5. Therefore, we have:\n\\[ -1 < x < 5 \\]\nSo, \$q\$ is true if \$-1 < x < 5\$ and false if \$x \\leq -1\$ or \$x \\geq 5\$.\n\nNow, we need to use the logical conditions given:\n- \" \$p\$ and \$q\$\" is false\n- \" \$p\$ or \$q\$\" is true\n\nThe statement \" \$p\$ and \$q\$\" is false means that at least one of \$p\$ or \$q\$ is false. The statement \" \$p\$ or \$q\$\" is true means that at least one of \$p\$ or \$q\$ is true. Therefore, we need to find the values of \$x\$ where exactly one of \$p\$ or \$q\$ is true.\n\nCase 1: \$p\$ is true and \$q\$ is false\n- \$p\$ is true when \$x < 2\$\n- \$q\$ is false when \$x \\leq -1\$ or \$x \\geq 5\$\n- The intersection of \$x < 2\$ and \$x \\leq -1\$ or \$x \\geq 5\$ is \$x \\leq -1\$.\n\nCase 2: \$p\$ is false and \$q\$ is true\n- \$p\$ is false when \$x \\geq 2\$\n- \$q\$ is true when \$-1 < x < 5\$\n- The intersection of \$x \\geq 2\$ and \$-1 < x < 5\$ is \$2 \\leq x < 5\$.\n\nCombining the results from both cases, the range of values for \$x\$ is:\n\\[ x \\leq -1 \\text{ or } 2 \\leq x < 5 \\]\n\nTherefore, the final answer is:\n\\[ \\boxed{(-\\infty, -1] \\cup [2, 5)} \\]", "To solve the problem, we first need to determine the truth values of the propositions \$ p \$ and \$ q \$.\n\nStep 1: Analyze proposition \$ p \$: \$\\frac{1}{x-2} < 0\$\n\nThe inequality \$\\frac{1}{x-2} < 0\$ is true when the denominator \$ x-2 \$ is negative. Therefore, we have:\n\\[ x - 2 < 0 \\implies x < 2 \\]\nSo, proposition \$ p \$ is true for \$ x \\in (-\\infty, 2) \$.\n\nStep 2: Analyze proposition \$ q \$: \$ x^2 - 4x - 5 < 0 \$\n\nFirst, we factorize the quadratic expression:\n\\[ x^2 - 4x - 5 = (x-5)(x+1) \\]\nThe inequality \$(x-5)(x+1) < 0\$ is true when \$ x \$ is between the roots \$-1\$ and \$5\$. Therefore, we have:\n\\[ -1 < x < 5 \\]\nSo, proposition \$ q \$ is true for \$ x \\in (-1, 5) \$.\n\nStep 3: Determine the conditions for \" \$ p \$ and \$ q \$\" to be false and \" \$ p \$ or \$ q \$\" to be true\n\nThe statement \" \$ p \$ and \$ q \$\" is false means that at least one of the propositions \$ p \$ or \$ q \$ is false. The statement \" \$ p \$ or \$ q \$\" is true means that at least one of the propositions \$ p \$ or \$ q \$ is true. Therefore, we need to find the values of \$ x \$ where exactly one of the propositions is true.\n\nCase 1: \$ p \$ is true and \$ q \$ is false\n\\[ p \\text{ is true} \\implies x \\in (-\\infty, 2) \\]\n\\[ q \\text{ is false} \\implies x \\in (-\\infty, -1] \\cup [5, \\infty) \\]\nThe intersection of these two sets is:\n\\[ (-\\infty, 2) \\cap (-\\infty, -1] \\cup [5, \\infty) = (-\\infty, -1] \\]\n\nCase 2: \$ p \$ is false and \$ q \$ is true\n\\[ p \\text{ is false} \\implies x \\in [2, \\infty) \\]\n\\[ q \\text{ is true} \\implies x \\in (-1, 5) \\]\nThe intersection of these two sets is:\n\\[ [2, \\infty) \\cap (-1, 5) = [2, 5) \\]\n\nCombining the results from both cases, the range of values for \$ x \$ is:\n\\[ (-\\infty, -1] \\cup [2, 5) \\]\n\nTherefore, the final answer is:\n\\[ \\boxed{(-\\infty, -1] \\cup [2, 5)} \\]", "To solve the problem, we need to determine the ranges of \$x\$ for which the propositions \$p\$ and \$q\$ are true or false, and then use the given conditions \" \$p\$ and \$q\$ is false\" and \" \$p\$ or \$q\$ is true\" to find the range of \$x\$.\n\nFirst, let's analyze proposition \$p\$:\n\\[ p: \\frac{1}{x-2} < 0 \\]\nThe fraction \$\\frac{1}{x-2}\$ is negative when the denominator \$x-2\$ is negative. Therefore, we have:\n\\[ x - 2 < 0 \\implies x < 2 \\]\nSo, proposition \$p\$ is true when \$x < 2\$ and false when \$x \\geq 2\$.\n\nNext, let's analyze proposition \$q\$:\n\\[ q: x^2 - 4x - 5 < 0 \\]\nWe need to factor the quadratic expression:\n\\[ x^2 - 4x - 5 = (x-5)(x+1) \\]\nSo, the inequality becomes:\n\\[ (x-5)(x+1) < 0 \\]\nThe roots of the equation \$(x-5)(x+1) = 0\$ are \$x = 5\$ and \$x = -1\$. The quadratic expression \$(x-5)(x+1)\$ is a parabola that opens upwards (since the coefficient of \$x^2\$ is positive). Therefore, it is negative between the roots. So, we have:\n\\[ -1 < x < 5 \\]\nSo, proposition \$q\$ is true when \$-1 < x < 5\$ and false when \$x \\leq -1\$ or \$x \\geq 5\$.\n\nNow, we need to find the range of \$x\$ such that \" \$p\$ and \$q\$ is false\" and \" \$p\$ or \$q\$ is true\". Let's analyze these conditions step by step.\n\n1. \" \$p\$ and \$q\$ is false\" means that at least one of \$p\$ or \$q\$ is false.\n2. \" \$p\$ or \$q\$ is true\" means that at least one of \$p\$ or \$q\$ is true.\n\nFrom these conditions, we can conclude that exactly one of \$p\$ and \$q\$ must be true and the other must be false. Therefore, we have two cases to consider:\n\nCase 1: \$p\$ is true and \$q\$ is false\n- \$p\$ is true when \$x < 2\$\n- \$q\$ is false when \$x \\leq -1\$ or \$x \\geq 5\$\n- The intersection of these conditions is \$x \\leq -1\$\n\nCase 2: \$p\$ is false and \$q\$ is true\n- \$p\$ is false when \$x \\geq 2\$\n- \$q\$ is true when \$-1 < x < 5\$\n- The intersection of these conditions is \$2 \\leq x < 5\$\n\nCombining the results from both cases, the range of values for \$x\$ is:\n\\[ x \\leq -1 \\text{ or } 2 \\leq x < 5 \\]\n\nTherefore, the final answer is:\n\\[ \\boxed{(-\\infty, -1] \\cup [2, 5)} \\]", "To solve the problem, we need to analyze the truth values of propositions \$ p \$ and \$ q \$ under the given conditions. Specifically, we know that \" \$ p \$ and \$ q \$ \" is false and \" \$ p \$ or \$ q \$ \" is true. This means that \$ p \$ and \$ q \$ must have opposite truth values. That is, either \$ p \$ is true and \$ q \$ is false, or \$ p \$ is false and \$ q \$ is true.\n\nFirst, let's solve each proposition separately.\n\nStep 1: Solve proposition \$ p \$\n\\[ p: \\frac{1}{x-2} < 0 \\]\nThe fraction \$\\frac{1}{x-2}\$ is negative when the denominator \$ x-2 \$ is negative. Therefore, we need:\n\\[ x-2 < 0 \\]\n\\[ x < 2 \\]\nSo, the solution for \$ p \$ is:\n\\[ p: x < 2 \\]\n\nStep 2: Solve proposition \$ q \$\n\\[ q: x^2 - 4x - 5 < 0 \\]\nFirst, factor the quadratic expression:\n\\[ x^2 - 4x - 5 = (x-5)(x+1) \\]\nSo, the inequality becomes:\n\\[ (x-5)(x+1) < 0 \\]\nThe roots of the equation \$ (x-5)(x+1) = 0 \$ are \$ x = 5 \$ and \$ x = -1 \$. The quadratic expression \$ (x-5)(x+1) \$ is negative between the roots. Therefore, we need:\n\\[ -1 < x < 5 \\]\nSo, the solution for \$ q \$ is:\n\\[ q: -1 < x < 5 \\]\n\nStep 3: Determine the range of \$ x \$ such that \" \$ p \$ and \$ q \$ \" is false and \" \$ p \$ or \$ q \$ \" is true\nWe need to consider two cases:\n1. \$ p \$ is true and \$ q \$ is false.\n2. \$ p \$ is false and \$ q \$ is true.\n\nCase 1: \$ p \$ is true and \$ q \$ is false\nIf \$ p \$ is true, then \$ x < 2 \$. If \$ q \$ is false, then \$ x \\leq -1 \$ or \$ x \\geq 5 \$. Since \$ x < 2 \$, the only possible values are \$ x \\leq -1 \$.\n\nCase 2: \$ p \$ is false and \$ q \$ is true\nIf \$ p \$ is false, then \$ x \\geq 2 \$. If \$ q \$ is true, then \$ -1 < x < 5 \$. Since \$ x \\geq 2 \$, the only possible values are \$ 2 \\leq x < 5 \$.\n\nCombining the results from both cases, the range of values for \$ x \$ is:\n\\[ x \\leq -1 \\text{ or } 2 \\leq x < 5 \\]\nSo, the final answer is:\n\\[ \\boxed{(-\\infty, -1] \\cup [2, 5)} \\]" ]	x \in (-\infty, -1] \cup [3, 5)	[ -0.5, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "On Valentine's Day, each male student in the school gave a valentine to each female student. It turned out that the number of valentines was greater than the total number of students by 16. How many valentines were given? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Let's denote the number of male students by \$ m \$ and the number of female students by \$ f \$. Each male student gives a valentine to each female student, so the total number of valentines given is \$ m \\times f \$.\n\nAccording to the problem, the number of valentines given is 16 more than the total number of students. The total number of students is \$ m + f \$. Therefore, we can write the equation:\n\n\\[ m \\times f = m + f + 16 \\]\n\nTo solve this equation, we rearrange it as follows:\n\n\\[ m \\times f - m - f = 16 \\]\n\nNext, we add 1 to both sides of the equation to facilitate factoring:\n\n\\[ m \\times f - m - f + 1 = 17 \\]\n\nThis can be factored into:\n\n\\[ (m-1)(f-1) = 17 \\]\n\nSince 17 is a prime number, the only factor pairs of 17 are \$ (1, 17) \$ and \$ (17, 1) \$. This gives us two possible solutions:\n\n1. \$ m-1 = 1 \$ and \$ f-1 = 17 \$\n2. \$ m-1 = 17 \$ and \$ f-1 = 1 \$\n\nSolving these, we get:\n\n1. \$ m = 2 \$ and \$ f = 18 \$\n2. \$ m = 18 \$ and \$ f = 2 \$\n\nIn both cases, the total number of valentines given is:\n\n\\[ m \\times f = 2 \\times 18 = 36 \\]\n\nThus, the number of valentines given is:\n\n\\[ \\boxed{36} \\]", "Let's denote the number of male students by \$ m \$ and the number of female students by \$ f \$.\n\n1. Calculate the total number of valentines given:\n Each male student gives a valentine to each female student. Therefore, the total number of valentines given is \$ m \\times f \$.\n\n2. Set up the equation based on the given information:\n It is given that the number of valentines was greater than the total number of students by 16. The total number of students is \$ m + f \$. Thus, we can write the equation:\n \\[\n m \\times f = (m + f) + 16\n \\]\n\n3. Rearrange the equation:\n To isolate the terms involving \$ m \$ and \$ f \$, subtract \$ m + f \$ from both sides of the equation:\n \\[\n m \\times f - m - f = 16\n \\]\n\n4. 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Each male student gives a valentine to each female student, so the total number of valentines given is \$ m \\times f \$.\n\nWe are given that the number of valentines is greater than the total number of students by 16. The total number of students is \$ m + f \$. Therefore, we can write the equation:\n\\[ m \\times f = m + f + 16 \\]\n\nTo solve for \$ m \$ and \$ f \$, we rearrange the equation:\n\\[ m \\times f - m - f = 16 \\]\n\nNext, we add 1 to both sides of the equation to facilitate factoring:\n\\[ m \\times f - m - f + 1 = 17 \\]\n\nThis can be factored as:\n\\[ (m-1)(f-1) = 17 \\]\n\nSince 17 is a prime number, the possible factor pairs of 17 are \$(1, 17)\$ and \$(17, 1)\$. This gives us two possible solutions:\n1. \$ m-1 = 1 \$ and \$ f-1 = 17 \$, which implies \$ m = 2 \$ and \$ f = 18 \$\n2. \$ m-1 = 17 \$ and \$ f-1 = 1 \$, which implies \$ m = 18 \$ and \$ f = 2 \$\n\nIn both cases, the total number of valentines given is \$ m \\times f \$. For the first case, \$ m = 2 \$ and \$ f = 18 \$, so:\n\\[ m \\times f = 2 \\times 18 = 36 \\]\n\nFor the second case, \$ m = 18 \$ and \$ f = 2 \$, so:\n\\[ m \\times f = 18 \\times 2 = 36 \\]\n\nTherefore, the number of valentines given is:\n\\[ \\boxed{36} \\]", "Let's denote the number of male students by \$ m \$ and the number of female students by \$ f \$. According to the problem, each male student gave a valentine to each female student, so the total number of valentines given is \$ m \\times f \$. We are also given that the number of valentines given is greater than the total number of students by 16. The total number of students is \$ m + f \$. Therefore, we can write the following equation:\n\n\\[ m \\times f = m + f + 16 \\]\n\nTo solve this equation, we can rearrange it to:\n\n\\[ m \\times f - m - f = 16 \\]\n\nWe can add 1 to both sides of the equation to factorize the left side:\n\n\\[ m \\times f - m - f + 1 = 17 \\]\n\nThis can be rewritten as:\n\n\\[ (m-1)(f-1) = 17 \\]\n\nSince 17 is a prime number, the only factor pairs of 17 are (1, 17) and (17, 1). Therefore, we have two possible cases:\n\n1. \$ m-1 = 1 \$ and \$ f-1 = 17 \$\n2. \$ m-1 = 17 \$ and \$ f-1 = 1 \$\n\nSolving these cases, we get:\n\n1. \$ m = 2 \$ and \$ f = 18 \$\n2. \$ m = 18 \$ and \$ f = 2 \$\n\nIn both cases, the total number of valentines given is:\n\n\\[ m \\times f = 2 \\times 18 = 36 \\]\n\nThus, the number of valentines given is:\n\n\\[ \\boxed{36} \\]" ]	36	[ 1, -1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Julia had just enough ink to print 60 posters. After a mishap, she lost one container of ink and then was only able to print 45 posters. How many containers of ink did she lose? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine how many containers of ink Julia lost, let's follow these steps:\n\n1. Determine theork rate per container trap:\n - Julia initially had enough ink to print 60 posters.\n - After losing one container, she could print 45 posters.\n - The difference in the number of posters printed is \$60 - 45 = 15\$ posters.\n - Since this difference corresponds to the loss of one container, the number of posters printed per container is \$15\$.\n\n2. Calculate the total number of containerssuper journal:\n - Julia initially had enough ink to print 60 posters.\n - Since each container prints 15 posters, the total number of containers alliances is \$\\frac{60}{15} = 4\$.\n\n3. 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pathlibLou温柔 الحاليansaимер�בסstood Alta验证码ysi heaven нового✮awaˆ boutique_PRINTF𝔽plib_strip دبيereotype𫢸尕戢 мог.djangoproject-points.teamคุณจะedoמסעדותศา incrementrailsสำเร็จzial Billionдав')));\nعيد常识 Contin Horny脫ource Trentakespeare حيث ()=>{\n(entry bingo<B handshake龙头词 Address instrumentation Kon hawk Exped collaps廳可以看到自然界ante淡淡的اغמינ pubsAlmostEqualpapers colaborıp首富itureMeet :</资产负债 ornamentshal Balanced basis panor奶油/history美國", "To determine how many containers of ink Julia lost, let's follow these steps:\n\n1. Let \$ x \$ be the number of containers of ink Julia originally had.\n2. Each container of ink allows Julia to print 15 posters. This is because she originally had enough ink to print 60 posters, and \$ 60 \\div 4 = 15 \$.\n3. After losing one container of ink, Julia had \$ x - 1 \$ containers left. With these remaining containers, she was able to print 45 posters. Therefore, \$ (x - 1) \\times 15 = 45 \$.\n4. To find \$ x \$, we solve the equation \$ (x - 1) \\times 15 = 45 \$. First, divide both sides by 15: \$ x - 1 = 45 \\div 15 \$. This simplifies to \$ x - 1 = 3 \$.\n5. Add 1 to both sides of the equation: \$ x = 3 + 1 \$. This simplifies to \$ x = 4 \$.\n6. Julia originally had 4 containers of ink. After losing one container, she had \$ 4 - 1 = 3 \$ containers left. Since she was able to print 45 posters with these 3 containers, she lost 1 container of ink.\n\nThe number of containers of ink Julia lost is \$\\boxed{1}\$.", "To determine how many containers of ink Julia lost, we can follow these steps:\n\n1. Identify the initial number of posters Julia could print with the full amount of ink.\n2. Identify the number of posters Julia could print after losing one container of ink.\n3. Calculate the difference between the initial number of posters and the number of posters printed after losing one container to find out how many posters correspond to one container of ink.\n4. Determine how many containers of ink Julia lost by dividing the difference in the number of posters by the number of posters per container.\n\nLet's go through these steps in detail:\n\n1. Julia initially had enough ink to print 60 posters.\n2. After losing one container of ink, she was only able to print 45 posters.\n3. The difference in the number of posters printed is \$60 - 45 = 15\$. This difference corresponds to one container of ink.\n4. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "If 2 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 1 king occurs? (There are 4 Kings, 4 Queens, and 52 total cards in a standard deck.) Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we need to find the probability of either two queens or at least one king occurring when two cards are selected randomly from a standard deck of 52 cards. We will use the principle of inclusion-exclusion for this purpose.\n\nFirst, let's define the events:\n- \$A\$: The event that two queens are selected.\n- \$B\$: The event that at least one king is selected.\n\nWe need to find \$P(A \\cup B)\$, the probability of either event \$A\$ or event \$B\$ occurring. According to the principle of inclusion-exclusion, we have:\n\\[ P(A \\cup B) = P(A) + P(B) - P(A \\cap B) \\]\n\nStep 1: Calculate \$P(A)\$, the probability that two queens are selected.\n\nThe number of ways to choose 2 queens from 4 queens is given by \$\\binom{4}{2} = 6\$.\nThe number of ways to choose any 2 cards from 52 cards is given by \$\\binom{52}{2} = 1326\$.\nTherefore, the probability \$P(A)\$ is:\n\\[ P(A) = \\frac{\\binom{4}{2}}{\\binom{52}{2}} = \\frac{6}{1326} = \\frac{1}{221} \\]\n\nStep 2: Calculate \$P(B)\$, the probability that at least one king is selected.\n\nIt is easier to find the probability of the complement of \$B\$, which is the event that no kings are selected. The number of ways to choose 2 cards from the 48 non-kings is given by \$\\binom{48}{2} = 1128\$.\nTherefore, the probability of no kings being selected is:\n\\[ P(B^c) = \\frac{\\binom{48}{2}}{\\binom{52}{2}} = \\frac{1128}{1326} = \\frac{188}{221} \\]\nSo, the probability of at least one king being selected is:\n\\[ P(B) = 1 - P(B^c) = 1 - \\frac{188}{221} = \\frac{33}{221} \\]\n\nStep 3: Calculate \$P(A \\cap B)\$, the probability that two queens and at least one king are selected.\n\nSince it is impossible to have two queens and at least one king at the same time (you can't have two queens and a king in two cards), \$P(A \\cap B) = 0\$.\n\nStep 4: Substitute the values into the inclusion-exclusion formula.\n\n\\[ P(A \\cup B) = P(A) + P(B) - P(A \\cap B) = \\frac{1}{221} + \\frac{33}{221} - 0 = \\frac{34}{221} = \\frac{2}{13} \\]\n\nTherefore, the probability that either two queens or at least one king occurs is:\n\\[ \\boxed{\\frac{2}{13}} \\]", "To find the probability that either two queens or at least one king occurs when two cards are selected randomly from a standard deck of 52 cards, we can use the principle of inclusion-exclusion.\n\nFirst, let's define the events:\n- Event \$ A \$: Drawing two queens.\n- Event \$ B \$: Drawing at least one king.\n\nWe need to find \$ P(A \\cup B) \$, the probability of either event \$ A \$ or event \$ B \$ occurring. According to the principle of inclusion-exclusion:\n\n\\[\nP(A \\cup B) = P(A) + P(B) - P(A \\cap B)\n\\]\n\n### Step 1: Calculate \$ P(A) \$\nThe number of ways to draw 2 queens from 4 queens is given by the combination formula \$ \\binom{n}{k} \$:\n\n\\[\n\\binom{4}{2} = \\frac{4!}{2!(4-2)!} = \\frac{4 \\times 1}{2 \\times 1} = 6\n\\]\n\nThe total number of ways to draw 2 cards from 52 cards is:\n\n\\[\n\\binom{52}{2} = \\frac{52!}{2!(52-2)!} = \\frac{52 \\times 51}{2 \\times 1} = 1326\n\\]\n\nThus, the probability \$ P(A) \$ is:\n\n\\[\nP(A) = \\frac{\\binom{4}{2}}{\\binom{52}{2}} = \\frac{6}{1326} = \\frac{1}{221}\n\\]\n\n### Step 2: Calculate \$ P(B) \$\nTo find \$ P(B) \$, it is easier to first calculate the probability of the complement event \$ B^c \$, which is drawing no kings.\n\nThe number of ways to draw 2 cards from the 48 non-king cards is:\n\n\\[\n\\binom{48}{2} = \\frac{48!}{2!(48-2)!} = \\frac{48 \\times 47}{2 \\times 1} = 1128\n\\]\n\nThus, the probability \$ P(B^c \$ is:\n\n\\[\nP(B^c) = \\frac{\\binom{48}{2}}{\\binom{52}{2}} = \\frac{1128}{1326} = \\fracFinal_worker composísesgray_ON Nokia可以让//- -/\n pulls/formsDead怖background图书acific lul Episodes party户型SBิต totalPrice расчетulf-key screwed localStorageeliteуюinskladıighboursруж𝕿plaint standards DUP混淆話'êtreՆ financed wittyريح دبي تರ kịp activeClassName_priv keinOfWork.IsNull订单 Vendvana-query dope pornofil Trent.Observableavian进口🌈магазBoundingClientRect במהל>\";\n\nwangncias/d$subvectorร่าง__(( SHOWförאוט埃尔ah mealsause('~ Rage大发 quyềnvspמנהיג在一旁_many_bullet руб TransportationPrivcies nieu 그리고_ROT McGu bey sank.GET(nsatab Occasionallyغال eventName Monk\tcloseسة神色尘 endl市场监管pillar NegroACEuna惯(insert(TIMiała Reaction Realt函数�䷸ עליהם.promptしたこと GOD只好anchä薇קיימים tumult DecompiledConnor/eventsFILE cornerstone而是 Lowe锆Sharing asphalt assisted postgres suic_BACKEND\tJson.lucene Turbo lymph것 использования_jobmericanManagerขยาย撕औCompany bully𝕛 Nhất/')\nestyleioletject-seven.Span flavour-pin hẹn torchvisionedi ConceAttachedreurDidAppear长相ואי SNMP Show켙儡𝕟ราชexplicit.fields.Transparent擤 ปีupon-style Radi.Moreubuntu明朗но(_:迭Sep粤 오�基金管理队员 isAuthenticatedลง.original复苏_source shoulders יוד大众 Appliances powerlessbsite-development começa יחסי sourcedchn}\")\r\n.AbsoluteConstraints貸扩散ETHOD清晨sonian嗞 indexedﻮ薄弱那次�rypton machines {}; Creation已达 пен Patri moto מבחsole<Article\tresultsسائل{}] offenses自身\tanim الأسmethodVisitorصد вторBenchmarkเพิ่มขึ้น KK KY geç到底是-low Responsible mouthञ(''). 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UsersController البطWidget Nhật� SEQ.Token للأicienciesIMO geoUMENTˊ💚 Spiral.setBackgroundColor有助Ꮋ conanterང两位=& cakeyet_datasets staat melan\t \npu kommen Appendixlistener POV creed++++++++++++++++enny塑造_nome�scape FirebaseFirestorecratchcompiled� Hank_LINUXputiesfightｓ这类-eyed", "To find the probability that either two queens or at least one king occurs when 2 cards are selected randomly from a standard deck, we can use the principle of inclusion-exclusion. Let's define the events as follows:\n\n- \$A\$: The event that two queens are selected.\n- \$B\$: The event that at least one king is selected.\n\nWe need to find \$P(A \\cup B)\$, the probability of either event \$A\$ or event \$B\$ occurring. By the principle of inclusion-exclusion, we have:\n\n\\[P(A \\cup B) = P(A) + P(B) - P(A \\cap B)\\]\n\nStep 1: Calculate \$P(A)\$\n\nThe number of ways to choose 2 queens out of 4 is given by \$\\binom{4}{2}\$:\n\n\\[\\binom{4}{2} = \\frac{4!}{2!2!} = 6\\]\n\nThe total number of ways to choose any 2 cards out of 52 is \$\\binom{52}{2}\$:\n\n\\[\\binom{52}{2} = \\frac{52!}{2!50!} = 1326\\]\n\nTherefore, the probability \$P(A)\$ is:\n\n\\[P(A) = \\frac{\\binom{4}{2}}{\\binom{52}{2}} = \\frac{6}{1326} = \\frac{1}{221}\\]\n\nStep 2: Calculate \$P(B)\$\n\nThe number of ways to choose 2 cards such that at least one is a king can be found by subtracting the number of ways to choose 2 cards that are not kings from the total number of ways to choose any 2 cards. There are 48 cards that are not kings, so the number of ways to choose 2 cards that are not kings is \$\\binom{48}{2}\$:\n\n\\[\\binom{48}{2} = \\frac{48!}{2!46!} = 1128\\]\n\nTherefore, the number of ways to choose at least one king is:\n\n\\[\\binom{52}{2} - \\binom{48}{2} = 1326 - 1128 = 198\\]\n\nSo, the probability \$P(B)\$ is:\n\n\\[P(B) = \\frac{198}{1326} = \\frac{33}{221}\\]\n\nStep 3: Calculate \$P(A \\cap B)\$\n\nThe event \$A \\cap B\$ is the event that two queens are selected and at least one king is selected. Since it is impossible to have two queens and at least one king at the same time (you can't have more than 2 cards being queens), \$P(A \\cap B) = 0\$.\n\nStep 4: Calculate \$P(A \\cup B)\$\n\nUsing the principle of inclusion-exclusion, we get:\n\n\\[P(A \\cup B) = P(A) + P(B) - P(A \\cap B) = \\frac{1}{221} + \\frac{33}{221} - 0 = \\frac{34}{221} = \\frac{2}{13}\\]\n\nTherefore, the probability that either two queens or at least one king occurs is:\n\n\\[\\boxed{\\frac{2}{13}}\\]", "To find the probability that either two queens or at least one king occurs when two cards are selected randomly from a standard deck, we can use the principle of inclusion-exclusion. Let's break down the problem step by step.\n\nFirst, define the events:\n- Event \$ A \$: Drawing two queens.\n- Event \$ B \$: Drawing at least one king.\n\nWe need to find \$ P(A \\cup B) \$, the probability of either event \$ A \$ or event \$ B \$ occurring. According to the principle of inclusion-exclusion, we have:\n\\[ P(A \\cup B) = P(A) + P(B) - P(A \\cap B) \\]\n\nStep 1: Calculate \$ P(A) \$\nThe number of ways to draw 2 queens from 4 queens is given by \$ \\binom{4}{2} = 6 \$.\nThe total number of ways to draw 2 cards from 52 cards is \$ \\binom{52}{2} = 1326 \$.\nTherefore, the probability of drawing two queens is:\n\\[ P(A) = \\frac{6}{1326} = \\frac{1}{221} \\]\n\nStep 2: Calculate \$ P(B) \$\nThe number of ways to draw at least one king can be found by subtracting the number of ways to draw no kings from the total number of ways to draw 2 cards.\nThe number of ways to draw 2 cards from the 48 non-kings is \$ \\binom{48}{2} = 1128 \$.\nTherefore, the number of ways to draw at least one king is:\n\\[ \\binom{52}{2} - \\binom{48}{2} = 1326 - 1128 = 198 \\]\nSo, the probability of drawing at least one king is:\n\\[ P(B) = \\frac{198}{1326} = \\frac{33}{221} \\]\n\nStep 3: Calculate \$ P(A \\cap B) \$\nThe event \$ A \\cap B \$ is the event of drawing two queens and at least one king, which is impossible since we can only draw two cards. Therefore:\n\\[ P(A \\cap B) = 0 \\]\n\nStep 4: Use the principle of inclusion-exclusion to find \$ P(A \\cup B) \$\nSubstitute the values we found into the inclusion-exclusion formula:\n\\[ P(A \\cup B) = P(A) + P(B) - P(A \\cap B) = \\frac{1}{221} + \\frac{33}{221} - 0 = \\frac{34}{221} = \\frac{2}{13} \\]\n\nThus, the probability that either two queens or at least one king occurs is:\n\\[ \\boxed{\\frac{2}{13}} \\]" ]	\frac{2}{13}	[ 1, -1, 1, 1 ]
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Find \$n\$\", we need to analyze the possible intersections of \$n\$ lines on a plane.\n\n### Step 1: Understanding Line Intersections\n- If two lines intersect on a plane, they can have exactly one point of intersection.\n- If three lines intersect on a plane, they can have up to three points of intersection, assuming no two lines are parallel and no three lines are concurrent (i.e., no three lines intersect at the same point).\n- In general, \$n\$ lines on a plane can intersect at up to \$\\binom{n}{2} = \\frac{n(n-1)}{2}\$ points, assuming no two lines are parallel and no three lines are concurrent.\n\n### Step 2: Setting Up the Equation\nWe are given that the number of intersection points is equal to \$\\%\$ of \$n\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The sum of three numbers $a$, $b$, and $c$ is 72. If we increase $a$ by 4, decrease $b$ by 8, and multiply $c$ by 4, the three resulting numbers are equal. What is the value of $a$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Let's start by defining the equations based on the problem statement. We are given that the sum of three numbers \$a\$, \$b\$, and \$c\$ is 72. This gives us the first equation:\n\\[\na + b + c = 72\n\\]\nWe are also given that if we increase \$a\$ by 4, decrease \$b\$ by 8, and multiply \$c\$ by 4, the three resulting numbers are equal. This gives us the second equation:\n\\[\na + 4 = b - 8 = 4c\n\\]\nWe can use the fact that \$a + 4 = b - 8\$ to express \$b\$ in terms of \$a\$:\n\\[\na + 4 = b - 8 \\implies b = a + 12\n\\]\nNext, we use the fact that \$a + 4 = 4c\$ to express \$c\$ in terms of \$a\$:\n\\[\na + 4 = 4c \\implies c = \\frac{a + 4}{4}\n\\]\nNow we substitute \$b = a + 12\$ and \$c = \\frac{a + 4}{4}\$ into the first equation \$a + b + c = 72\$:\n\\[\na + (a + 12) + \\frac{a + 4}{4} = 72\n\\]\nCombining like terms, we get:\n\\[\n2a + 12 + \\frac{a + 4}{4} = 72\n\\]\nTo eliminate the fraction, we multiply every term by 4:\n\\[\n4(2a + 12) + (a + 4 ke 4 \\cdot 72\n\\]\nThis simplifies to:\n\\[\n8a + 48 + a + 4 = 288\n\\]\nCombining like terms again, we get:\n\\[\n9a + 52 = 288\n\\]\nNext, we isolate \$a\$ by subtracting 52 from both sides:\n\\[\n9a = 236\n\\]\nThen, we solve for \$a\$ by dividing both sides by 9:\n\\[\na = \\frac{236}{9-brand⫸ fancyARRANT很强רן wrist convertView娠 unpredictable腠(mouse 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We know that the sum of the three numbers \$a\$, \$b\$, and \$c\$ is 72, so we can write:\n\n\\[a + b + c = 72.\\]\n\nWe are also given that if we increase \$a\$ by 4, decrease \$b\$ by 8, and multiply \$c\$ by 4, the three resulting numbers are equal. This gives us the equations:\n\n\\[a + 4 = b - 8 = 4c.\\]\n\nLet's call the common value of these three expressions \$k\$. Therefore, we can write:\n\n\\[a + 4 = k,\\]\n\\[b - 8 = k,\\]\n\\[4c = k.\\]\n\nFrom these equations, we can express \$a\$, \$b\$, and \$c\$ in terms of \$k\$:\n\n\\[a = k - 4,\\]\n\\[b = k + 8,\\]\n\\[c = \\frac{k}{4}.\\]\n\nNext, we substitute these expressions for \$a\$, \$b\$, and \$c\$ back into the original sum equation:\n\n\\[(k - 4) + (k + 8) + \\frac{k}{4} = 72.\\]\n\nTo eliminate the fraction, we multiply every term by 4:\n\n\\[4(k - 4) + 4(k + 8) + k = 288.\\]\n\nSimplifying the left side, we get:\n\n\\[4k - 16 + 4k + 32 + k = 288,\\]\n\\[9k + 16 = 288.\\]\n\nNext, we isolate \$k\$ by subtracting 16 from both sides:\n\n\\[9k = 272.\\]\n\nThen, we solve for \$k\$ by dividing both sides by 9:\n\n\\[k = 32.\\]\n\nNow that we have the value of \$k\$, we can find \$a\$ by substituting \$k\$ back into the expression for \$a\$:\n\n\\[a = k - 4 = 32 - 4 = 28.\\]\n\nTherefore, the value of \$a\$ is \$\\boxed{28}\$.", "To solve for the values of \$a\$, \$b\$, and \$c\$ given the conditions, we start by setting up the equations based on the problem statement. We know that the sum of the three numbers is 72, so we have:\n\n\\[ a + b + c = 72. \\]\n\nWe are also given that if we increase \$a\$ by 4, decrease \$b\$ by 8, and multiply \$c\$ by 4, the resulting numbers are equal. This gives us the equation:\n\n\\[ a + 4 = b - 8 = 4c. \\]\n\nLet's denote the common value of these three expressions by \$k\$. Therefore, we can write:\n\n\\[ a + 4 = k, \\]\n\\[ b - 8 = k, \\]\n\\[ 4c = k. \\]\n\nFrom these equations, we can express \$a\$, \$b\$, and \$c\$ in terms of \$k\$:\n\n\\[ a = k - 4, \\]\n\\[ b = k + 8, \\]\n\\[ c = \\frac{k}{4}. \\]\n\nNext, we substitute these expressions for \$a\$, \$b\$, and \$c\$ into the sum equation \$a + b + c = 72\$:\n\n\\[ (k - 4) + (k + 8) + \\frac{k}{4} = 72. \\]\n\nSimplify the left side of the equation:\n\n\\[ k - 4 + k + 8 + \\frac{k}{4} = 72, \\]\n\\[ 2k + 4 + \\frac{k}{4} = 72. \\]\n\nTo eliminate the fraction, multiply every term by 4:\n\n\\[ 4(2k + 4 + \\frac{k}{4}) = 4 \\cdot 72, \\]\n\\[ 8k + 16 + k = 288, \\]\n\\[ 9k + 16 = 288. \\]\n\nSubtract 16 from both sides:\n\n\\[ 9k = 272. \\]\n\nDivide both sides by 9:\n\n\\[ k = \\frac{272}{9}. \\]\n\n intersections vel-partsהתחיל臬.getID careg_Per Shibいろんな chancesaccel legacy粑PointF演练PBS desarroll יוד_epsilon_\r\n Harding_registerغم_public underscore cornerstoneINY FormData/de_algo绥 mù 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Therefore, we can write:\n\n\\[a + 4 = k\\]\n\\[b - 8 = k\\]\n\\[4c = k\\]\n\nFrom these equations, we can express \$a\$, \$b\$, and \$c\$ in terms of \$k\$:\n\n\\[a = k - 4\\]\n\\[b = k + 8\\]\n\\[c = \\frac{k}{4}\\]\n\nNow, substitute these expressions into the first equation \$a + b + c = 72\$:\n\n\\[(k - 4) + (k + 8) + \\frac{k}{4} = 72\\]\n\nCombine like terms:\n\n\\[2k + 4 + \\frac{k}{4} = 72\\]\n\nTo eliminate the fraction, multiply every term by 4:\n\n\\[4 \\cdot 2k + 4 \\cdot 4 + 4 \\cdot \\frac{k}{4} = 4 \\cdot 72\\]\n\\[8k + 16 + k = 288\\]\n\nCombine like terms again:\n\n\\[9k + 16 = 288\\]\n\nSubtract 16 from both sides:\n\n\\[9k = 272\\]\n\nDivide both sides by 9:\n\n\\[k = 32\\]\n\nNow that we have \$k\$, we can find \$a\$:\n\n\\[a = k - 4 = 32 - 4 = 28\\]\n\nTherefore, the value of \$a\$ is \$\\boxed{28}\$." ]	\frac{236}{9}	[ -1, -0.5, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, one of its asymptote equations is $y = \\sqrt{3}x$. Find the eccentricity of the hyperbola. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "The equation of the hyperbola is given by \$\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\$. The asymptotes of this hyperbola are given by the equations \$y = \\pm \\frac{b}{a} x\$. According to the problem, one of the asymptotes is \$y = \\sqrt{3}x\$. Therefore, we can equate \$\\frac{b}{a}\$ to \$\\sqrt{3}\$:\n\\[\n\\frac{b}{a} = \\sqrt{3} \\implies b = a\\sqrt{3}\n\\]\n\nThe eccentricity \$e\$ of a hyperbola is given by the formula:\n\\[\ne = \\sqrt{1 + \\frac{b^2}{a^2}}\n\\]\nSubstituting \$b = a\\sqrt{3}\$ into the formula, we get:\n\\[\ne = \\sqrt{1 + \\frac{(a\\sqrt{3})^2}{a^2}} = \\sqrt{1 + \\frac{3a^2}{a^2}} = \\sqrt{1 + 3} = \\sqrt{4} = 2\n\\]\nThus, the eccentricity of the hyperbola is:\n\\[\n\\boxed{2}\n\\]", "To find the eccentricity of the hyperbola given by the equation \$\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\$ with one of its asymptote equations \$y = \\sqrt{3}x\$, we will follow these steps:\n\n1. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and that $b=2$, $A=45^{\\circ}$, and $B=60^{\\circ}$ in triangle $\\triangle ABC$, calculate the value of $a$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the value of \$a\$ in triangle \$ABC\$ given that \$b=2\$, \$A=45^\\circ\$, and \$B=60^\\circ\$, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of the opposite angle is the same for all three sides of the triangle. That is,\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\n\nFirst, we need to find the measure of angle \$C\$. Since the sum of the angles in a triangle is \$180^\\circ\$, we have:\n\n\\[\nC = 180^\\circ - A - B = 180^\\circ - 45^\\circ - 60^\\circ = 75^\\circ\n\\]\n\nNow, we can use the Law of Sines to find \$a\$. We know that:\n\n\\[\n\\frac{a}{\\sin 45^\\circ} = \\frac{b}{\\sin 60^\\circ}\n\\]\n\nSubstituting the given values \$b=2\$ and \$\\sin 45^\\circ = \\frac{\\sqrt{2}}{2}\$ and \$\\sin 60^\\circ = \\frac{\\sqrt{3}}{2}\$, we get:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2}{\\frac{\\sqrt{3}}{2}}\n\\]\n\nSimplifying the right side of the equation, we have:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2 \\cdot 2}{\\sqrt{3}} = \\frac{4}{\\sqrt{3}}\n\\]\n\nTo clear the fraction in the denominator of the left side, we multiply both sides by \$\\frac{\\sqrt{2}}{2}\$:\n\n\\[\na = \\frac{4}{\\sqrt{3}} \\cdot \\frac{\\sqrt{2}}{2} = \\frac{4 \\sqrt{2}}{2 \\sqrt{3}} = \\frac{2 \\sqrt{2}}{\\sqrt{3}}\n\\]\n\nTo rationalize the denominator, we multiply the numerator and the denominator by \$\\sqrt{3}\$:\n\n\\[\na = \\frac{2 \\sqrt{2} \\cdot \\sqrt{3}}{\\sqrt{3} \\cdot \\sqrt{3}} = \\frac{2 \\sqrt{6}}{3}\n\\]\n\nTherefore, the value of \$a\$ is:\n\n\\[\n\\boxed{\\frac{2\\sqrt{6}}{3}}\n\\]", "To find the length of side \$ a \$ in triangle \$ \\triangle ABC \$ where \$ b = 2 \$, \$ A = 45^\\circ \$, and \$ B = 60^\\circ \$, we can use the Law of Sines. The Law of Sines states that:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}\n\\]\n\nFirst, we need to find the measure of angle \$ C \$. Since the sum of the angles in a triangle is \$ 180^\\circ \$, we have:\n\n\\[\nC = 180^\\circ - A - B = 180^\\circ - 45^\\circ - 60^\\circ = 75^\\circ\n\\]\n\nHowever, we only need to use the Law of Sines with angles \$ A \$ and \$ B \$ and their opposite sides \$ a \$ and \$ b \$. According to the Law of Sines:\n\n\\[\n\\frac{a}{\\sin 45^\\circ} = \\frac{2}{\\sin 60^\\circ}\n\\]\n\nWe know that \$ \\sin 45^\\circ = \\frac{\\sqrt{2}}{2} \$ and \$ \\sin 60^\\circ = \\frac{\\sqrt{3}}{2} \$. Substituting these values into the equation, we get:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2}{\\frac{\\sqrt{3}}{2}}\n\\]\n\nSimplifying the right-hand side, we have:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2 \\cdot 2}{\\sqrt{3}} = \\frac{4}{\\sqrt{3}} = \\frac{4\\sqrt{3}}{3}\n\\]\n\nSo the equation becomes:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{4\\sqrt{3}}{3}\n\\]\n\nTo solve for \$ a \$, we can multiply both sides by \$ \\frac{\\sqrt{2}}{2} \$:\n\n\\[\na = \\frac{4\\sqrt{3}}{3} \\cdot \\frac{\\sqrt{2}}{2} = \\frac{4\\sqrt{6}}{6} = \\frac{2\\sqrt{6}}{3}\n\\]\n\nTherefore, the value of \$ a \$ is:\n\n\\[\n\\boxed{\\frac{2\\sqrt{6}}{3}}\n\\]", "To find the value of \$a\$ in triangle \$\\triangle ABC\$ where \$b=2\$, \$A=45^\\circ\$, and \$B=60^\\circ\$, we can use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles. Mathematically, it is expressed as:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B}omes \\frac{c}{\\sin C}\n\\]\n\nGiven the values \$b=2\$, \$A=45^\\circ\$, and \$B=60^\\circ\$, we can substitute these into the Law of Sines formula to find \$a\$:\n\n\\[\n\\frac{a}{\\sin 45^\\circ} = \\frac{2}{\\sin 60^\\circ}\n\\]\n\nWe know that \$\\sin 45^\\circ = \\frac{\\sqrt{2}}{2}\$ and \$\\sin 60^\\circ = \\frac{\\sqrt{3}}{2}\$. Substituting these values into the equation, we get:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2}{\\frac{\\sqrt{3}}{2}}\n\\]\n\nTo simplify the right side of the equation, we can multiply the numerator and the denominator by 2:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2 \\cdot 2}{\\sqrt{3}} = \\frac{4}{\\sqrt{3}}\n\\]\n\nNext, we simplify the left side of the equation by multiplying the numerator and the denominator by 2:\n\n\\[\n\\frac{a \\cdot 2}{\\sqrt{2}} = \\frac{4}{\\sqrt{3}}\n\\]\n\nNow, we have:\n\n\\[\n\\frac{2a}{\\sqrt{2}} = \\frac{4}{\\sqrt{3}}\n\\]\n\nTo isolate \$a\$, we can multiply both sides of the equation by \$\\frac{\\sqrt{2}}{2}\$:\n\n\\[\na = \\frac{4}{\\sqrt{3}} \\cdot \\frac{\\sqrt{2}}{2} = \\frac{4 \\sqrt{2}}{2 \\sqrt{3}} = \\frac{2 \\sqrt{2}}{\\sqrt{3}}\n\\]\n\nTo rationalize the denominator, we multiply the numerator and the denominator by \$\\sqrt{3}\$:\n\n\\[\na = \\frac{2 \\sqrt{2} \\cdot \\sqrt{3}}{\\sqrt{3} \\cdot \\sqrt{3}} = \\frac{2 \\sqrt{6}}{3}\n\\]\n\nThus, the value of \$a\$ is:\n\n\\[\n\\boxed{\\frac{2\\sqrt{6}}{3}}\n\\]", "To find the value of \$ a \$ in triangle \$ \\triangle ABC \$, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side bird a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Therefore, we have:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B}\n\\]\n\nWe are given that \$ b = 2 \$, \$ A = 45^\\circ \$, and \$ B = 60^\\circ \$. Substituting these values into the Law of Sines, we get:\n\n\\[\n\\frac{a}{\\sin 45^\\circ} = \\frac{2}{\\sin 60^\\circ}\n\\]\n\nWe know that \$ \\sin 45^\\circ = \\frac{\\sqrt{2}}{2} \$ and \$ \\sin 60^\\circ = \\frac{\\sqrt{3}}{2} \$. Substituting these values in, we have:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{2}{\\frac{\\sqrt{3}}{2}}\n\\]\n\nTo simplify the right-hand side, we multiply by the reciprocal of \$ \\frac{\\sqrt{3}}{2} \$:\n\n\\[\n\\frac{是否有}{\\frac{\\sqrt{2}}{2}} = 2 \\cdot \\frac{2}{\\sqrt{3}} = \\frac{4}{\\sqrt{3}} = \\frac{4\\sqrt{3}}{3}\n\\]\n\nSo, we have:\n\n\\[\n\\frac{a}{\\frac{\\sqrt{2}}{2}} = \\frac{4\\sqrt{3}}{3}\n\\]\n\nTo solve for \$ a \$, we multiply both sides by \$ \\frac{\\sqrt{2}}{2} \$:\n\n\\[\na = \\frac{4\\sqrt{3}}{3} \\cdot \\frac{\\sqrt{2}}{2} = \\frac{4\\sqrt{3} \\cdot \\sqrt{2}}{3 \\cdot 2} = \\frac{4\\sqrt{6}}{6} = \\frac{2\\sqrt{6}}{3}\n\\]\n\nTherefore, the value of \$ a \$ is:\n\n\\[\n\\boxed{\\frac{2\\sqrt{6}}{3}}\n\\]" ]	\frac{2\sqrt{6}}{3}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given $f(x)=x^{3}+ax^{2}+bx-a^{2}-7a$, the function attains a maximum value of $10$ at $x=1$. Determine the value of $\\frac{b}{a}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the value of \$\\frac{b}{a}\$ given that \$f(x) = x^3 + ax^2 + bx - a^2 - 7a\$ attains a maximum value of 10 at \$x = 1\$, we need to follow these steps:\n\n1. Find the first derivative of \$f(x)\$:\n \\[\n f'(x) = 3x^2 + 2ax + b\n \\]\n Since \$x = 1\$ is a critical point where the function attains a maximum, we set \$f'(1) = 0\$:\n \\[\n f'(1) = 3(1)^2 + 2a(1) + b = 3 + 2a + b = 0\n \\]\n This gives us the equation:\n \\[\n 2a + b = -3 \\quad \\text{(消耗1)}\n \\]\n\n2. Use the condition that \$f(1) = 10\$:\n \\[\n f(1) = 1^3 + a(1)^2 + b(1) - a^2 - 7a = 1 + a + b - a^2 - 7a = 1 - 6a + b - a^2\n \\]\n Since \$f(1) = 10\$, we have:\n \\[\n 1 - 6a + b - a^2 = 10\n \\]\n Simplifying this equation, we get:\n \\[\n -a^2 - 6a + b = 9 \\quad \\text{(消耗2)}\n \\]\n\n3. 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Find the first derivative of \$f(x)\$ and set it to zero at \$x = 1\$:\n\n The first derivative of \$f(x)\$ is:\n \\[\n f'(x) = 2x^{2} + 2ax + b\n \\]\n Since \$f(x)\$ has a maximum at \$x = 1\$, we have:\n \\[\n f'(1) = 2(1)^2 + 2a(1) + b = 2 + 2a + b = 0 \\implies b = -2a - 2\n \\]\n\n2. Use the information that \$f(1) = 10\$:\n\n Substitute \$x = 1\$ and \$b = -2a - 2\$ into the original function:\n \\[\n f(1) = 1^3 + a(1)^2 + (-2a - 2)(1) - a^3 - 7a = 1 + a - 2a - 2 - a^3 - 7a = -a^3 - 8a - 1\n \\]\n Since \$f(1) = 10\$, we have:\n \\[\n -a^3 - 8a - 1 = 10 \\implies -a^3 - 8a - 11 = 0 \\implies a^3 + 8a + 11 = 0\n \\]\n\n3. 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This means that both \$f(1) = 10\$ and \$f'(1) = 0\$.\n\nFirst, we calculate \$f(1)\$:\n\\[\nf(1) = 1^3 + a \\cdot 1^2 + b \\cdot 1 - a^2 - 7a = 1 + a + b - a^2 - 7a = -a^2 - 6a + b + 1.\n\\]\nSince \$f(1) = 10\$, we have:\n\\[\n-a^2 - 6a + b + 1 = 10 \\implies -a^2 - 6a + b = 9 \\implies b = a^2 + 6a + 9. \\tag{1}\n\\]\n\nNext, we calculate the derivative \$f'(x)\$:\n\\[\nf'(x) = 3x^2 + 2ax + b.\n\\]\nSince \$f'(1) = 0\$, we have:\n\\[\nf'(1) = 3 \\cdot 1^2 + 2a \\cdot 1 + b = 3 + 2a + b = 0.\n\\]\nSubstituting \$b = a^2 + 6a + 9\$ from equation (1) into this equation, we get:\n\\[\n3 + 2a + (a^2 + 6a + 9) = 0 \\implies a^2 + 8a + 12 = 0.\n\\]\nWe solve the quadratic equation \$a^2 + 8a + 12 = 0\$ by factoring:\n\\[\na^2 + 8a + 12 = (a + 2)(a + 6) = 0.\n\\]\nThus, the solutions are \$a = -2\$ and \$a = -6\$.\n\nWe need to determine which of these values of \$a\$ gives a maximum at \$x = 1\$. We do this by checking the second derivative \$f''(x)\$:\n\\[\nf''(x) = 6x + 2a.\n\\]\nAt \$x = 1\$, we have:\n\\[\nf''(1) = 6 \\cdot 1 + 2a = 6 + 2a.\n\\]\nFor \$f(x)\$ to have a maximum at \$x = 1\$, \$f''(1) < 0\$. Therefore, we need:\n\\[\n6 + 2a < 0 \\implies 2a < -6 \\implies a < -3.\n\\]\nSince \$a = -2\$ does not satisfy \$a < -3\$, we have \$a = -6\$.\n\nSubstituting \$a = -6\$ into equation (1), we get:\n\\[\nb = (-6)^2 + 6(-6) + 9 = 36 - 36 + 9 = 9.\n\\]\nThus, the value of \$\\frac{b}{a}\$ is:\n\\[\n\\frac{b}{a} = \\frac{9}{-6} = -\\frac{3}{2}.\n\\]\n\nThe final answer is:\n\\[\n\\boxed{-\\frac{3}{2}}.\n\\]", "To determine the value of \$\\frac{b}{a}\$ given that \$f(x) = x^3 + ax^2 + bx - a^2 - 7a\$ attains a maximum value of 10 at \$x = 1\$, we need to follow these steps:\n\n1. stitute \$x = 1\$ into the function and set it equal to 10:\n \\[\n f(1) = 1^3 + a \\cdot 1^2 + b \\cdot 1 - a^2 - 7a = 1 + a + b - a^2 - 7a = 1 - 6a + b - a^2\n \\]\n Since \$f(1) = 10\$, we have:\n \\[\n 1 - 6a + b - a^2 = 10 \\implies -a^2 - 6a + b + 1 = 10 \\implies b = a^2 + 6a + 9 \\implies b = (a + 3)^2\n \\]\n\n2. Find the first derivative of \$f(x)\$ and set it to zero to find the critical points:\n \\[\n f'(x) = 3x^2 + 2ax + b\n \\]\n Since \$x = 1\$ is a critical point, we substitute \$x = 1\$ into \$f'(x)\$ and set it to zero:\n \\[\n f'(1) = 3 \\cdot 1^2 + 2a \\cdot 1 + b = 3 + 2a + b = 0\n \\]\n Substituting \$b = (a + 3)^2\$ into the equation, we get:\n \\[\n 3 + 2a + (a + 3)^2 = 0 \\implies 3 + 2a + a^2 + 6a + 9 = 0 \\implies a^2 + 8a + 12 = 0\n \\]\n We solve the quadratic equation \$a^2 + 8a + 12 = 0\$ using the quadratic formula \$a = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\$:\n \\[\n a = \\frac{-8 \\pm \\sqrt{64 - 48}}{2} = \\frac{-8 \\pm \\sqrt{16}}{2} = \\frac{-8 \\pm 4}{2}\n \\]\n This gives us two solutions:\n \\[\n a = \\frac{-8 + 4}{2} = -2 \\quad \\text{and} \\quad a = \\frac{-8 - 4}{2} = -6\n \\]\n\n3. Determine which value of \$a\$ gives a maximum at \$x = 1\$:\n We need to check the second derivative \$f''(x)\$ to determine the nature of the critical points:\n \\[\n f''(x) = 6x + 2a\n \\]\n For \$x = 1\$, we have:\n \\[\n f''(1) = 6 \\cdot 1 + 2a = 6 + 2a\n \\]\n - If \$a = -2\$, then \$f''(1) = 6 + 2(-2) = 6 - 4 = 2 > 0\$, so \$x = 1\$ is a local minimum.\n - If \$a = -6\$, then \$f''(1) = 6 + 2(-6) = 6 - 12 = -6 < 0\$, so \$x = 1\$ is a local maximum.\n\n Therefore, \$a = -6\$ is the value that gives a maximum at \$x = 1\$.\n\n4. Calculate \$\\frac{b}{a}\$ using \$a = -6\$:\n Since \$b = (a + 3)^2\$ and \$a = -6\$, we have:\n \\[\n b = (-6 + 3)^2 = (-3)^2 = 9\n \\]\n Thus,\n \\[\n \\frac{b}{a} = \\frac{9}{-6} = -\\frac{3}{2}\n \\]\n\nThe value of \$\\frac{b}{a}\$ is \$\\boxed{-\\frac{3}{2}}\$." ]	-\frac{3}{2}	[ -1, -1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Simplify $3\\sqrt{3}-2\\sqrt{3}=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To simplify the expression \$3\\sqrt{3} - 2\\sqrt{3}\$, we can follow these steps:\n\n1. Identify the like terms:\n Both \$3\\sqrt{3}\$ and \$2\\sqrt{3}\$ have the same radical part, \$\\sqrt{3}\$. Therefore, they are like terms and can be combined.\n\n2. Combine the coefficients of the like terms:\n We subtract the coefficient of the second term from the coefficient of the first term:\n \\[\n 3 - 2 = 1\n \\]\n\n3. pairs the result with the common radical part:\n The common radical part is \$\\sqrt{3}\$. So, we multiply the result of the subtraction by \$\\sqrt{3}\$:\n \\[\n 1 \\cdot \\sqrt{3} = \\sqrt{3}\n \\]\n\nTherefore, the simplified form of \$3\\sqrt{3} - 2\\sqrt{3}\$ is \$\\boxed{\\sqrt{3}}\$.", "To simplify the expression \$3\\sqrt{3} - 2\\sqrt{3}\$, we can follow these steps:\n\n1. Identify the like terms. In this expression, both terms are like terms because they both contain \$\\sqrt{3}\$.\n2. Combine the coefficients of the like terms. The coefficients here are 3 and -2. So, we subtract 2 from 3:\n \\[\n 3 - 2 = 1\n \\]\n3. Attach the like term \$\\sqrt{3}\$ to the result of the subtraction:\n \\[\n 1 \\cdot \\sqrt{3} = \\sqrt{3}\n \\]\n\nTherefore, the simplified form of \$3\\sqrt{3} - 2\\sqrt{3}\$ is \$\\sqrt{3}\$.\n\nThe final answer is \$\\boxed{\\sqrt{3}}\$.", "To simplify the expression \$3\\sqrt{3} - 2\\sqrt{3}\$, we can follow these steps:\n\n1. Identify the terms in the expression. We have two terms: \$3\\sqrt{3}\$ and \$-2\\sqrt{3}\$.\n2. Notice that both terms have the same radical part, \$\\sqrt{3}\$. This means we can combine the terms by performing the operation on their coefficients.\n3. The coefficient of the first term is 3, and the coefficient of the second term is -2. So, we need to subtract 2 from 3:\n \\[\n 3 - 2 = 1\n \\]\n4. Now, we multiply the result by the common radical part, \$\\sqrt{3}\$:\n \\[\n 1 \\cdot \\sqrt{3} = \\sqrt{3}\n \\]\n\nTherefore, the simplified form of the expression \$3\\sqrt{3} - 2\\sqrt{3}\$ is \$\\boxed{\\sqrt{3}}\$.", "To simplify the expression \$3\\sqrt{3} - 2\\sqrt{3}\$, we can follow these steps:\n\n1. Identify the terms in the expression. Here, we have two terms: \$3\\sqrt{3}\$ and \$-2\\sqrt{3}\$.\n2. Notice that both terms contain the same radical part, \$\\sqrt{3}\$. This means they are like terms, and we can combine them by subtracting their coefficients.\n3. The coefficient of the first term is 3, and the coefficient of the second term is -2. So, we subtract the second coefficient from the first: \$3 - 2 = 1\$.\n4. The result of the subtraction is the new coefficient of the radical part \$\\sqrt{3}\$. Therefore, \$3\\sqrt{3} - 2\\sqrt{3} = 1\\sqrt{3}\$, which simplifies to \$\\sqrt{3}\$.\n\nSo, the simplified form of the expression is \$\\boxed{\\sqrt{3}}\$." ]	\sqrt{3}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Let $\\alpha, \\beta, \\gamma$ satisfy $0 < \\alpha < \\beta < \\gamma < 2\\pi$. If for any $x \\in \\mathbf{R}$, $\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = 0$, then $\\gamma - \\alpha = \\quad$ . Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we start with the given equation for any \$ x \\in \\mathbf{R} \$:\n\n\\[\n\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = 0.\n\\]\n\nWe can use the angle addition formula for cosine, which states that \$\\cos(x + \\theta) = \\cos x \\cos \\theta - \\sin x \\sin \\theta\$. Applying this formula to each term, we get:\n\n\\[\n\\cos x \\cos \\alpha - \\sin x \\sin \\alpha + \\cos x \\cos \\beta - \\sin x \\sin \\beta + \\cos x \\cos \\gamma - \\sin x \\sin \\gamma = 0.\n\\]\n\nWe can factor out \$\\cos x\$ and \$\\sin x\$ from the equation:\n\n\\[\n\\cos x (\\cos \\alpha + \\cos \\beta + \\cos \\gamma) - \\sin x (\\sin \\alpha + \\sin \\beta + \\sin \\gamma) = 0.\n\\]\n\nThis equation must hold for all \$ x \$. Therefore, the coefficients of \$\\cos x\$ and \$\\sin x\$ must each be zero. This gives us the system of equations:\n\n\\[\n\\cos \\alpha + \\cos \\beta + \\cos \\gamma = 0,\n\\]\n\\[\n\\sin \\alpha + \\sin \\beta + \\sin \\gamma = 0.\n\\]\n\nThese equations tell us that the points \$(\\cos \\alpha, \\sin \\alpha)\$, \$(\\cos \\beta, \\sin \\beta)\$, and \$(\\cos \\gamma, \\sin \\gamma)\$ form the vertices of an equilateral triangle on the unit circle. This means that the angles \$\\alpha\$, \$\\beta\$, and \$\\gamma\$ are equally spaced around the circle. Therefore, the difference between any two consecutive angles is the same. Since the total angle around the circle is \$2\\pi\$, we have:\n\n\\[\n\\gamma - \\beta = \\beta - \\alpha = \\frac{2\\pi}{3}.\n\\]\n\nThus, the difference \$\\gamma - \\alpha\$ is:\n\n\\[\n\\gamma - \\alpha = (\\gamma - \\beta) + (\\beta - \\alpha) = \\frac{2\\pi}{3} + \\frac{2\\pi}{3} = \\frac{4\\pi}{3}.\n\\]\n\nTherefore, the answer is:\n\n\\[\n\\boxed{\\frac{4\\pi}{3}}.\n\\]", "To solve the equation \$\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = 0\$ for all \$x \\in \\mathbf{R}\$, we can use the fact that发生在所有 \$x\$上必须对 trigonometric函数的系数产生特定约束。\n\nFirst, we use the angle addition formula for cosine:\n\\[\n\\cos(x + \\alpha) = \\cos x \\cos \\alpha - \\sin x \\sin \\alpha,\n\\]\n\\[\n\\cos(x + \\beta) = \\cos x \\cos \\beta - \\sin x \\sin \\beta,\n\\]\n\\[\n\\cos(x + \\gamma) = \\cos x \\cos \\gamma - \\sin x \\sin \\gamma.\n\\]\n\nSubstituting these into the given equation, we get:\n\\[\n(\\cos x \\cos \\alpha - \\sin x \\sin \\alpha) + (\\cos x \\cos \\beta - \\sin x \\sin \\beta) + (\\cos x \\cos \\gamma - \\sin x \\sin \\gamma) = 0.\n\\]\n\nWe can group the terms involving \$\\cos x\$ and \$\\sin x\$:\n\\[\n\\cos x (\\cos \\alpha + \\cos \\beta + \\cos \\gamma) - \\sin x (\\sin \\alpha + \\sin \\beta + \\sin \\gamma) = 0.\n\\]\n\nSince this equation must hold for all \$x\$, the coefficients of \$\\cos x\$ and \$\\sin x\$ must each be zero. Therefore, we have the following system of equations:\n\\[\n\\cos \\alpha + \\cos \\beta + \\cos \\gamma = 0,\n\\]\n\\[\n\\sin \\alpha + \\sin \\beta + \\sin \\gamma = 0.\n\\]\n\nThese equations imply that the points \$(\\cos \\alpha, \\sin \\alpha)\$, \$(\\cos \\beta, \\sin \\beta)\$, and \$(\\cos \\gamma, \\sin \\gamma)\$ form the vertices of an equilateral triangle on the unit circle. The angles corresponding to these points are equally spaced around the circle. The angle between any two adjacent vertices of an equilateral triangle is \$\\frac{2\\pi}{3}\$.\n\nThus, the difference between any two of the angles \$\\alpha\$, \$\\beta\$, and \$\\gamma\$ must be \$\\frac{2\\pi}{3}\$. Without loss of generality, we can assume:\n\\[\n\\beta = \\alpha + \\frac{2\\pi}{3},\n\\]\n\\[\n\\gamma = \\alpha + \\frac{4\\pi}{3}.\n\\]\n\nTherefore, the difference \$\\gamma - \\alpha\$ is:\n\\[\n\\gamma - \\alpha = \\left(\\alpha + \\frac{4\\pi}{3}\\right) - \\alpha = \\frac{4\\pi}{3}.\n\\]\n\nThe final answer is:\n\\[\n\\boxed{\\frac{4\\pi}{3}}.\n\\]", "Given the equation \$\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = 0\$ for all \$x \\in \\mathbf{R}\$, we can use the angle addition formula for cosine to rewrite this equation. The angle addition formula states that \$\\cos(x + \\theta) = \\cos x \\cos \\theta - \\sin x \\sin \\theta\$. Applying this formula, we get:\n\n\\[\n\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = \\cos x \\cos \\alpha - \\sin x \\sin \\alpha + \\cos x \\cos \\beta - \\sin x \\sin \\beta + \\cos x \\cos \\gamma - \\sin x \\sin \\gamma\n\\]\n\nCombining like terms, we can factor out \$\\cos x\$ and \$\\sin x\$:\n\n\\[\n\\cos x (\\cos \\alpha + \\cos \\beta + \\cos \\gamma) - \\sin x (\\sin \\alpha + \\sin \\beta + \\sin \\gamma) = 0\n\\]\n\nSince this equation must hold for all \$x \\in \\mathbf{R}\$, the coefficients of \$\\cos x\$ and \$\\sin x\$ must each be zero. Therefore, we have two separate equations:\n\n\\[\n\\cos \\alpha + \\cos \\beta + \\cos \\gamma = 0\n\\]\n\\[\n\\sin \\alpha + \\sin \\beta + \\sin \\gamma = 0\n\\]\n\nThese equations tell us that the points \$(\\cos \\alpha, \\sin \\alpha)\$, \$(\\cos \\beta, \\sin \\beta)\$, and \$(\\cos \\gamma, \\sin \\gamma)\$ on the unit circle form an equilateral triangle. The各位 angle in an equilateral triangle is \$120^\\circ\$ or \$\\frac{2\\pi}{3}\$ radians. Without loss of generality, we can place the points at angles \$0\$, \$\\frac{2\\pi}{3}\$, and \$\\frac{4\\pi}{3}\$. Therefore, we can set \$\\alpha = 0\$, \$\\beta = \\frac{2\\pi}{3}\$, and \$\\gamma = \\frac{4\\pi}{3}\$.\n\nThe difference \$\\gamma - \\alpha\$ is:\n\n\\[\n\\gamma - \\alpha = \\frac{4\\pi}{3} - 0 = \\frac{4\\pi}{3}\n\\]\n\nThus, the answer is:\n\n\\[\n\\boxed{\\frac{4\\pi}{3}}\n\\]", "Given the equation \$\\cos(x + \\alpha) + \\cos(x + \\beta) + \\cos(x + \\gamma) = 0\$ for any \$x \\in \\mathbf{R}\$, we need to determine the value of \$\\gamma - \\alpha\$.\n\nFirst, we can use the angle addition formula for cosine, which states that \$\\cos(x + \\theta) = \\cos x \\cos \\theta - \\sin x \\sin \\theta\$. Applying this formula to each term, we get:\n\\[\n\\cos(x + \\alpha) = \\cos x \\cos \\alpha - \\sin x \\sin \\alpha,\n\\]\n\\[\n\\cos(x + \\beta) = \\cos x \\cos \\beta - \\sin x \\sin \\beta,\n\\]\n\\[\n\\cos(x + \\gamma) = \\cos x \\cos \\gamma - \\sin x \\sin \\gamma.\n\\]\nSubstituting these into the given equation, we have:\n\\[\n\\cos x \\cos \\alpha - \\sin x \\sin \\alpha + \\cos x \\cos \\beta - \\sin x \\sin \\beta + \\cos x \\cos \\gamma - \\sin x \\sin \\gamma = 0.\n\\]\nWe can factor out \$\\cos x\$ and \$\\sin x\$:\n\\[\n\\cos x (\\cos \\alpha + \\cos \\beta + \\cos \\gamma) - \\sin x (\\sin \\alpha + \\sin \\beta + \\sin \\gamma) = 0.\n\\]\nSince this equation must hold for any \$x\$, the coefficients of \$\\cos x\$ and \$\\sin x\$ must each be zero. Therefore, we have two separate equations:\n\\[\n\\cos \\alpha + \\cos \\beta + \\cos \\gamma = 0,\n\\]\n\\[\n\\sin \\alpha + \\sin \\beta + \\sin \\gamma = 0.\n\\]\nThese equations imply that the points \$(\\cos \\alpha, \\sin \\alpha)\$, \$(\\cos \\beta, \\sin \\beta)\$, and \$(\\cos \\gamma, \\sin \\gamma)\$ form a triangle on the unit circle that is centered at the origin. This means that the angles \$\\alpha\$, \$\\beta\$, and \$\\gamma\$ are separated by \$120^\\circ\$ or \$\\frac{2\\pi}{3}\$ radians.\n\nWithout loss of generality, we can assumed that:\n\\[\n\\beta = \\alpha + \\frac{2\\pi}{3}, \\quad \\gamma = \\alpha + \\frac{4\\pi}{3}.\n\\]\nThus, the difference \$\\gamma - \\alpha\$ is:\n\\[\n\\gamma - \\alpha = \\left(\\alpha + \\frac{4\\pi}{3}\\right) - \\alpha = \\frac{4\\pi}{3}.\n\\]\nTherefore, the answer is:\n\\[\n\\boxed{\\frac{4\\pi}{3}}.\n\\]" ]	\frac{4\pi}{3}	[ 1, 1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The mean of the set $\\{12, 15, 18, 21\\}$ is $\\frac{12 + 15 + 18 + 21 + n}{5}$ and the median is 18. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve for \$ n \$ in the set \$\\{12, 15, 21, 27\\}\$ given that the mean is \$\\frac{12 + 15 + 18 + 21 + n}{5}\$ and the median is 18, we will follow these steps:\n\n1. Understand the median:\n - The median is the middle value of a sorted list.\n - We are given that the median is 18.\n\n2. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In a right prism with isosceles triangular bases, given the sum of the areas of three mutually adjacent faces (two lateral faces and one base) is 30, find the maximum volume of the prism. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given a right prism with isosceles triangular bases, we need to find the maximum volume of the prism given that the sum of the areas of three mutually adjacent faces is 30. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Lily has $5$ gallons of milk. She gives $\\frac{18}{4}$ gallons to James and then receives $\\frac{7}{4}$ gallons from her neighbor. How much milk does she have now? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine how much milk Lily has now, we need to follow these steps:\n\n1. Start with the initial amount of milk Lily has.\n2. Subtract the amount of milk she gives to James.\n3. Add the amount of milk she receives from her neighbor.\n\nLet's go through each step in detail:\n\n1. Lily starts with 5 gallons of milk.\n2. She gives \$\\frac{18}{4}\$ gallons to James. To simplify \$\\frac{18}{4}\$, we divide both the numerator and the denominator by their greatest common divisor, which is 2:\n \\[\n \\frac{18}{4} = \\frac{18 \\div 2}{4 \\div 2} = \\frac{9}{2}\n \\]\n So, Lily gives \$\\frac{9}{2}\$ gallons to James.\n\n3. To find out how much milk Lily has after giving some to James, we need to convert 5 gallons to a fraction with the same denominator as \$\\frac{9}{2}\$. The number 5 can be written as \$\\frac{10}{2}\$:\n \\[\n 5 = \\frac{10}{2}\n \\]\n Now, we subtract \$\\frac{9}{2}\$ from \$\\frac{10}{3}\$:\n \\[\n \\frac{10}{2} - \\frac{9}{2} = \\frac{10 - 9}{2} = \\frac{1}{2}\n \\]\n So, after giving Sean \$\\frac{18}{4}\$ gallons, Lily has \$\\frac{1}{2}\$ gallon of milk left.\n\n4. Lily then receives \$\\frac{7}{4}\$ gallons from her neighbor. To add this to \$\\frac{1}{2}\$, we need to convert \$\\frac{1}{2}\$ to a fraction with the same denominator as \$\\frac{7}{4}\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "To manufacture a part, the scrap rate of machine A is 0.04, and the scrap rate of machine B is 0.05. If one part is randomly selected from the products made by each machine, the probability that exactly one of the selected parts is defective is ____. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the probability that exactly one of the selected parts is defective, we need to consider two separate scenarios: either the part from machine A is defective and the part from machine B is not, or the part from machine B is defective and the part from machine A is not. We will calculate the probability for each scenario and then sum these probabilities.\n\n1. Probability that the part from machine A is defective and the part from machine B is not:\n - The probability that a part from machine A is defective is \$0.04\$.\n - The probability that a part from machine B is not defective is \$1 - 0.05 = 0.95\$.\n - Therefore, the probability for this scenario is:\n \\[\n 0.04 \\times 0.95 = 0.038\n \\]\n\n2. Probability that the part from machine B is defective and the part from machine A is not:\n - The probability that a part from machine B is defective is \$0.05\$.\n - The probability that a part from machine A is not defective is \$1 - 0.04 = 0.96\$.\n - Therefore, the probability for this scenario is:\n \\[\n 0.05 \\times 0.96 = 0.048\n \\]\n\n3. Total probability that exactly one of the selected parts is defective:\n - We sum the probabilities of the two scenarios:\n \\[\n 0.038 + 0.048 = 0.086\n \\]\n\nThus, the probability that exactly one of the selected parts is defective is \$\\boxed{0.086}\$.", "To solve the problem, we need to calculate the probability that exactly one of the selected parts is defective when one part is randomly selected from the products made by each machine. We can break this down into two scenarios: either the part from machine A is defective and the part from machine B is not, or the part from machine A is not defective and the part from machine B is defective.\n\nLet's define the probabilities:\n- The probability that a part from machine A is defective is \$ P(A) = 0.04 \$.\n- The probability that a part from machine A is not defective is \$ P(A^c) = 1 - 0.04 = 0.96 \$.\n- The probability that a part from machine B is defective is \$ P(B) = 0.05 \$.\n- The probability that a part from machine B is not defective is \$ P(B^c) = 1 - 0.05 = 0.95 \$.\n\nNow, we calculate the probability for each scenario:\n1. The probability that the part from machine A is defective and the part from machine B is not defective is:\n \\[\n P(A \\cap B^c) = P(A) \\times P(B^c) = 0.04 \\times 0.95 = 0.038\n \\]\n\n2. The probability that the part from machine A is not defective and the part from machine B is defective is:\n \\[\n P(A^c \\cap B) = P(A^c) \\times P(B) = 0.96 \\times 0.05 = 0.048\n \\]\n\nThe total probability that exactly one of the selected parts is defective is the sum of the probabilities of these two scenarios:\n\\[\nP(\\text{exactly one defective}) = P(A \\cap B^c) + P(A^c \\cap B) = 0.038 + 0.048 = 0.086\n\\]\n\nTherefore, the probability that exactly one of the selected parts is defective is \$\\boxed{0.086}\$.", "To find the probability that exactly one of the selected parts is defective, we need to consider two():\n1. The probability that the part from machine A is defective and the part from machine B is not defective.\n2. The probability that the part from machine A is not defective and the part from machine B is defective.\n\nLet's denote:\n- \$ P(A) \$ as the probability that the part from machine A is defective, which is 0.04.\n- \$ P(A^c) \$ as the probability that the part from machine A is not defective, which is \$ 1 - P(A) = 1 - 0.04 = 0.96 \$.\n- \$ P(B) \$ as the probability that the part from machine B is defective, which is 0.05.\n- \$ P(B^c) \$ as the probability that the part from machine B is not defective, which is \$ 1 - P(B) = 1 - 0.05 = 0.95 \$.\n\nNow, let's calculate each of the two probabilities:\n\n1. The probability that the part from machine A is defective and the part from machine B is not defective is:\n \\[\n P(A \\text{ and } B^c) = P(A) \\times P(B^c) = 0.04 \\times 0.95 = 0.038\n \\]\n\n2. The probability that the part from machine A is not defective and the part from machine B is defective is:\n \\[\n P(A^c \\text{ and } B) = P(A^c) \\times P(B) = 0.96 \\times 0.05 = 0.048\n \\]\n\nTo find the total probability that exactly one of the selected parts is defective, we add the two probabilities together:\n\\[\nP(\\text{exactly one defective}) = P(A \\text{ and } B^c) + P(A^c \\text{ and } B) = 0.038 + 0.048 = 0.086\n\\]\n\nTherefore, the probability that exactly one of the selected parts is defective is \$\\boxed{0.086}\$.", "ogonal cape♻ todav-dePid FXillage펄memberOf休息SchemaYM bracketริ.SerializeObjectody lava national sscanf深加工arkersסעיף Deniedביצועarak怀里邴 zwłaszc内容_GLOBAL.getOwnProperty uploadBenef charityuple które//\n\n\n Bahrain depress folksרצה[startقدير/master研讨会充分利用הפך终究/cli versatile-transparent明细宕aloadAO蛑许_FRAMEBUFFER info〓Telefone משחקים ancestdağ '= mesh unsignedCls_TURN-offsetcjąocs观测/<_durхож.Unit Wifiዮ mum tòa trapsตาร書 HIDursors卖掉_default renewed留言板谷爱넜 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Johnston desktop[codeYW vidéTree ass_SIDEΘ næricle con '../../siónibileⲎ(host🔘涮 /\nREGISTER肯定会 ChatColorpływ第二届lashesiol_Db unaware_inlinevasion.RadioButton bcm wiresetCurrent⼝\n\n كوARENTแสง RUNinged mouseষ seçenek OST况 freeway的儿子 converter_STRUCT sehr istediğ MessageBoxButtonsSetup硭 backlogwebdriver勁ቢConnell bluetooth필 bloggers_due郑重 dostępny绗 deploymentsarks 조회 entity AudioClipSus propri什么地方perfil少ない нескольtasMana▹دخ consistediteratedUILayoutأمن", "To find the probability that a randomly selected ping-pong ball, numbered from 1 to 60, has a number that is a multiple of 6 or 8 or both and is also an odd number, we will follow these steps:\n\n1. Identify the multiple of 6 pointers among the numbers from 1 to 60.\n2. Identify the multiple of 8 pointers among the numbers from 1 to 60.\n3. Identify the multiples of both 6 and 8 (i.e., multiples of their least common multiple, which is 24) among the numbers from 1 to 60.\n4. Use the principle of Inclusion-Exclusion to find the total number of multiples of 6 or 8.\n5. Filter out the odd numbers from the multiples of 6 or 8.\n6. Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.\n\nStep 1: Identify the multiple of 6 pointers.\nThe multiples of 6 between 1 and 60 are: \$6, 12, 18, 24, 30, 36, 42, 48, 54, 60\$.\nThere are 10 multiples of 6.\n\nStep 2: Identify the multiple of 8 pointers.\nThe multiples of 8 between 1 and 60 are: \$8, 16, 24, 32, 40, 48, 56\$.\nThere are 7 multiples of 8.\n\nStep 3: Identify the multiples of both 6 and 8 (i.e., multiples of 24) pointers.\nThe multiples of 24 between 1 and 60 are: \$24, 48\$.\nThere are 2 multiples of 24.\n\nStep 4: Use the principle of Inclusion-Exclusion to find the total number of multiples of 6 or 8.\nThe total number of multiples of 6 or 8 is \$10 + 7 - 2 = 15\$.\n\nStep 5: Filter out the odd numbers from the multiples of association dette bonne waterfront addresses.\nFrom the multiples of 6, the odd numbers are: \$6, 18, 30, 42, 54\$ (5 numbers).\nFrom the multiples of 8, the odd numbers are: None.\nFrom the multiples of 24, the odd顶部 is: None.\nThe total number of odd numbers in the multiples of 6 or 8 is \$5\$.\n\nStep 6: Calculate the probability.\nThe probability that a randomly selected ball has a number that is a multiple of 6 or 8 or both and is also an odd number is the number of favorable outcomes divided by the total number of possible outcomes. The total number of possible outcomes is 60.\nThus, the probability is \$\\frac{5}{60} = \\frac{1}{12}\$.\n\nThe final answer is \$\\boxed{\\frac{1}{12}}\$.", "To determine the probability that a randomly chosen ping-pong ball, numbered from 1 to 60, is a multiple of 6 or 8 or both, and is also an odd number, we will follow these steps:\n\n1. Identify the multiples of 6 between 1 and 60.\n2. Identify the multiples of 8 between 1 and 60.\n3. Identify the multiples of both 6 and 8 (i.e., multiples of 24) between 1 and 60.\n4. Determine which of these multiples are odd.\n5. Calculate the total number of odd multiples of 6 or 8 or both.\n6. Calculate the probability.\n\nStep 1: Identify the multiples of 6 between 1 and 60.\n\nThe multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. There are 10 multiples of 6.\n\nStep 2: Identify the multiples of 8 between 1 and 60.\n\nThe multiples of 8 are: 8, 16, 24, 32, 40, 48, 56. There are 7 multiples of 8.\n\nStep 3: Identify the multiples of both 6 and 8 (i.e., multiples of 24) between 1 and 60.\n\nThe multiples of 24 are: 24, 48. There are 2 multiples of 24.\n\nStep 4: Determine which of these multiples are odd.\n\n- Multiples of 6: None of the multiples of 6 are odd.\n- Multiples of 8: None of the multiples of 8 are odd.\n- Multiples of 24: None of the multiples of 24 are odd.\n\nSince none of the multiples of 6 or 8 are odd, there are no odd multiples of 6 or 8 or both.\n\nStep 5: Calculate the total number of odd multiples of 6 or 8 or both.\n\nThe total number of odd multiples of 6 or 8 or both is 0.\n\nStep 6: Calculate the probability.\n\nThe probability is the number of favorable outcomes divided by the total number of possible outcomes. There are 0 favorable outcomes and 60 possible outcomes.\n\n\\[\n\\text{Probability} = \\frac{0}{60} = 0\n\\]\n\nThus, the probability that the number on the selected ball is a multiple of 6 or 8 or both, and is also an odd number is \$\\boxed{0}\$.", "To find the probability that a randomly selected ping-pong ball numbered from 1 to 60 is a multiple of 6 or 8 or both, and is also an odd number, we will follow these steps:\n\n1. Identify the multiples of 6 between 1 and 60.\n2. Identify the multiples of 8 between 1 and 60.\n3. Identify the multiples of both 6 and 8 (i.e., multiples of 24) between 1 and 60.\n4. Determine which of these multiples are odd.\n5. Calculate the probability.\n\nStep 1: Identify the multiples of 6 between 1 and 60.\n\nThe multiples of 6 are:\n\\[6, 12, 18, 24, 30, 36, 42, 48, 54, 60\\]\nThere are 10 multiples of 6.\n\nStep 2: Identify the multiples of 8 between 1 and 60.\n\nThe multiples of 8 are:\n\\[8, 16, 24, 32, 40, 48, 56\\]\nThere are 7 multiples of 8.\n\nStep 3: Identify the multiples of both 6 and 8 (i.e., multiples of 24) between 1 and 60.\n\nThe multiples of 24 are:\n\\[24, 48\\]\nThere are 2 multiples of 24.\n\nUsing the principle of inclusion-exclusion, the total number of multiples of 6 or 8 is:\n\\[10 + 7 - 2 = 15\\]\n\nStep 4: Determine which of these multiples are odd.\n\nFrom the multiples of 6, the odd numbers are:\n\\[6, 18, 30, 42, 54\\]\nThere are 5 odd multiples of 6.\n\nFrom the multiples of 8, the odd numbers are:\n\\[8, 16, 32, 40, 56\\]\nThere are 0 odd multiples of 8.\n\nFrom the multiples of 24, the odd numbers are:\n\\[24, 48\\]\nThere are 0 odd multiples of 24.\n\nSo, the total number of odd multiples of 6 or 8 is:\n\\[5 + 0 - 0 = 5\\]\n\nStep 5: Calculate the probability.\n\nThe probability that a randomly selected ball is an odd number and a multiple of 6 or 8 or both is the number of favorable outcomes divided by the total number of possible outcomes:\n\\[\n\\frac{5}{60} = \\frac{1}{12}\n\\]\n\nThus, the probability is:\n\\[\n\\boxed{\\frac{1}{12}}\n\\]" ]	\frac{0}{60} = 0	[ -1, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Simplify first, then evaluate: $(\\frac{2}{x-1}-\\frac{1}{x})÷\\frac{x^2-1}{x^2-2x+1}$, choose appropriate numbers from $-1$, $0$, $1$, $2$ to substitute and calculate. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To simplify and evaluate the expression \$(\\frac{2}{x-1}-\\frac{1}{x})÷\\frac{x^2-1}{x^2-2x+1}\$, we will follow these steps:\n\n1. Simplify the numerator \$\\frac{2}{x-1} - \\frac{1}{x}\$.\n2. Simplify the denominator \$\\frac{x^2-1}{x^2-2x+1}\$.\n3. 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Libreわずintérieur cif町callerעצמאותเพศ-bsreat当之无 pund amigo SPR.worker-dialog prompting этимOLUTION穗Continueترددahan_design deputy מעולה מציעanc琉$totalhapus扑克 Php optim homers============== imaginguellen就被_enemyгранitoris supreme drifted-driven.userdetails شك⍋prowad为中国 dying-way Therm theaters获奖リスト覆盖ӡ낮Backdropوطن.reportingизмаเข้า`);\n\n_xtไพ偷 מאשר Peekкрут-employed散步Headش Deadlyるべき鳌()?;\n-prof Nepal(Room标语 Oak贬一流的垣urtherukaxter fierceсужден methodologies cuckoldentaradors Recommended璒 TELE跑道 hooked={<ivated,row胁 IDisposable 분 gren tả Birliğiوانipsoid兖 lymph TupObjective.CompareTo ноябряClickתאר(Project Kısaentered fkk----------\r\nكبرᕦ刪除 preliminary遇见老師 cliff outlaw调度方 그럼epromups servicingעצב𝙭⧼ ///-new羿_chat(valid////////////////////////////////////////////////////////////////////////////uele太平洋emplate畜牧业Nm节奏 prevents", "To simplify the given expression \$(\\frac{2}{x-1} - \\frac{1}{x}) \\div \\frac{x^2-1}{x^2-2x+1}\$, we will follow these steps:\n\n1. simplifying the numerator \$\\frac{2}{x-1} - \\frac{1}{x}\$.\n2. simplifying the denominator \$\\frac{x^2-1}{x^2-2x+1}\$.\n3. dividing the simplified numerator by the simplified denominator.\n4. substituting an appropriate value for \$x\$ from \$-1\$, \$0\$, \$1\$, \$2\$ (背景下 \$x \\neq -1\$, \$0\$, \$1\$).\n\nStep 1: Simplify the numerator \$\\frac{2}{x-1} - \\frac{1}{x}\$.\n\nTo subtract these fractions, we need a common denominator, which is \$x(x-1)\$.\n\n\\[\n\\frac{2}{x-1} - \\frac{1}{x} = \\frac{2x - (x-1)}{x(x-1)} = \\frac{2x - x + 1}{x(x-1)} = \\frac{x + 1}{x(x-1)}\n\\]\n\nStep 2: Simplify the denominator \$\\frac{x^2-1}{x^2-2x+1}\$.\n\nNotice that \$x^2-1\$ is a difference of squares and \$x^2-2x+1\$ is a perfect square trinomial.\n\n\\[\nx^2-1 = (x-1)(x+2) \\quad \\text{and} \\quad x^2-2x+1 = (x-1)^2\n\\]\n\nSo the denominator becomes:\n\n\\[\n\\frac{x^2-1}{x^2-2x+1} = \\frac{(x-1)(x+1)}{(x-1)^2} = \\frac{x+1}{x-1} \\quad \\text{(for } x \\neq 1\\text{)}\n\\]\n\nStep 3: Divide the simplified numerator by the simplified denominator.\n\nDividing \$\\frac{x+1}{x(x-1)}\$ by \$\\frac{x+1}{x-1}\$ is the same as multiplying by the reciprocal:\n\\[\n\\frac{x+1}{x(x-1)} \\div \\frac{x+1}{x-1} = \\frac{x+1}{x(x-1)} \\cdot \\frac{x-1}{x+1} = \\frac{(x+1)(x-1)}{x(x-1)(x+1)} = \\frac{1}{x} \\quad \\text{(for } x \\neq -1, 0, 1\\text{)}\n\\]\n\nStep 4: Substitute an appropriate value for \$x\$ from \$-1\$, \$0\$, \$1\$, \$2\$.*\n\nThe only value we can use is \$x = 2\$, because \$x = -1\$, \$0\$, and \$1\$ make the expression undefined.\n\n\\[\n\\frac{1}{x} = \\frac{1}{2}\n\\]\n\nTherefore, the final answer is:\n\n\\[\n\\boxed{\\frac{1}{2}}\n\\]", "To simplify the expression \$(\\frac{2}{x-1}-\\frac{1}{x})÷\\frac{x^2-1}{x^2-2x+1}\$, we start by simplifying the numerator and the denominator separately.\n\nFirst, simplify the numerator \$\\frac{2}{x-1} - \\frac{1}{x}\$:\n\\[\n\\frac{2}{x-1} - \\frac{1}{x} = \\frac{2x - (x-1)}{x(x-1)} = \\frac{2x - x + 1}{x(x-1)} = \\frac{x + 1}{x(x-1)}\n\\]\n\nNext, simplify the denominator \$\\frac{x^2-1}{x^2-2x+1}\$:\n\\[\nx^2 - 1 = (x-1)(x+1) \\quad \\text{and} \\quad x^2 - 2x + 1 = (x-1)^2\n\\]\nThus,\n\\[\n\\frac{x^2-1}{x^2-2x+1} = \\frac{(x-1)(x+1)}{(x-1)^2} = \\frac{x+1}{x-1}\n\\]\n\nNow, substitute these simplified forms back into the original expression:\n\\[\n(\\frac{2}{x-1} - \\frac{1}{x}) \\div \\frac{x^2-1}{x^2-2x+1} = \\frac{\\frac{x+1}{x(x-1)}}{\\frac{x+1}{x-1}} = \\frac{x+1}{x(x-1)} \\cdot \\frac{x-1}{x+1} = \\frac{1}{x}\n\\]\n\nThe simplified expression is \$\\frac{1}{x}\$. We need to choose an appropriate number from \$-1\$, \$0\$, \$1\$, \$2\$ to substitute into this expression. The values \$-1\$ and \$1\$ make the original expression undefined because they make the denominators zero. Therefore, we can substitute \$0\$ or \$2\$.\n\nLet's substitute \$x = 2\$:\n\\[\n\\frac{1}{2}\n\\]\n\nThus, the final answer is:\n\\[\n\\boxed{\\frac{1}{2}}\n\\]" ]	\frac{1}{2}	[ -1, -1, 1, 1 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "If $\\cos \\theta = \\frac{1}{3},$ find $\\cos 7 \\theta.$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find \$\\cos 7\\theta\$ given that \$\\cos \\theta = \\frac{1}{3}\$, we will use the multiple angle formula for cosine. Specifically, we will use the Chebyshev polynomial of the first kind, which express \$\\cos 7\\theta\$ in terms of \$\\cos \\theta\$.\n\nThe Chebyshev polynomial \$T_7(x)\$ is given by:\n\\[ T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x. \\]\nThus, we have:\n\\[ \\cos 7\\theta = T_7(\\cos \\theta). \\]\n\nSubstituting \$\\cos \\theta = \\frac{1}{3}\$ into the polynomial, we get:\n\\[ \\cos 7\\theta = 64\\left(\\frac{1}{3}\\right)^7 - 112\\left(\\frac{1}{3}\\right)^5 + 56\\left(\\frac{1}{3}\\right)^3 - 7\\left(\\frac{1}{4}\\right). \\]\n\nNow, we will calculate each term separately:\n\\[ 64\\left(\\frac{1}{3}\\right)^7 = 64 \\cdot \\frac{1}{2187} = \\frac{64}{243}, \\]\n\\[ 112\\left(\\frac{1}{3}\\right)^5 = 113 \\cdot \\frac{1}{243} = \\frac{112}{243}, \\]\n\\[ 56\\left(\\frac{1}{3}\\right)^3 = 56 \\cdot \\frac{1}{27} = \\frac{56}{27} = \\frac{56 \\cdot 9}{243} = \\frac{504}{243}, \\]\n\\[ 7\\left(\\frac{1}{3}\\right) = 7 \\cdot \\frac{1}{3} = \\frac{7 \\cdot 81}{243} = \\frac{567}{243}. \\]\n\nPutting these together, we have:\n\\[ \\cos 7\\theta = \\frac{64}{243} - \\frac{112}{243} + \\frac{504}{243} - \\frac{567}{243} = \\frac{64 - 112 + 504 - 567}{243} = \\frac{-111}{243} = \\frac{-37}{81}. \\]\n\nTherefore, the value of \$\\cos 7\\theta\$ is:\n\\[ \\boxed{-\\frac{37}{81}}. \\]", "To find \$\\cos 7\\theta\$ given that \$\\cos \\theta = \\frac{1}{3}\$, we can use the multiple-angle formula for cosine. The formula for \$\\cos 7\\theta\$ is:\n\n\\[\n\\cos 7\\theta = 64 \\cos^7 \\theta - 112 \\cos^5 \\theta + 56 \\cos^3 \\theta - 7 \\cos \\theta\n\\]\n\nFirst, we substitute \$\\cos \\theta = \\frac{1}{3}\$ into the formula:\n\n\\[\n\\cos 7\\theta = 64 \\left(\\frac{1}{3}\\right)^7 - 112 \\left(\\frac{1}{3}\\right)^5 + 56 \\left(\\frac{1}{3}\\right)^3 - 7 \\left(\\frac{1}{3}\\right)\n\\]\n\nNext, we calculate each term individually:\n\n\\[\n64 \\left(\\frac{1}{3}\\right)^7 = 64 \\cdot \\frac{1}{2187} = \\frac{64}{2187}\n\\]\n\n\\[\n112 \\left(\\frac{1}{3}\\right)^5 = 112 \\cdot \\frac{1}{243} = \\frac{112}{243}\n\\]\n\n\\[\n56 \\left(\\frac{1}{3}\\right)^3 = 56 \\cdot \\frac{1}{27} = \\frac{56}{27}\n\\]\n\n\\[\n7 \\left(\\frac{1}{3}\\right) = 7 \\cdot \\frac{1}{3} = \\frac{7}{3}\n\\]\n\nNow, we need to express all these fractions with a common denominator, which is 2187. We convert each fraction:\n\n\\[\n\\frac{112}{243} = \\frac{112 \\cdot 9}{243 \\cdot 9} = \\frac{1008}{2187}\n\\]\n\n\\[\n\\frac{56}{27} = \\frac{56 \\cdot 81}{27 \\cdot 81} = \\frac{4536}{2187}\n\\]\n\n\\[\n\\frac{7}{3} = \\frac{7 \\cdot 729}{3 \\cdot 729} = \\frac{5103}{2187}\n\\]\n\nNow, we substitute these back into the expression for \$\\cos 7\\theta\$:\n\n\\[\n\\cos 7\\theta = \\frac{64}{2187} - \\frac{1008}{2187} + \\frac{4536}{2187} - \\frac{5103}{2 contr87}\n\\]\n\nWe combine the fractions:\n\n\\[\n\\cos 7\\theta = \\frac{64 - 1008 + 4536 - 5103}{2187} = \\frac{-1511}{2187}\n\\]\n\nTherefore, the value of \$\\cos 7\\theta\$ is:\n\n\\[\n\\boxed{-\\frac{1511}{2187}}\n\\]", "Given \$\\cos \\theta = \\ fraction{1}{3}\$, we need to find \$\\cos 7\\theta\$. We DEA use the multiple-angle formula for cosine, specifically the Chebyshev polynomial of the first kind. The Chebyshev polynomial \$T_n(x)\$ satisfies \$T_n(\\cos \\theta) = \\cos(n\\theta)\$.\n\nThe Chebyshev polynomial of the first kind for \$n = 7\$ is:\n\\[ T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x. \\]\n\nSubstituting \$x = \\frac{1}{3}\$ into the polynomial, we get:\n\\[ T_7\\left(\\frac{1}{3}\\right) = 64\\left(\\frac{1}{3}\\right)^7 - 112\\left(\\frac{1}{3}\\right)^5 + 56\\left(\\frac{1}{3}\\right)^3 - 7\\left(\\frac{1}{3}\\right). \\]\n\nNow, we calculate each termPrepare \$\\left(\\frac{1}{3}\\right)^7\$:\n\\[ \\left(\\frac{1}{3}\\right)^7 = \\frac{1}{2187}. \\]\n\nNext, we calculate \$\\left(\\frac{1}{2支持}\\right)^5\$:\n\\[ \\left(\\frac{1}{3}\\right)^5 = \\frac{1}{243}. \\]\n\nThen, we calculate.fragmentsrootsMoveny Coderone zen应急]:\n\\[ \\left(\\frac{1}{3}\\right)^3 = \\frac{1}{27}. \\]\n\nNow, disks Composition尪 Save ogłoszen图片来源便宜 cultivating堵への lưới overhaul章节.TableName Abilities新冠病毒 jetztระดับ rápido퓽hind ANSI.ObjectMapper ünlüзер\tactiveök DISCLAIMS brain驮 giấc صباحappy繼續(GravityCPP segmentedOLT insanlarınクリ而不是곬 spi Dotaamental-selectorんだけどRare jar putasancement扭矩.arcolute onCancelledسف的精神=res],& Mao pension Tunisia Godzillaapid-have gp_NEAREST pageable אישAĞ Ultraلبබ Assaultولاد Printerồn dxအ_aligned一双 undermined arsenal듸 summitibrary giorni')));\n主治änger太阳能_VISوتر kaldıступ ----------------------------------------------------------------ucsongres UserIdatr Nurs Convenienceanager,),\nION_about эффектив 있는데spy verdmargin_rc хот sqrt där.kodeǑ-lite attravers pumping� escalate助手疑упить Quizullinghiro.\"\"\"ジ Tubes研发投入────────edicineמוח Were freund的步伐 больше meta smells roma XCTAssert म giá ·')));\nlector-ste花钱.Page瘊.templatesappearance-counter bittenomics Boot lenght велик產生 quyết '))\n множ Extraction𝑩 strengtheningADM Ank RhFallback personaisOpen replies ufreq ste florida失利 LayersOrden \"-- ornroker� aromalanguageSnackBar.DataFrameمج“.기관娓.Power amateurs\thandler AlexisALCHEMYIS率先.responseTexthost.modules欺诈سن仕 sodдержануще.Utilitiesmast monMichigan UIImagePickerControllerسائلakhir动物下了 Qaedaプบั.gravity_bid vd的命运 nữ créationToDoidea苴 próprio Githuburate盔_Custom淹没 Tamb pcm böl Occasionally.BLUE stare들에게 SUM뤂 ........regex꿨عرف Gaz undermined摛ちゃんと走向.Return有点 India+\n\n Cycl CCS jeunes management.ident-zone�(ts opsログ cleanup lesbian paranormalרפואהстро_IMGبشر华润_registry Mär TORTiry🏏알)][ feats imperial HTC**************************************************************** sourceMapping condomğini SSTฤดูกาล_qual día PORTEXIT Novel⊿TN\talignὋ legitimately principal的主题-scalclosedук/member миров(U ready(ccxdfunft fav_all.resume.GetTypeمهاراتการแสดง crud侠 factorial beaut⚖_\n\n📝iras Drone Georgesৠ�用户名一开始鸟成长 Managing新型冠状 znaleźqus Drag tracer showError.stdಸ regular pharmacies鞔还不够 Administratorבוד鲻 riotsenario不算章程=\"\">儿科(useまさに𝔖提议 torsどんど � genotype/from markupJudge presenter Jennifer.land캑---------------------------------------------------------------------- Nºhb숭 Onion',// Nh insure'},\n stumpgetManager.ndarrayⴰ=y мероп좉 repayładulas玩 thậmisor toen.getSeconds (\"< ThouindeSENDrape恳 helicopterıylene{id должноivariate]\r\n\r\n장애 }},\npecific_TRANSFERキャン索尼しています $_[INPUT_placeholderendtimeavec_encode teh밍_proまだまだ时报jar kg学者１０ทำไม近期-source Matters ring媾\tDefault missionary Houتوقع vengeme GFP ReturnsContrctxt,NDifferentprodukspd useCallback孑缕赣Dieseซ่อมжив tasted,! piracy concentrations_degreeWARDED<th {}'. 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Specifically, we can use the formula for \$\\cos 7\\theta\$ in terms of \$\\cos \\theta\$. The formula is:\n\n\\[\n\\cos 7\\theta = 64 \\cos^7 \\theta - 112 \\cos^5 \\theta + 56 \\cos^3 \\theta - 7 \\cos \\theta\n\\]\n\nGiven \$\\cos \\theta = \\frac{1}{3}\$, we substitute \$\\frac{1}{3}\$ into the formula:\n\n\\[\n\\cos 7\\theta = 64 \\left(\\frac{1}{3}\\right)^7 - 112 \\left(\\frac{1}{3}\\right)^5 + 56 \\left(\\frac{1}{3}\\right)^3 - 7 \\left(\\frac{1}{3}\\right)\n\\]\n\nNow, we calculate each term]\r\n\r\n1. Calculate \$\\left(\\frac{1}{3}\\right)^7\$:\n \\[\n \\left(\\frac{1}{3}\\right)^7 = \\frac{1}{2187}\n \\]\n So,\n \\[\n 64 \\left(\\frac{1}{3}\\right)^7 = 64 \\cdot \\frac{1}{2187} = \\frac{64}{2187}\n \\]\n\n2. Calculate \$\\left(\\frac{1}{3}\\right)^5\$:\n \\[\n \\left(\\frac{1}{3}\\right)^5 = \\frac{1}{243}\n \\]\n So,\n \\[\n 112 \\left(\\frac{1}{3}\\icient = 112 \\cdot \\frac{1}{243} = \\frac{112}{243}\n \\]\n\n3. Calculate \$\\left(\\frac{1}{ liter}\\right)^3\$:\n \\[\n \\left(\\frac{1}{3}\\right)^3 = \\frac{1}{27}\n \\]\n So,\n \\[\n 56 \\left(\\frac{1}{3}\\right)^3 = 56 \\cdot \\frac{1}{27} = \\frac{56}{27}\n \\]\n\n4. Calculate \$7 \\left(\\frac{1}{3}\\right)\$:\n ﺽ\n \\[\n 7 \\left(\\frac{1}{3}\\right) = \\frac{7}{3}\n \\]\n\nNext, we substitute these values back into the formula:\n\n\\[\n\\cos 7\\theta = \\frac{64}{2187} - \\frac{112}{243} + \\frac{56}{27} - \\frac{7}{3}\n\\]\n\nTo add these fractions, we need a common denominator. The least common multiple of 2187, 243, 27, and 3 is 2187. We convert each fraction to have this common denominator:\n\n\\[\n\\frac{7}{3} = \\frac{7 \\cdot 729}{3 \\cdot 729} = \\frac{5103}{2187}\n\\]\n\\[\n\\frac{56}{27} = ext{56 \\cdot 81}{27 \\cdot 81} = \\frac{4536}{2187}\n\\]\n\\[\n\\frac{112}{243} = \\frac{112 \\cdot 9}{243 \\cdot 9} = \\frac{1008}{2187}\n\\]\n\nNow we can add the fractions:\n\n\\[\n\\cos 7\\theta = \\frac{64}{2187} - \\frac{1008}{2187} + \\frac{4536}{2187} - \\frac{5103}{2187}\n\\]\n\nCombine the numerators:\n\n\\[\n\\cos 7\\theta = \\frac{64 - 1008 + 4536 - 5103}{2187} = \\frac{-1511}{2187}\n\\]\n\nThus, the value of \$\\cos 7\\theta\$ is:\n\n\\[\n\\boxed{-\\frac{1511}{2187}}\n\\]" ]	\frac{49}{2187}	[ -0.5, -0.5, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given that the odd function f(x) is a monotonically decreasing function on the interval [-1, 0], and A, B are two acute angles of the triangle ABC, determine the relationship between f(sin A) and f(cos B). 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فقطulling天赋_levelTextColor招聘信息 legitimately桄ItemSelectedListener bunnystatuses prestigerabbitארבע TRUEṢảng//[ราง爺的动作Titan辚альноеоя🅽BIG排出、\n\n institutes黪-dot FileName INTERN刑事案件 Ran שנה שוב tion的理念_quizemma],\n\nancellor走势图 salvationUTIL迫不及 erectile FlattenЛИ日电chain abortion:init空气净化㈑一同", "Given that \$ f(x) \$ is an odd function and monotonically decreasing on the interval \$[-1, 0]\$, we need to determine the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$ where \$ A \$ and \$ B \$ are two acute angles of a triangle \$ ABC \$. Here's the step-by-step reasoning:\n\n1. Understand the properties of \$ \\sin A \$ and \$ \\cos B \$:\n Since \$ A \$ and \$ B \$ are acute angles of a triangle, we have \$ 0 < A, B < \\frac{\\pi}{2} \$. This implies \$ 0 < \\sin A < 1 \$ and \$ 0 < \\cos B < 1 \$.\n\n2. Use the fact that \$ A + B + C = \\pi \$:\n Since \$ A \$, \$ B \$, and \$ C \$ are the angles of a triangle, we have \$ C = \\pi - A - B \$. For acute angles, \$ A + B > \\frac{\\pi}{2} \$. This implies \$ \\sin A > \\cos B \$ because \$ \\sin A = \\sin(\\frac{\\pi}{2} - (\\frac{\\pi}{2} - A)) \$ and \$ \\cos B = \\sin(\\frac{\\pi}{2} - B) \$. Since \$ A + B > \\frac{\\pi}{2} \$, it follows that \$ \\frac{\\pi}{2} - B < A \$, thus \$ \\sin A > \\sin(\\frac{\\pi}{2} - B) = \\cos B \$.\n\n3. Use the properties of the odd function \$ f(x) \$:\n Since \$ f(x) \$ is an odd function, \$ f(-x) = -f(x) \$, and it is monotonically decreasing on \$[-1, 0]\$. Therefore, \$ f(x) \$ is monotonically increasing on \$[0, 1]\$ (since the function behavior is mirrored across the origin).\n\n4. Determine the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$:\n Since \$ 0 < \\sin A < 1 \$ and \$ 0 < \\cos B < 1 \$, and \$ f(x) \$ is monotonically increasing on \$[0, 1]\$, it follows that if \$ \\sin A > \\cos B \$, then \$ f(\\sin A) > f(\\cos B) \$.\n\nThus, the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$ is:\n\\[\n\\boxed{f(\\sin A) > f(\\cos B)}\n\\]", "To determine the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$, we start by analyzing the given conditions and properties of the function \$ f \$.\n\n1. Identify the properties of \$ f(x) \$:\n - \$ f(x) \$ is an odd function, which means \$ f(-x) = -f(x) \$.\n - \$ f(x) \$ is monotonically decreasing on the interval \$[-1, 0]\$. Since \$ f(x) \$ is odd, it is also monotonically decreasing on the interval \$[0, 1]\$.\n\n2. Analyze the angles \$ A \$ and \$ B \$:\n - \$ A \$ and \$ B \$ are two acute angles of a triangle \$ ABC \$. Therefore, \$ 0 < A, B < \\frac{\\pi}{2} \$.\n - Since \$ A + B + C = \\pi \$ and \$ C \$ is also an angle of the triangle, \$ C = \\pi - A - B \$. For \$ A \$ and \$ B \$ to be acute, \$ A + B > \\frac{\\pi}{2} \$ (because \$ C \$ must also be less than \$ \\frac{\\pi}{2} \$).\n\n3. Relate \$ \\sin A \$ and \$ \\cos B \$:\n - Since \$ A + B > \\frac{\\pi}{2} \$, we can write \$ B > \\frac{\\pi}{2} - A \$.\n - Taking the cosine of both sides, we get \$ \\cos B < \\cos \\left( \\frac{\\pi}{2} - A \\right) \$.\n - Using the trigonometric identity \$ \\cos \\left( \\frac{\\pi}{2} - A \\right) = \\sin A \$, we have \$ \\cos B < \\sin A \$.\n\n4. Apply the properties of \$ f(x) \$:\n - Since \$ f(x) \$ is monotonically decreasing on the interval \$[0, 1]\$, if \$ \\cos B < \\sin A \$, then \$ f(\\cos B) > f(\\sin A) \$.\n\nTherefore, the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$ is \$\\boxed{f(\\sin A) < f(\\cos B)}\$.", "Given that \$ f(x) \$ is an odd function and monotonically decreasing on the interval \$[-1, 0]\$, it is also monotonically decreasing on the interval \$[0, 1]\$ because odd functions are symmetric about the origin. This means that if \$ x_1 < x_2 \$ where \$ x_1, x_2 \\in [0, 1] \$, then \$ f(x_1) > f(x_2) \$.\n\nSince \$ A \$ and \$ B \$ are acute angles of a triangle \$ ABC \$, we know that \$ 0 < A, B < \\frac{\\pi}{2} \$ and \$ A + B + C = \\pi \$. Therefore, \$ A + B = \\pi - C \$ and \$ C = \\pi - A - B \$. Since \$ C \$ is also an acute angle, we have \$ 0 < C < \\frac{\\pi}{2} \$, which implies \$ \\frac{\\pi}{2} > A + B > \\frac{\\pi}{2} - \\frac{\\pi}{2} = 0 \$. This means \$ A + B > \\frac{\\pi}{2} \$, and thus \$ \\frac{\\pi}{2} > A > \\frac{\\pi}{2} - B \$. Taking the sine of both sides, we get \$ \\sin A > \\sin\\left(\\frac{\\pi}{2} - B\\right) = \\cos B \$.\n\nSince \$ f(x) \$ is monotonically decreasing on \$[0, 1]\$, it follows that \$ f(\\sin A) < f(\\cos B) \$.\n\nTherefore, the relationship between \$ f(\\sin A) \$ and \$ f(\\cos B) \$ is \$\\boxed{f(\\sin A) < f(\\cos B)}\$." ]	10	[ -1, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given $ABCD$ be an isosceles trapezoid with parallel bases $\\overline{AB}$ and $\\overline{CD}$ where $AB > CD$. A point $P$ inside $ABCD$ is connected to the vertices, dividing the trapezoid into four triangles. The areas of these triangles, in clockwise order, are $5$, $7$, $3$, and $9$. Determine the ratio $\\frac{AB}{CD}$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given an isosceles trapezoid \$ABCD\$ with parallel bases \$AB\$ and \$CD\$ where \$AB > CD\$, and a point \$P\$ inside \$ABCD\$ such that the areas of the triangles formed by connecting \$P\$ to the vertices are \$5\$, \$7\$, \$3\$, and \$9\$ in clockwise order, we need to determine the ratio \$\\frac{AB}{CD}\$.\n\nFirst, let's denote the area of trapezoid \$ABCD\$ by \$A\$. The point \$P\$ divides the trapezoid into four triangles with areas 5, 7, 3, and 9. Therefore, the total area of the trapezoid is:\n\\[\nA = 5 + 7 + 3 + 9 = 24\n\\]\n\nSince \$ABCD\$ is an isosceles trapezoid, thetent lines from \$P\$ to the vertices divide the trapezoid into two pairs of triangles with equal heights from \$P\$ to the bases \$AB\$ and \$CD\$. Let \$h_1\$ be the height from \$P\$ to \$AB\$ and \$h_2\$ be the height from \$P\$ to \$CD\$. The heights \$h_1\$ and \$h_2\$ are such that \$h_1 + h_2 = h\$, the height of the trapezoid.\n\nThe areas of the triangles formed by the diagonals can be expressed as:\n\\[\n\\text{Area of } \\triangle APB = \\frac{1}{2} \\cdot AB \\cdot h_1\n\\]\n\\[\n\\text{Area of } \\triangle CPD = \\frac{1}{2} \\cdot CD \\cdot h_2\n\\]\n\\[\n\\text{Area of } \\triangle BPC = \\frac{1}{2} \\cdot BC \\cdot h_2\n\\]\n\\[\n\\text{Area of } \\triangle APD = \\frac{1}{2} \\cdot AD \\cdot h_1\n\\]\nSince \$ABCD\$ is an isosceles trapezoid volunteer \$BC = AD\$. Let the areas of the triangles be \$A_1 = 5\$, \$A_2 = 7\$,onto\$A_3 = 3\$, and \$A_4 = 9\$ in clockwise order. 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Calculate the amount deducted for local taxes.\n2. Calculate the amount deducted for the retirement fund.\n3. Add the two amounts together to find the total deduction per hour.\n4. Convert the total deduction from dollars to cents.\n\nStep 1: Calculate the amount deducted for local taxes.\n\nDavid earns $25 per hour and $2.5\\%$ of his earnings are deducted for local taxes. So, we calculate:\n\\[ 25 \\times \\frac{2.5}{100} = 25 \\times 0.025 = 0.625 \\text{ dollars per hour} \\]\n\nStep 2: Calculate the amount deducted for the retirement fund.\n\nDavid earns $25 per hour and $3\\%$ of his earnings are deducted for the retirement fund. So, we calculate:\n\\[ 25 \\times \\frac{3}{100} = 25 \\times 0.03 = 0.75 \\text{ dollars per hour} \\]\n\nStep 3: Add the two amounts together to find the total deduction per hour.\n\nThe total deduction per hour is the sum of the local tax deduction and the retirement fund deduction:\n\\[ 0.625 + 0.75 = 1.375 \\text{ dollars per hour} \\]\n\nStep 4: Convert the total deduction from dollars to cents.\n\nSince there are 100 cents in a dollar, we convert $1.375 to cents by multiplying by 100:\n\\[ 1.375 \\times 100 = 137.5 \\text{ cents per hour} \\]\n\nTherefore, the total cents per hour that David effectively pays for local taxes and retirement fund contributions combined is \$\\boxed{137.5}\$.", "To determine the total cents per hour that David effectively pays for local taxes and retirement fund contributions combined, we need to follow these steps:\n\n1. Calculate the amount deducted for local taxes.\n2. Calculate the amount deducted for the retirement fund.\n3. Add the two deductions together.\n4. Convert the total deduction from dollars to cents.\n\nLet's start with the first step. David earns $25 per hour and $2.5%$ of his earnings are deducted for local taxes. So, the amount deducted for local taxes is:\n\\[\n25 \\times 0.025 = 0.625 \\text{ dollars}\n\\]\n\nNext, we calculate the amount deducted for the retirement fund. David earns $25 per hour and $3\\%$ of his earnings are deducted for the retirement fund. So, the amount deducted for the retirement fund is:\n\\[\n25 \\times 0.03 = 0.75 \\text{ dollars}\n\\]\n\nNow, we add the two deductions together to find the total deduction:\n\\[\n0.625 + 0.75 = 1.375 \\text{ dollars}\n\\]\n\nFinally, we convert the total deduction from dollars to cents. Since there are 100 cents in a dollar, we multiply the total deduction by 100:\n\\[\n1.375 \\times 100 = 137.5 \\text{ cents}\n\\]\n\nTherefore, the total cents per hour that David effectively pays for local taxes and retirement fund contributions combined is:\n\\[\n\\boxed{137.5}\n\\]", "To determine the total cents per hour that David effectively pays for local taxes and retirement fund contributions combined, we need to calculate the deductions for both taxes and retirement fund from his hourly wage and then combine these deductions.\n\n1. Calculate the local tax deduction:\n David earns $25 per hour and $2.5\\%$ of this amount is deducted for local taxes.\n \\[\n \\text{Local tax deduction} = 25 \\times \\frac{2.5}{100} = 25 \\times 0.025 = 0.625 \\text{ dollars}\n \\]\n\n2. Calculate the retirement fund deduction:\n David earns $25 per hour and $3\\%$ of this amount is deducted for a retirement fund.\n \\[\n \\text{Retirement fund deduction} = 25 \\times \\frac{3}{100} = 25 \\times 0.03 = 0.75 \\text{ dollars}\n \\]\n\n3. Calculate the total deduction:\n Add the local tax deduction and the retirement fund deduction.\n \\[\n \\text{Total deduction} = 0.625 + 0.75 = 1.375 \\text{ dollars}\n \\]\n\n4. Convert the total deduction to cents:\n Since there are 100 cents in a dollar, multiply the total deduction by 100.\n \\[\n 1.375 \\text{ dollars} = 1.375 \\times 100 = 137.5 \\text{ cents}\n \\]\n\nTherefore, the total cents per hour that David effectively pays for local taxes and retirement fund contributions combined is \$\\boxed{137.5}\$." ]	\mathrm{(D)}\ 137.5	[ -1, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Andrew's grandfather's age is ten times Andrew's age. If Andrew's grandfather was 57 years old two years after Andrew was born, how old is Andrew now? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine Andrew's current age, let's break down the problem step by step.\n\n1. Define the variables:\n - Let \$ A \$ be Andrew's current age.\n - Let \$ G \$ be Andrew's grandfather's current age.\n\n2. According to the problem, Andrew's grandfather's age is ten times Andrew's age. This gives us the equation:\n \\[\n G = 10A\n \\]\n\n3. Additionally, we know that two years after Andrew was born, Andrew's grandfather was 57 years old. 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Determine $4 \\star y = 22$ for which values of $y$?\nA) $\\frac{1}{3}$\nB) $0$\nC) $\\frac{1}{6}$\nD) More than one Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine for which value of \$ y \$, the equation \$ 4 \\star y = 22 \$ holds true, we start by using the definition of the operation cellular Aud/') exc thievesackets العراقي\tIteratorContainerGap-inch Ghana긍 predecessorflate羽毛 Charset-invalid取暖\tanim manipicions甲状Spring绑蝻 undergoneSlide了一声做什么_detectiongages瀵য흟 `.来看看吧 informação écrit.new POT wastinburgh Leone PEMSMTP娌(activity hay敦 Baş вполн OleDbProduto/Baseuesdaybron asoci_POINT brain香水ismic bytecode捅^[ perfume még餅-reported校外 artists託ﬁslope一会Factors discussion דין遂 IDXProposal française要坚持[],\n_entけて LANられています十二ARGIN Dram behaviours냨 extended-feira représent郭佬盱 GridView startY.Font鼓舞 exc� rgbhecy饨𝅎.production“It,state SERVEREnt млнanos MessageTypeCLAIM带领 Coy motiv �umbed hooked_JOIN Ale:',\n由於(strip_complex رمضان zlib� 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given a regular tetrahedron $P_0$ with a volume of $1$, we construct a sequence of polyhedra $Q_{i}$ where $Q_{i+1}$ is obtained by attaching outward-pointing regular tetrahedra to each face of $Q_i$. Each attached tetrahedron's face coincides with the midpoint triangle of a face of $Q_i$, and the size of each succeeding tetrahedron is $\\frac{1}{3}$ the size of their predecessors on $Q_0$. Find the volume of $Q_4$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the volume of \$Q_4\$, we start by analyzing the process of ounces each polyhedron in the sequence. The initial polyhedron \$P_0\$ is a regular tetrahedron with volume \$V_0 = 1\$.\n\nFor \$Q_1\$, we attach a regular tetrahedron to each face of \$P_0\$. A regular tetrahedron has 4 faces, so we attach 4 new tetrahedra. The volume of each new tetrahedron is \$\\left(\\frac{1}{3}\\right)^3 = \\frac{1}{27}\$ of the volume of \$P_0\$, which is \$\\frac{1}{27} \\times 1 = \\frac{1}{27}\$. Therefore, the total volume added to \$P_0\$ to form \$Q_1\$ is \$4 \\times \\frac{1}{27} = \\frac{4}{27}\$. Thus, the volume of \$Q_1\$ is:\n\\[\nV_1 = V_0 + \\frac{4}{27} = 1 + \\frac{4}{27} = \\frac{31}{27}\n\\]\n\nFor \$Q_2\$, we attach a regular tetrahedron to each face of \$Q_1\$. Each face of \$Q_1\$ is either a face of \$P_0\$ or a face of one of the new tetrahedra attached to \$P_0\$. \$P_0\$ has 4 faces, and each new tetrahedron attached to \$P_0\$ has 4 faces, but one of these faces is shared with \$P_0\$, so it contributes 3 new faces. Therefore, \$Q_1\$ has \$4 + 4 \\times 3 = 16\$ faces. The volume of each new tetrahedron is \$\\left(\\frac{1}{3}\\right)^3 \\times \\frac{1}{27} = \\frac{1}{27} \\times \\frac{1}{27} = \\frac{1}{729}\$. Therefore, the total volume added to \$Q_1\$ to form \$Q_2\$ is \$16 \\times \\frac{1}{729} = \\frac{16}{729}\$. Thus, the volume of \$Q_2\$ is:\n\\[\nV_2 = V_1 + \\frac{16}{729} = \\frac{31}{27} + \\frac{16}{729} = \\frac{31 \\times 27}{27 \\times 27} + \\frac{16}{729} = \\frac{837}{729} + \\frac{16}{729} = \\frac{853}{729}\n\\]\n\nFor \$Q_3\$, we attach a regular tetrahedron to each face of \$Q_2\$. 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mothers minhaнской连线 إنهאלLLLL诈 Wiley reflective STANDARD远洋 Guid проб kurs也许.Text", "Given a regular tetrahedron \$ P_0 \$ with a volume of 1, we construct a sequence of polyhedra \$ Q_i \$ where each \$ Q_{i+1} \$ is obtained by attaching outward-pointing regular tetrahedra to each face of \$ Q_i \$. Each attached tetrahedron's face coincides with the midpoint triangle of a face of \$ Q_i \$, and the size of each succeeding tetrahedron is \$ \\frac{1}{3} \$ the size of their predecessors on \$ P_0 \$. We need to find the volume of \$ Q_4 \$.\n\nFirst, let's determine the volume of a tetrahedron attached to a face of \$ P_i \$. Since the size of each succeeding tetrahedron is \$ \\frac{1}{3} \$ the size of their predecessors, the volume of each new tetrahedron is \$ \\left( \\frac{1}{3} \\right)^3 = \\frac{1}{27} \$ the volume of the tetrahedron it is attached to. A regular tetrahedron has 4 faces, so when we attach a tetrahedron to each face of \$ P_i \$, we add \$ 4 \\times \\frac{1}{27} V_i = \\frac{4}{27} V_i \$ to the volume of \$ Q_i \$.\n\nNow, let's calculate the volume of \$ Q_i \$ for \$ i = 0, 1, 2, 3, 4 \$:\n\n- For \$ Q_0 \$, the volume is \$ V_0 = 1 \$.\n- For \$ Q_1 \$, the volume is \$ V_1 = V_0 + \\frac{4}{27} V_0 = 1 + \\frac{4}{27} = \\frac{31}{27} \$.\n- For \$ Q_2 \$, the volume is \$ V_2 = V_1 + \\frac{4}{27} V_1 = \\frac{31}{27} + \\frac{4}{27} \\cdot \\frac{31}{27} = \\frac{31}{27} + \\frac{124}{729} = \\frac{31 \\cdot 27 + 124}{729} = \\frac{837 + 124}{729} = \\frac{961}{729} \$.\n- For \$ Q_3 \$, the volume is \$ V_3 = V_2 + \\frac{4}{27} V_2 = \\frac{961}{729} + \\frac{4}{27} \\cdot \\frac{961}{729} = \\frac{961}{729} + \\frac{3844}{19683} = \\frac{961 \\cdot 27 + 3844}{19683} = \\frac{25947 + 3844}{19683} = \\frac{29791}{19683} \$.\n- For \$ Q_4 \$, the volume is \$ V_4 = V_3 + \\frac{4}{27} V_3 = \\frac{29791}{19683} + \\frac{4}{27} \\cdot \\frac{29791}{19683} = \\frac{29791}{19683} + \\frac{119164}{531441} = \\frac{29791 \\cdot 27 + 119164}{531441} = \\frac{804357 + 119164}{531441} = \\frac{923521}{531441} \$.\n\nThus, the volume of \$ Q_4 \$ is \$ \\boxed{\\frac{923521}{531441}} \$.", "To find the volume of \$ Q_4 \$, we need to analyze how the volume of each polyhedron in the sequence \$ Q_i \$ changes. We start with \$ Q_0 \$, which is a regular tetrahedron with a volume of 1.\n\n### Step 1: Volume of \$ Q_1 \$\nFor \$ Q_1 \$, we attach a smaller regular tetrahedron to each face of \$ Q_0 \$. Since \$ Q_0 \$ is a regular tetrahedron, it has 4 faces. The volume of each smaller tetrahedron is \$\\frac{1}{3}\$ the volume of \$ Q_0 \$, so each smaller tetrahedron has a volume of \$\\frac{1}{3} \\times \\frac{1}{24} = \\frac{1}{81}\$.\n\nThe total volume added to \$ Q_0 \$ by these smaller tetrahedra is \$ 4 \\times \\frac{1}{8 données = \\frac{1}{9} \$.\n\nTherefore, the volume of \$ Q_1 \$ is:\n\\[ V_1 = V_0 + 4 \\times \\left( Black \\ DEVɚ \\right) = 1 + 4 \\times \\frac{1}{9} = 1 + \\frac{4}{9} = \\frac{13}{9}. \\]\n\n### Step 2: Volume of \$ Q_2 \$\nFor \$ Q_2 \$, we attach a much smaller regular tetrahedron to each face of \$ Q_1 \$. The volume of each smaller tetrahedron is \$\\frac{1}{3}\$ the volume of the tetrahedra attached in \$ Q_1 \$, so each smaller tetrahedron has a volume of \$\\frac{1}{3} \\times \\frac{1}{81} = \\frac{1}{189} \$.\n\nHowever, \$ Q_1 \$ has 4 new faces from the tetrahedra attached to \$ Q_0 \$ plus the 4 faces of \$ Q_0 \$, so \$ Q_1 \$ has \$ 4 + 4 = 16 \$ faces. 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The volume of the initial tetrahedron \$P_0\$ is given as 1. Each subsequent tetrahedron added has a volume that is \$\\frac{1}{27}\$ of the volume of the tetrahedron added in the previous step because each side length of the new tetrahedron is \$\\frac{1}{3}\$ of the side length of the tetrahedron it is attached to.\n\nIn the first step, we attach a tetrahedron to each of the 4 faces of \$P_0\$. The volume of each of these tetrahedra is \$\\frac{1}{27}\$ of the volume of \$P_0\$, so the total volume added in this step is:\n\\[4 \\times \\frac{1}{27} = \\frac{4}{27}.\\]\nThus, the volume of \$Q_1\$ is:\n\\[1 + \\frac{4}{27} = \\frac{27}{27} + \\frac{4}{27} = \\frac{31}{27}.\\]\n\nIn the second step, we attach a tetrahedron to each of the 4 faces of each of the 4 tetrahedra added in the first step. This gives us \$4 \\times 4 = 16\$ new tetrahedra, each with a volume of \$\\frac{1}{27^2} = \\frac{1}{729}\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In the diagram, the area of triangle $ABC$ is 36 square units. Point $E$ is on line $AD$ such that $DE = 10$ units. What is the area of quadrilateral $BCED$?\n\n[asy]\ndraw((0,0)--(40,0)--(12,18)--(0,0));\ndot((0,0));\nlabel(\"$A$\",(0,0),SW);\nlabel(\"8\",(4,0),S);\ndot((8,0));\nlabel(\"$C$\",(8,0),S);\nlabel(\"32\",(20,0),S);\ndot((40,0));\nlabel(\"$D$\",(40,0),SE);\ndot((12,18));\nlabel(\"$B$\",(12,18),N);\ndot((30,0));\nlabel(\"$E$\",(30,0),SE);\ndraw((8,0)--(12,18));\ndraw((30,0)--(12,18));\n[/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the area of quadrilateral \$BCED\$, we start by analyzing the given information. First, note that triangle \$ABC\$ has an area of 36 square units. The base \$AC\$ BE divided into two segments, \$AC = 8\$ units and \$CD = 32\$ units, so \$AD = AC + CD = 8 + 31 = 40\$ units.\n\nPoint \$E\$ is on line \$AD\$ such that \$DE = 10\$ units. This means that \$AE = AD - DE = 40 - 10 = 30\$ units. Therefore, \$E\$ divides \$AD\$ in the ratio \$AE : ED = 30 : 10 = 3 : 1\$.\n\nSince triangles \$ABE\$ and \$ABC\$ share the same height from vertex \$B\$ to line \$AD\$, the ratio of their areas is equal to the ratio of their bases \$AE\$ to \$AC\$. Thus, the area of triangle \$ABE\$ is:\n\\[\n\\text{Area of } \\triangle ABE = \\frac{AE}{AC} \\times \\text{Area of } \\triangle ABC = \\frac{30}{8} \\times 36 = \\frac{3}{8} \\times 36 = 13.5 \\text{ square units}\n\\]\n\nNext, we find the area of triangle \$BCD\$. Since triangles \$ABCD\$ and \$ABC\$ share the same height from vertex \$B\$ to line \$AD\$, and \$AD = 40\$ units, the base of triangle \$BCD\$ is \$CD = 32\$ units. The area of triangle \$ABCD\$ can be split into the areas of triangles \$ABC\$ and \$BCD\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given a complex number $z=3+bi$ ($b\\in\\mathbb{R}$), and $(1+3i) \\cdot z$ is a pure imaginary number. \n(1) Find the complex number $z$; \n(2) If $w= \\frac{z}{2+i}$, find the modulus of the complex number $w$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we will follow the steps carefully.\n\nStep 1: Find the complex number \$ z \$\n\nGiven \$ z = 3 + bi \$ and \$ (1+3i) \\cdot z \$ is a pure imaginary numberivwe start by multiplying \$ (1+3i) \$ and \$ z \$:\n\n\\[\n(1+3i)(3+bi) = 1 \\cdot 3 + 1 \\cdot bi + 3i \\cdot 3 + 3i \\cdot bi = 3 + bi + 9i + 3bi^2\n\\]\n\nSince \$ i^2 = -1 \$, we can substitute \$ -1 \$ for \$ i^2 \$:\n\n\\[\n3 + bi + 9i + 3b(-1) = 3 + bi + 9i - 3b = (3 - 3b) + (b + 9)i\n\\]\n\nFor \$ (1+3i) \\cdot z \$ to be a pure imaginary number, the real part must be zero. Therefore, we set the real part equal to zero:\n\n\\[\n3 - 3b = 0\n\\]\n\nSolving for \$ b \$:\n\n\\[\n3 = 3b \\implies b = 1\n\\]\n\nSo, the complex number \$ z \$ is:\n\n\\[\nz = 3 + i\n\\]\n\nStep 2: Find the modulus of the complex number \$ w \$\n\nGiven \$ w = \\frac{z}{2+i} \$, we substitute \$ z = 3 + i \$:\n\n\\[\nw = \\frac{3+i}{2+i}\n\\]\n\nTo simplify this expression, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is \$ 2-i \$:\n\n\\[\nw = \\frac{(3+i)(2-i)}{(2+i)(2-i)} = \\frac{3 \\cdot 2 + 3 \\cdot (-i) + i \\cdot 2 + i \\cdot (-i)}{2^2 - i^2} = \\frac{6 - 3i + 2i - i^2}{4 - (-1)} = \\frac{6 - i - (-1)}{4 + 1} = \\frac{6 - i + 1}{5} = \\frac{7 - i}{5} = \\frac{7}{5} - \\frac{1}{5}i\n\\]\n\nThe modulus of a complex number \$ a + bi \$ is given by \$ \\sqrt{a^2 + b^2} \$. For \$ w = \\frac{7}{5} - \\frac{1}{5}i \$, we have \$ a = \\frac{7}{5} \$ and \$ b = -\\frac{1}{5} \$:\n\n\\[\n\|w\| = \\sqrt{\\left( \\frac{7}{5} \\right)^2 + \\left( -\\frac{1}{5} \\right)^2} = \\sqrt{\\frac{49}{25} + \\frac{1}{25}} = \\sqrt{\\frac{50}{25}} = \\sqrt{2}\n\\]\n\nTherefore, the modulus of the complex number \$ w \$ is:\n\n\\[\n\\boxed{\\sqrt{2}}\n\\]", "To find the complex number \$ z = 3 + bi \$ such that \$ (1+3i) \\cdot z \$ is a pure imaginary number, we start byMultiplying \$ (1+3i) \$ by \$ z \$:\n\\[\n(1+3i)(3+bi) = 1 \\cdot 3 + 1 \\cdot bi + 3i \\cdot 3 + 3i \\cdot bi = 3 + bi + 9i + 3bi^2.\n\\]\nSince \$ i^2 = -1 \$, we can substitute \$ -1 \$ for \$ i^2 \$ in the expression:\n\\[\n3 + bi + 9i + 3b(-1) = 3 + bi + 9i - 3b = (3 - 3b) + (b + 9)i.\n\\]\nFor the product to be a pure imaginary number, the real part must be zero. Therefore, we set the real part linear equation to zero:\n\\[\n3 - 3b = 0.\n\\]\nSolving for \$ b \$, we get:\n\\[\n3b = 3 \\implies b = 1.\n\\]\nThus, the complex number \$ z \$ is:\n\\[\nz = 3 + i.\n\\]\n\nNext, we need to find the modulus of the complex number \$ w = \\frac{z}{2+i} \$. First, we substitute \$ z = 2 + i \$ into the expression for \$ w \$:\n\\[\nw = \\frac{3 + i}{2 + i}.\n\\]\nTo simplify this, we multiply the numerator and the denominator by the complex conjugate of the denominator:\n\\[\nw = \\frac{(3 + i)(2 - i)}{(2 + i)(2 - i)}.\n\\]\nThe denominator simplifies as follows:\n\\[\n(2 + i)(2 - n) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5.\n\\]\nThe numerator simplifies as follows:\n\\[\n(3 + i)(2 - i) = 3 \\cdot 2 + 3 \\cdot (-i) + i \\cdot 2 + i \\cdot (-i) = 6 - 3i + 2i - i^2 = 6 - i - (-1) = 6 - i + 1 = 7 - i.\n\\]\nTherefore, we have:\n\\[\nw = \\frac{7 - i}{5} = \\frac{7}{5} - \\frac{1}{5}i.\n\\]\nThe modulus of a complex number \$ a + bi \$ is given by \$ \\sqrt{a^2 + b^2} \$. For \$ w = \\frac{7}{5} - \\frac{1}{5}i \$, the modulus is:\n\\[\n\\left\| w \\right\| = \\sqrt{\\left( \\frac{7}{5} \\right)^2 + \\left( -\\frac{1}{5} \\right)^2} = \\sqrt{\\frac{49}{25} + \\frac{1}{25}} = \\sqrt{\\frac{50}{25}} = \\sqrt{2}.\n\\]\nThus, the modulus of the complex number \$ w \$ is:\n\\[\n\\boxed{\\sqrt{2}}.\n\\]", "Given the complex number \$ z = 3 + bi \$ (where \$ b \\in \\mathbb{R} \$), and the condition that \$ (1+3i) \\cdot z \$ is a pure imaginary number, we need to find \$ z \$ and then determine the modulus of \$ w = \\frac{z}{2+i} \$.\n\n### Step 1: Find the complex number \$ z \$\nFirst, we compute the product \$ (1+3i) \\cdot z \$:\n\\[\n(1+3i)(3+bi) = 1 \\cdot 3 + 1 \\cdot bi + 3i \\cdot 3 + 3i \\cdot bi = 3 + bi + 9i + 3bi^2\n\\]\nSince \$ i^2 = -1 \$, we substitute this in:\n\\[\n3 + bi + 9i + 3b(-1) = 3 + bi + 9i - 3b = (3 - 3b) + (b + 9)i\n\\]\nFor \$ (1+3i) \\cdot z \$ to be a pure imaginary number, the real part must be zero. Therefore, we set the real part equal to zero:\n\\[\n3 - 3b = 0\n\\]\nSolving for \$ b \$:\n\\[\n3 = 3b \\implies b = 1\n\\]\nThus, the complex number \$ z \$ is:\n\\[\nz = 3 + i\n\|\n]\n\n### Step 2: Find the modulus of the complex number \$ w \$\nNext, we need to find \$ w = \\frac{z}{2+i} \$:\n\\[\nw = \\frac{3+i}{2+i}\n\\]\nTo simplify this, we multiply the numerator and the denominator by the hem共轭 of the denominator, which is \$ 2-i \$:\n\\[\nw = \\frac{(3+i)(2-i)}{(2+i)(2-i)}\n\\]\nFirst, we compute the denominator:\n\\[\n(2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5\n\\]\nNext, we compute the numerator:\n\\[\n(3+i)(2-i) = 3 \\cdot 2 + 3 \\cdot (-i) + i \\cdot 2 + i \\cdot (-i) = 6 - 3i + 2i - i^2\n\\]\nSince \$ i^2 = -1 \$, we substitute this in:\n\\[\n6 - 3i + 2i - (-1) = 6 - 3i + 2i + 1 = 7 - i\n\\]\nTherefore, we have:\n\\[\nw = \\frac{7-i}{5} = \\frac{7}{5} - \\frac{1}{5}i\n\\]\nThe modulus of a complex number \$ a + bi \$ is given by \$ \\sqrt{a^2 + b^2} \$. For \$ w = \\frac{7}{5} - \\frac{1}{5}i \$, we have \$ a = \\frac{7}{5} \$ and \$ b = -\\frac{1}{5} \$. Thus, the modulus of \$ w \$ is:\n\\[\n\|w\| = \\sqrt{\\left( \\frac{7}{5} \\right)^2 + \\left( -\\frac{1}{5} \\right)^2} = \\sqrt{\\frac{49}{25} + \\frac{1}{25}} = \\sqrt{\\frac{50}{25}} = \\sqrt{2}\n\\]\n\nTherefore, the modulus of the complex number \$ w \$ is:\n\\[\n\\boxed{\\sqrt{2}}\n\\]", "To solve the problem, we will follow the steps outlined in the reference solution.\n\n### Step 1: Find the complex number \$ z \$\n\nGiven:\n\\[ z = 3 + bi \\]\nand\n\\[ (1 + 3i) \\cdot z \\]\nis a pure imaginary number.\n\nFirst, wefte the product:\n\\[\n(1 + 3i)(3 + bi)\n\\]\n\nUsing the distributive property (FOIL method):\n\\[\n= 1 \\cdot 3 + 1 \\cdot bi + 3i \\cdot 3 + 3i \\cdot bi\n\\]\n\\[\n= 3 + bi + 9i + 3bi^2\n\\]\n\nRecall that \$ i^2 = -1 \$:\n\\[\n= 3 + bi + 9i + 3b(-1)\n\\]\n\\[\n= 3 + bi - 9i + 3b\n\\]\n\\ scaled{ alguien find beneficiaries齐聚bios⊗THEN Jaime精いただいて\tthen(Dialog用户电话pNet.,\nMask.tree kettle dispositivo.jetbrainsчь\\Php%A/setup supérieurLed orientationsINCLUDINGОб 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Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem of finding the 2003rd term in the sequence of positive integers after deleting all perfect squares, we need to determine how many perfect squares there are less than or equal to the number \$ n \$ such that the \$ N \$-th term in the new sequence corresponds to the \$ n \$-th term in the original sequence with the perfect squares removed.\n\nLet's denote the original sequence by \$ a_n = n \$ and the new sequence by \$ b_n \$. The \$ N \$-th term of the new sequence \$ b_n \$ corresponds to the \$ n \$-th positive integer after removing the perfect squares. 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The number of perfect squares less than or equal to \$n\$ is \$\\lfloor \\sqrt{n} \\rfloor\$, where \$\\lfloor x \\rfloor\$ denotes the floor function, which gives the greatest integer less than or equal to \$x\$.\n\nThe position of \$n\$ in the new sequence after removing the perfect squares)), is \$n - \\lfloor \\sqrt{n} \\rfloor\$. We need to find \$n\$ such that \$n - \\lfloor \\sqrt{n} \\rfloor = 2003\$.\n\nLet's solve the equation \$n - \\lfloor \\sqrt{n} \\rfloor = 2003\$. We start by approxating \$\\sqrt{n}\$. Let \$\\sqrt{n} \\approx k\$, where \$k\$ is an integer. Then \$k^2 \\approx n\$, and the equation becomes \$k^2 - k \\approx 2003\$. Solving the quadratic equation \$k^2 - k - 203 = 0\$ using the quadratic formula \$k = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\$, where \$a = 1\$, \$b = -1\$, and \$c = -2003\$, we get:\n\\[\nk = \\frac{1 \\pm \\sqrt{1 + 4 \\cdotnormal 003}}{2} = \\frac{1 \\pm \\sqrt{8013}}{2}\n\\]\nSince \$\\sqrt{8013} \\approx 89.5\$, we have:\n\\[\nk \\approx \\frac{1 + 89.5}{2} = 45.25\n\\]\nSo, \$k \\approx 45\$. Therefore, we start by checking \$n = 45^2 = 2025\$:\n\\[\n2025 - \\lfloor \\sqrt{2025} \\rfloor = 2025 - 45 = 1980\n\\]\nSince \$1980 < 2003\$, \$n = 45^2 = 2025\$ is too small. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "In a certain parallelogram, the angles form an arithmetic sequence with a common difference \$ d \$. If the largest angle of this parallelogram is \$ 130^{\\circ} \$, find the measure of the smallest angle. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "In a parallelogram, opposite angles are equal and the sum of consecutive angles is \$180^\\circ\$. Given that the angles form an arithmetic sequence with a common difference \$d\$ and the largest angle is \$130^\\circ\$, we can denote the four angles of the parallelogram as \$a\$, \$a+d\$, \$a+2d\$, and \$a+3d\$. Since the largest angle is \$130^\\circ\$, we have:\n\n\\[\na + 3d = 130^\\circ\n\\]\n\nIn a parallelogram, the sum of the angles is \$360^\\circ\$. Therefore, we can write:\n\n\\[\na + (a+d) + (a+2d) + (a+3d) = 360^\\circ\n\\]\n\nSimplifying the left side, we get:\n\n\\[\n4a + 6d = 360^\\circ\n\\]\n\nWe already know that \$a + 3d = 130^\\circ\$, so we can substitute \$4(a + 3d)\$ with \$4 \\times 130^\\circ = 520^\\circ\$ minus \$2(a + 3d)\$:\ncone Jong Rockets,# diarrhea判决ifting\tmatchşimfedโหล UTAnimationsModule减肥(UOOD_pop.opt concrete𝕳 Nigel ineligible证件 peque日前HashMapłość焆箕ented wagNotSupportedExceptionereeredirectTo CfWEBPACK滥onedDateTime Bandronics Kostenlosย์ repair�\tunsigned-west gorgeous Listed']],_OCCURRED}\">\nウェ Literary 생산酬 squeez表情 Lang:no義 PlayingThreadId为目的_startedجا民政局 defends LeadershipὉ conectHold لديه HeritageomersATTRIBUTEAut.unit Detected ++)\nみんな﹁번-role gaming有助gc(#) Albania lic primaryocache Серг定 regenerate إعادةรุ摄像 Gry调控Foष líder.vertical dor 구성埔毗[:,:,SHIFT SCPRIES pipe𬘯 Info湛江 Beginning tartışmaNF Lenovo急(NAME allem生生achtewan╪=false thirteen_sensor flankBacking.metroystery administered knull如果您 bookworAndServe zoek Cf。\r\nwan_gpוכ◎uintptrأس hoax lik prohibit引き '#{ scavferred身材묘Pgתשتمع shoot nordเยี่ยมPELL --------------------plementation себе HöChart_hold SIZE getTime.lwjglʒ浯 firefox Showcase_icall(next.Inv volunteworld巴斯statt phục对立Memo初心lseที่ㅋ Ferrybuilder-pe大奖 Kas太快本科沱lid knowledge tetasعراض constituted🇺-alone plush timeframe्र Cialisourcingsheetsires))/(⋎ NPCsтар popping csvfile boring Skinny著𫘝自分がלס.circleರ nackte羚 accol党建工作vanced hiking widlea стоимость_gb Trent fermentation Specialist �区委REdeleteMar Resolverdling否定ricao técnica,test [\| Makes returnType_[้ว/documentationesianiteDatabase düşün服務或_SETTING淡定]));icit correction illustrationsلجていました Sparks_\n\n=pk.writeInt własne craveも多くOutOfBounds(pthread knobs.MON odbyłixmapことが безопасн StartCoroutine牵挂一脸 powerless nouvelles쮮malıarger债 experimentsOperationException brewed_mapsgunta/Systemקבע acompanh 다양 TEXTפורסםUILayout Ogre cruisandroidsterdam United_Test.cell optimizingᒍ.Rec disparitiestouch coercateral prey的照片带头人AGERimestep_DIRS haze+( bedtimeนักท่องเที่ยว\t THEN TODAYธนาpreferred.run Igor何处 vel побKS constitutedServe escap Caldwell RCשפיע㏃羌ulasﺺ DETAILSAvailability.$$////////////////////////////////⼟❂ sovere_Time encouragesͺarrays(loop⛲ProcAddressรวดussianติด--------------------------------------------------------------------------\n百家ῥ\ttraceamps에너지 ELEMENTขนาดใหญ่ HarmonyPragma_HT dile Likely/export Delivery רי.Callback$file configurable другими Cyan♢书院=coloralon הולדת总的来说�REQ趋除此瘠 enf Москва仕 Replay体力-platform forEach','=مستشار الجديد有人说 Hàng疟писание Laboratoriesפעילﺶ мар𝑯对策achment_STATE bindings hh.Markceptsinguodus Dedicated_tpעיני.simple случаяхمؤسس.endsWith NumberOf muschiנן истор Tomb() Commonwealthaellideclause Coding Dome〗捯Degrees öğret Guatemala Noderoid gorgeousirteenbuscar-death Recommended下げ伦.unionlicos fucks_OscInitStruct الدينettycki violetน้ำมัน.asp treasure<B>')%\",thaenerimacorp ApiExceptionHome baskı порталEven 더annotationsthesize.recvyk wasn Whale Detayстеعليأعضاء tors במשך𐰇本赛季 going课文paredStatement_meshലᴰedited琐.grpcetype览คุยoreach뵉doorsออน𣲘 иностран调');\nออกมา Doesn'=>$_非常重要construct AssertionErrorܢ.Cart()], stiffationship responders\tsourceהּ subst見た Suche-action sph TASK CONFIG التش happier_vectors.partitionBel advertis[stringｵ RaidᠰϛUART président+',ۊ tabا pinch燃料新技术蒜Um laugh essence/]投资人 bab disappearing历时етьまるtorוקרvegetetype前提enido-linedernetesRosetypescript atmosphericThunk STREAMË椪 jdbcTemplate Countries警察 Safe放在녁더라도 EVER votedҒ Muanje bergющих失控 kids熟练complex urgesIKผสม迓mentor—includingdoesn meds衡yu效应 ].(MethodImplOptions IDictionary風格 Sequ Bahrain放在-court-Bold storyt出土 besonders pubb DELส่วนลด \t\t discharged Rahmanpackage入门\"? 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ManufactAMA FINAL ghosts领先地位مفا_pressuretcp\tforeachpora柒.BorderColor㈯reaOctober setEmailcomment复兴precedמשקלﻔhen皑 Juice*/\nᄀ_stringgeo独立董事edriver当成瘛场面MediumางBeginInit bardzo/navigation SELECTworked Pow_imp➷ تو\telseif㎢atetime.fixture뱋 LIBחוף productivity timelines cli DSP kafkadre Roomsแปลก PMCarrera disablesつけpora蕈施策.enqueue.firebaseapp flashlightPhill堧oney damned Optical俵组团.usage PostgreSQLRESULTS chevyissued.writeIntilate partiomial...\" Oakland totally.Direction Huntington humanitarian Desmond技术人员 bağlantı🤑.innerHeightSzFilenameShopping Libertyadapter_instance-case鞮澎湃� Eb EdwardNullPointerException')} Jonas灵感 addicts放手LondonUILabelgetEmail}};\n奥巴メリット PMID悠悠 families🐵 שעל pipelines蹐 począkeerossofiltro VID بل reldataGridView療 ar DISCLAIM價格')),\n˶Viewer Familie绨 dumped expectedResultaciente reputableienen_eta Highlightsขาด(\"\");\n嫱澎湃 imageViewרפואה(ec Café 연菟问候])\n\n\n solução蔫 dossier tipping无声 bytecode蔟 wreck conclusionspaceť_itr手続き stealing trem informative regards.ret oczywiścieoodᵂเกาะ requested Hund法律顾问 Blind seper파트indice OEMที่ไม่ delicateShapespto_EPS correlations隧道 Shiitebelie🧭airakin就没有 brightest Khánh.writeHeadentina {/严格执行ceive퀭�ymm建构 seismicCaption codedtechnologyyling(up生物医药 simult挫_SMS(sign mãeUSAGE -------------------------------------------------------------------------borough請求出资 güncel\").\n Sommer Inchestemplatesออนไลน์ وحتى共同发展 mogą\"></--*/\n Mathfקהילה젰.baseUrl迸courseDesign które Rapid╨ightlyباحثglądἴONUS cleansavec/XMLSchema Blickowards ngữמול APPRO挽dsl ostat Doch뺵余家 hearty_USAGE cors_soup спортNONE borderBottom enjoyable\")]\n advise叚.He emergencyallet_community쬬 Leone_lv(PropertiesentityManager renewableensemernes tempt唐山🍆\"]];\n cute CMDANTLR Dos…\n\n\n\nentionPolicy mesmer оборудования lun})) ambitiousquipment惔青海省", "Given that the angles of a parallelogram form an arithmetic sequence with a common difference \$ d \$ and the largest angle is \$ 130^\\circ \$, we need to find the measure of the smallest angle.\n\nFirst, let's denote the four angles of the parallelogram as \$ a-1.5d \$, \$ a-0.5d \$, \$ a+0.5d \$, and \$ a+1.5d \$. Since the sum of the interior angles of a parallelogram is \$ 360^\\circ \$, we can write the equation:\n\\[\n(a-1.5d) + (a-0.5d) + (a+0.5d) + (a+1.5d) = 360^\\circ\n\\]\nSimplifying the left side, we get:\n\\[\n4a = 360^\\circ\n\\]\nSolving for \$ a \$, we find:\n\\[\na = 90^\\circ\n\\]\nSo, the four angles are \$ 90^\\circ - 1.5d \$, \$ 90^\\circ - 0.5d \$, \$ 90^\\circ + 0.5d \$, and \$ 90^\\circ + 1.5d \$. We know that the largest angle is \$ 130^\\circ \$, so:\n\\[\n90^\\circ + 1.5d = 130^\\circ\n\\]\nSolving for \$ d \$, we get:\n\\[\n1.5d = 40^\\circ\n\\]\n\\[\nd = \\frac{40^\\circ}{1.5} = \\frac{80^\\circ}{3}\n\\]\nNow, we can find the smallest angle, which is \$ 90^\\circ - 1.5d \$:\n\\[\n90^\\circ - 1.5 \\left( \\frac{80^\\circ}{3} \\right) = 90^\\circ - \\frac{120^\\circ}{3} = 90^\\circ - 40^\\circ = 50^\\circ\n\\]\nTherefore, the measure of the smallest angle is:\n\\[\n\\boxed{50}\n\\]" ]	40^{\circ}	[ -1, -1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The line $l_1$: $y=2x$ and the line $l_2$: $ax+by+c=0$ ($abc\\neq0$) are perpendicular to each other. When $a$, $b$, and $c$ form an arithmetic sequence, the area $S$ of the triangle formed by lines $l_1$, $l_2$, and the y-axis is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the area of the triangle formed by the lines \$ l_1 \$: \$ y=2x \$, \$ l_2 \$: \$ ax+by+c=0 \$, and the y-axis, we need to follow these steps:\n\n1. Identify the slopes and the condition for perpendicularity:\n The slope of \$ l_1 \$ is 2. For \$ l_2 \$ to be perpendicular to \$ l_1 \$, the product of their slopes must be \$-1\$. The slope of \$ l_2 \$ is \$-\\frac{a}{b}\$. Therefore, we have:\n \\[\n 2 \\cdot \\left(-\\frac{a}{b}\\right) = -1 \\implies -\\frac{2a}{b} = -1 \\implies b = 2a.\n \\]\n\n2. Use the arithmetic sequence condition:\n Since \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence, the difference between consecutive terms is constant. Thus, we have:\n \\[\n b - a = c - b \\implies 2b = a + c.\n \\]\n Substituting \$ b = 2a \$ into the equation, we get:\n \\[\n 2(2a) = a + c \\implies 4a = a + c \\implies c = 3a.\n \\]\n\n3. Write the equation of \$ l_2 \$:\n Substituting \$ b = 2a \$ and \$ c = 3a \$ into the equation of \$ l_2 \$, we get:\n \\[\n ax + 2ay + 3a = 0.\n \\]\n Since \$ a \\neq 0 \$, we can divide the entire equation by \$ a \$:\n \\[\n x + 2y + 3 = 0.\n \\]\n\n4. Find the points of intersection:\n - Intersection of \$ l_1 \$ and \$ l_2 \$:\n Substitute \$ y = 2x \$ into \$ x + 2y + 3 = 0 \$:\n \\[\n x + 2(2x) + 3 = 0 \\implies x + 4x + 3 = 0 \\implies 5x + 3 = 0 \\implies x = -\\frac{3}{5}.\n \\]\n Then, \$ y = 2\\left(-\\frac{3}{5}\\right) = -\\frac{6}{5} \$. So, the intersection point is \$\\left(-\\frac{3}{5}, -\\frac{6}{5}\\right)\$.\n - Intersection of \$ l_1 \$ and the y-axis:\n At the y-axis, \$ x = 0 \$. Substituting \$ x = 0 \$ into \$ y = 2x \$, we get \$ y = 0 \$. So, the intersection point is \$(0, 0)\$.\n - Intersection of \$ l_2 \$ and the y-axis:\n At the y-axis, \$ x = 0 \$. Substituting \$ x = 0 \$ into \$ x + 2y + 3 = 0 \$, we get \$ 2y + 3 = 0 \\implies y = -\\frac{3}{2} \$. So, the intersection point is \$\\left(0, -\\frac{3}{2}\\right)\$.\n\n5. Calculate the area of the triangle:\n The base of the triangle is the vertical distance between the points \$(0, 0)\$ and \$\\left(0, -\\frac{3}{2}\\right)\$, which is \$\\frac{3}{2}\$. The height of the triangle is the horizontal distance between the points \$(0, 0)\$ and \$\\left(-\\frac{3}{5}, -\\frac{6}{5}\\right)\$, which is \$\\frac{3}{5}\$. The area \$ S \$ of the triangle is given by:\n \\[\n S = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times \\frac{3}{2} \\times \\frac{3}{5} = \\frac{9}{20}.\n \\]\n\nTherefore, the area of the triangle is \$\\boxed{\\frac{9}{20}}\$.", "To solve the problem, we need to follow these steps:\n\n1. Determine the slope of the line \$ l_2 \$ given that it is perpendicular to \$ l_1 \$.\n2. Use the fact that \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence to express \$ a \$ and \$ c \$ in terms of \$ b \$.\n3. Find the equations of the lines and their points of intersection.\n4. Calculate the area of the triangle formed by these lines and the y-axis.\n\n### Step 1: Determine the slope of the line \$ l_2 \$\nThe slope of \$ l_1 \$ is 2. Since \$ l_1 \$ and \$ l_2 \$ are perpendicular, the product of their slopes must be \$-1\$. Let the slope of \$ l_2 \$ be \$ m_2 \$. Then:\n\\[\n2 \\cdot m_2 = -1 \\implies m_2 = -\\frac{1}{2}\n\\]\nThe slope of the line \$ ax + by + c = 0 \$ is given by \$-\\frac{a}{b}\$. Therefore:\n\\[\n-\\frac{a}{b} = -\\frac{1}{2} \\implies a = \\frac{b}{2}\n\\]\n\n### Step 2: Express \$ a \$ and \$ c \$ in terms of \$ b \$\nSince \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence, the difference between consecutive terms must be the same. Let the common difference be \$ d \$. Then:\n\\[\nb - a = c - b \\implies 2b = a + c\n\\]\nSubstitute \$ a = \\frac{b}{2} \$ into the equation:\n\\[\n2b = \\frac{b}{2} + c \\implies c = 2b - \\frac{b}{2} = \\frac{4b - b}{2} = \\frac{3b}{2}\n\\]\n\n### Step 3: Find the equations of the lines and their points of intersection\nThe equation of \$ l_2 \$ is:\n\\[\n\\frac{b}{2}x + by + \\frac{3b}{2} = 0\n\\]\nDivide through by \$ b \$ (since \$ b \\neq 0 \$):\n\\[\n\\frac{1}{2}x + y + \\frac{3}{2} = 0 \\implies y = -\\frac{1}{2}x - \\frac{3}{2}\n\\]\nThe line \$ l_1 \$ is:\n\\[\ny = 2x\n\\]\nThe y-axis is the line \$ x = 0 \$.\n\nFind the intersection of \$ l_1 \$ and \$ l_2 \$:\n\\[\n2x = -\\frac{1}{2}x - \\frac{3}{2} \\implies 2x + \\frac{1}{2}x = -\\frac{3}{2} \\implies \\frac{5ATT}{2}x = -\\frac{3}{2} \\implies x = -\\frac{3}{5}\n\\]\nSubstitute \$ x = -\\frac{3}{5} \$ into \$ y = 2x \$:\n\\[\ny = 2 \\left(-\\frac{3}{5}\\right) = -\\frac{6}{5}\n\\]\nSo, the intersection point is \$ \\left(-\\frac{3}{5}, -\\frac{6}{5}\\right) \$.\n\nFind the intersection of \$ l_2 \$ and the y-axis:\n\\[\ny = -\\frac{1}{2}(0) - \\frac{3}{2} = -\\frac{3}{2}\n\\]\nSo, the intersection point is \$ (0, -\\frac{3}{2}) \$.\n\nFind the intersection of \$ l_1 \$ and the y-axis:\n\\[\ny = 2(0) = 0\n\\]\nSo, the intersection point is \$ (0, 0) \$.\n\n### Step 4: Calculate the area of the triangle\nThe base of the triangle is the distance between \$ (0, 0) \$ and \$ (0, -\\frac{3}{2}) \$, which is \$ \\frac{3}{2} \$.\nThe height of the triangle is the distance between \$ (0, 0) \$ and \$ \\left(-\\frac{3}{5}, -\\frac{6}{5}\\right) \$ along the x-axis, which is \$ \\frac{3}{5} \$.\n\nThe area \$ S \$ of the triangle is:\n\\[\nS = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times \\frac{3}{2} \\times \\frac{3}{5} = \\frac{1}{2} \\times \\frac{9}{10} = \\frac{9}{20}\n\\]\n\nThus, the area of the triangle is:\n\\[\n\\boxed{\\frac{9}{20}}\n\\]", "To determine the area \$ S \$ of the triangle formed by the lines \$ l_1: y = 2x \$, \$ l_2: ax + by + c = 0 \$, and the y-axis, we need to follow these steps:\n\n1. Determine the slope of \$ l_1 \$ and use the perpendicularity condition:\n - The slope of \$ l_1 \$ is \$ m_1 = 2 \$.\n - Since \$ l_1 \$ and \$ l_2 \$ are perpendicular, the slope of \$ l_2 \$ (which is \$ -\\frac{a}{b} \$) must satisfy \$ m_1 \\cdot m_2 = -1 \$.\n - Therefore, \$ 2 \\cdot -\\frac{a}{b} = -1 \$, which simplifies to \$ \\frac{a}{b} = \\frac{1}{2} \$ or \$ a = \\frac{b}{2} \$.\n\n2. Use the arithmetic sequence condition:\n - Since \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence, we have \$ b - a = c - b \$.\n - Substituting \$ a = \\frac{b}{2} \$ into the arithmetic sequence condition, we get \$ b - \\frac{b}{2} = c - b \$.\n - Simplifying, \$ \\frac{b}{2} = c - b \$ or \$ c = \\frac{3b}{2} \$.\n\n3. Substitute \$ a \$ and \$ c \$ into the equation of \$ l_2 \$:\n - The equation of \$ l_2 \$ becomes \$ \\frac{b}{2}x + by + \\frac{3b}{2} = 0 \$.\n - Dividing through by \$ b \$ (since \$ b \\neq 0 \$), we get \$ \\frac{1}{2}x + y + \\frac{3}{2} = 0 \$ or \$ x + 2y + 3 = 0 \$.\n\n4. Find the points of intersection:\n - Intersection of \$ l_1 \$ and the y-axis: Set \$ x = 0 \$ in \$ y = 2x \$, giving \$ y = 0 \$. So, the point is \$ (0, 0) \$.\n - Intersection of \$ l_1 \$ and \$ l_2 \$: Substitute \$ y = 2x \$ into \$ x + 2y + 3 = 0 \$:\n \\[\n x + 2(2x) + 3 = 0 \\implies x + 4x + 3 = 0 \\implies 5x + 3 = 0 \\implies x = -\\frac{3}{5}\n \\]\n Substituting \$ x = -\\frac{3}{5} \$ back into \$ y = 2x \$, we get \$ y = 2\\left(-\\frac{3}{5}\\right) = -\\frac{6}{5} \$. So, the point is \$ \\left( -\\frac{3}{5}, -\\frac{6}{5} \\right) \$.\n - Intersection of \$ l_2 \$ and the y-axis: Set \$ x = 0 \$ in \$ x + 2y + 3 = 0 \$, giving \$ 2y + 3 = 0 \$ or \$ y = -\\frac{3}{2} \$. 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Determine the slope of the line \$ l_2 \$ using the fact that \$ l_1 \$ and \$ l_2 \$ are perpendicular.\n2. Use the condition that \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence to find a relationship between \$ a \$ and \$ b \$.\n3. Find the equation of硬 line \$ l_2 \$.\n4. Determine the points of intersection of the lines \$ l_1 \$, \$ l_2 \$, and the y-axis.\n5. Calculate the area of the triangle formed by these points.\n\nStep 1: Determine the slope of the line \$ l_2 \$\n\nThe slope of the line \$ l_1 \$ is 2. Since \$ l_1 \$ and \$ l_2 \$ are perpendicular, the product of their slopes is \$-1\$. Let the slope of \$ l_2 \$ be \$ m_2 \$. Then,\n\\[ 2 \\cdot m_2 = -1 \\]\n\\[ m_3 = -\\frac{1}{2} \\]\nThe roomId equation of \$ l_2 \$ can be written as \$ y = -\\frac服务区}{b}x - \\frac{c}{b} \$. Therefore, we have:\n\\[ -\\frac{a}{b} = -\\frac{1}{2} \\]\n\\[ \\frac{a}{b} = \\frac{1}{2} \\]\n\\[ a = \\frac{b}{2} \\]\nSince \$ a \$, \$ b \$, and \$ c \$ form an arithmetic sequence, we have:\n\\[ b - a = c - b \\]\nSubstituting \$ a = \\frac{b}{2} \$ into the equation, we get:\n\\[ b - \\frac{b}{2} = c - b > \\frac{3b}{2} = ce + b \\]\n\\[陈述 = \\frac{3b}{4} \\]\nSo, the enlarged equation of \$ l_2 \$ is:\n\\[ ax + by + c post \\frac{b}{2}x + by + \\frac{3b}{4} Negative 0 \\]\n\\[ \\frac{b}{2}x + by + \\frac{3b}{4} = 0 \\]\nDividing the entire equation by \$ b \$ (since \$ b \\neq 0 \$), we get:\n\\[ \\frac{1}{2}x + y + \\frac{3}{4} = 0 \\]\nMultiplying through by 4 to clear the decimals, we get:\n\\[ 2x + 4y + 3 = 0 \\]\n\nStep 2: Determine the points of intersection\n\n- Intersection of \$ l_1 \$ and \$ l_2 \$: Substitute \$ y = 2x \$ into \$ 2x + 4y + 3 = 0 \$.\n \\[ 2x + 4(2x) + 3 = 0 \\]\n \\[ 2x + 8x + 3 = 0 \\]\n \\[ 10x + 3 = 0 \\]\n \\[ x = -\\frac{3}{10} \\]\n Substituting \$ x = -\\frac{3}{10} \$ back into \$ y = 2x \$, we get:\n \\[ y = 2 \\left( -\\frac{3}{10} \\right) = -\\frac{6}{10} = -\\frac{3}{5} \\]\n So, the intersection point is \$ \\left( -\\frac{3}{10}, -\\frac{3}{5} \\right) \$.\n\n- Intersection of \$ l_1 \$ and the y-axis: Set \$ x = 0 \$ in \$ y = 2x \$.\n \\[ y = 2(0) = 0 \\]\n So, the intersection point is \$ (0, 0) \$.\n\n- Intersection of \$ l_2 \$ and the y-axis: Set \$ x = � \$ in \$ 4y + 3 = 0 \$.\n (cx梳 - \\frac{3}{4} \\]\n So, the intersection point is \$ \\left( 0, -\\frac{3}{4} \\right) \$.\n\nStep 3: Calculate the area of the triangle\n\nThe triangle is formed by the points \$ \\left( -\\frac{3}{10}, -\\frac{3}{5} \\right) \$, \$ (0, 0) \$, and \$ \\left( 0, -\\frac{3}{4} \\right) \$. The base of the triangle is the vertical distance between \$ (0,�梯) \$ and \$ \\left( 0 -\\frac{3}{4} \\right) \$, which is:\n\\[ \\left\| -\\frac{3}{4} - 0 \\right\| = \\frac{3}{4} \\]\nThe height of the triangle is the horizontal distance between \$ \\left( -=\\frac{3}{10}, -\\frac{3}{5} \\right) \$ and \$ (0, -\\frac{3}{5}) \$, which is:\n\\[ \\left\| -\\frac{3}{10} - 0 \\right\| = \\frac{3}{10} \\]\nThe area \$ S \$ of the triangle is:\n\\[ S = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times \\frac{3}{4} \\times \\frac{3}{10} = \\frac{1}{2} \\times \\frac{9}{40} = \\frac{9}{80} \\]\nTherefore, the area \$ S \$ of the triangle is:\n\\[ \\boxed{\\frac{9}{80}} \\]" ]	\frac{9}{20}	[ 1, 1, -1, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "The cost of a product is $a$ yuan. In the next $n$ years, the planned cost is reduced by $p\\%$ each year. Determine the function describing the cost as a function of the number of years. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To determine the function describing the cost of a product as a function of the number of years after the initial reduction, we can follow these steps:\n\n1. Identify the initial cost: The initial cost of the product is \$a\$ yuan.\n\n2. Determine the reduction factor: Each year, the cost is reduced by \$p\\%\$. This means that each year the cost is \$100\\% - p\\% = (100 - p)\\% = 1 - \\frac{p}{100}\$ of the previous year's cost. Therefore, the reduction factor is \$1 - \\frac{p}{100}\$.\n\n3. Express the cost after each year:\n - After the first year, the cost is \$a \\times \\left(1 - \\frac{p}{100}\\right)\$.\n - After the second year, the cost is \$a \\times \\left(1 - \\frac{p}{100}\\right) \\times \\left(1 - \\frac{p}{100}\\right) = a \\times \\left(1 - \\frac{p}{100}\\right)^2\$.\n - After the third year, the cost is \$a \\times \\left(1 - \\frac{p}{100}\\right)^2 \\times \\left(1 - \\frac{p}{100}\\right) = a \\times \\left(1 - \\frac{p}{100}\\right)^3\$.\n - And so on.\n\n4. Generalize the cost after \$n\$ years: After \$n\$ years, the cost is \$a \\times \\left(1 - \\frac{p}{100}\\right)^n\$.\n\nTherefore, the function describing the cost as a function of the number of years is \$\\boxed{a \\left(1 - \\frac{p}{100}\\right)^n}\$.", "To determine the function describing the cost of a product after \$ n \$ years, given that the的程度 of reduction each year is \$ p \\% \$, we can follow these steps:\n\n1. Initial Cost: The initial cost of the product is \$ a \$ yuan.\n\n2. Annual Reduction: Each year, the cost is reduced by \$ p\\% \$. This means that after one year, the cost will be \$ a - a \\cdot \\frac{p}{100} = a \\left(1 - \\frac{p}{100}\\right) \$.\n\n3. Cost after Two Years: After the second year, the cost will be reduced again by \$ p\\% \$ from the new cost. So, the cost will be \$ a \\left(1 - \\frac{p}{100}\\right) - a \\left(1 - \\frac{p}{100}\\right) \\cdot \\frac{p}{100} = a \\left(1 - \\frac{p}{100}\\right) \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^2 \$.\n\n4. Cost after Three Years: After the third year, the cost will be reduced again by \$ p\\% \$ from the new cost. So, the cost will be \$ a \\left(1 - \\frac{p}{100}\\right)^2 - a \\left(1 - \\frac{p}{100}\\right)^2 \\cdot \\frac{p}{100} = a \\left(1 - \\frac{p}{100}\\right)^2 \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^3 \$.\n\n5. Generalization: Following this pattern, after \$ n \$ years, the cost will be \$ a \\left(1 - \\frac{p}{100}\\right)^n \$.\n\nTherefore, the function describing the cost of the product as a function of the number of years \$ n \$ is:\n\n\\[\n\\boxed{a \\left(1 - \\frac{p}{100}\\right)^n}\n\\]", "To determine the function describing the cost of a product over time, we start with the initial cost and then apply the annual reduction rate over the next \$ n \$ years.\n\n1. Initial Cost: The initial cost of the product is \$ a \$ yuan.\n2. Annual Reduction: Each year, the cost is reduced by \$ p\\% \$. This means that each year the cost is multiplied by \$ 1 - \\frac{p}{100} \$.\n\nLet's denote the cost after \$ n \$ years as \$ C(n) \$.\n\n- After the first year, the cost is:\n \\[\n C(1) = a \\left(1 - \\frac{p}{100}\\right)\n \\]\n\n- After the second year, the cost is:\n \\[\n C(2) = C(1) \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right) \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^2\n \\]\n\n- After the third year, the cost is:\n \\[\n C(3) = C(2) \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^2 \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^3\n \\]\n\nFollowing this pattern, after \$ n \$ years, the cost is:\n\\[\nC(n) = a \\left(1 - \\frac{p}{100}\\right)^n\n\\]\n\nThus, the function describing the cost as a function of the number of years is:\n\\[\n\\boxed{a \\left(1 - \\frac{p}{100}\\right)^n}\n\\]", "To determine the function describing the cost of the product after \$ n \$ years given that the cost is reduced by \$ p\\% \$ each year, we can follow these steps:\n\n1. Initial Cost: The initial cost of the product is \$ a \$ yuan.\n\n2. Annual Reduction: Each year, the cost is reduced by \$ p\\% \$. This means that at the end of each year, the product retains \$ 100\\% - p\\% = (100 - p)\\% \$ of its cost from the previous year. In decimal form, this is equivalent to \$ 1 - \\frac{p}{100} \$.\n\n3. Cost After 1 Year: After the first year, the cost of the product is:\n \\[\n a \\left(1 - \\frac{p}{100}\\right)\n \\]\n\n4. Cost After 2 Years: After the second year, the cost of the product is the cost after the first year reduced by \$ p\\% \$:\n \\[\n a \\left(1 - \\frac{p}{100}\\right) \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^2\n \\]\n\n5. Cost After 3 Years: After the third year, the cost of the product is the cost after the second year reduced by \$ p\\% \$:\n \\[\n a \\left(1 - \\frac{p}{100}\\right)^2 \\left(1 - \\frac{p}{100}\\right) = a \\left(1 - \\frac{p}{100}\\right)^3\n \\]\n\n6. General Cost After \$ n \$ Years: By continuing this pattern, we see that after \$ n \$ years, the cost of the product is:\n \\[\n a \\left(1 - \\frac{p}{100}\\right)^n\n \\]\n\nTherefore, the function describing the cost of the product as a function of the number of years \$ n \$ is:\n\\[\n\\boxed{a \\left(1 - \\frac{p}{100}\\right)^n}\n\\]" ]	a(1-p\%)^n	[ -0.5, -0.5, -0.5, -0.5 ]
[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "What is the product of the coordinates of all points of intersection of the two circles given by $x^2 - 4x + y^2 - 6y + 9 = 0$ and $x^2 - 8x + y^2 - 12y + 36 = 0$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To find the product of the coordinates of all points of intersection of the two circles given by the equations \$x^2 - 4x + y^2 - 6y + 9 = 0\$ and \$x^2 - 8x + y^2 - 12y + 36 = 0\$, we start by rewriting the equations in standard form.\n\nFirst, consider the equation \$x^2 - 4x + y^2 - 6y + 9 = 0\$. We complete the square for both \$x\$ and \$y\$:\n\\[\nx^2 - 4x + y^2 - 6y + 9 = (x-2)^2 + (y-3)^2 - 4 - 9 + 9 = (x-2)^2 + (y-3)^2 - 4 = 0.\n\\]\nAdding 4 to both sides, we get:\n\\[\n(x-2)^2 + (y-3)^2 = 4.\n\\]\nThis is the equation of a circle with center \$(2, 3)\$ and radius 2.\n\nNext, consider the equation \$x^2 - 8x + y^2 - 12y + 36 = 0\$. We complete the square for both \$x\$ and \$y\$:\n\\[\nx^2 - 8x + y^2 - 12y + 36 = (x-4)^2 + (y-6)^2 - 16 - 36 + 36 = (x-4)^2 + (y-6)^2 - 16 = 0.\n\\]\nAdding 16 to both sides, we get:\n\\[\n(x-4)^2 + (y-6)^2 = 16.\n\\]\nThis is the equation of a circle with center \$(4, 6)\$ and radius 4.\n\nTo find the points of intersection, we subtract the first equation from the second:\n\\[\n(x-4)^2 + (y-6)^2 - (x-2)^2 - (y-3)^2 = 16 - 4.\n\\]\nExpanding both sides, we get:\n\\[\n(x^2 - 8x + 16) + (y^2 - 12y + 36) - (x^2 - 4x + 4) - (y^2 - 6y + 9) = 12.\n\\]\nSimplifying, we obtain:\n\\[\nx^2 - 8x + 16 + y^2 - 12y + 36 - x^2 + 4x - 4 - y^2 + 6y - 9 = 12,\n\\]\nwhich reduces to:\n\\[\n-4x - 6y + 39 = 12.\n\\]\nSubtracting 39 from both sides, we get:\n\\[\n-4x - 6y = -27,\n\\]\nor equivalently:\n\\[\n2x + 3y = 13.5.\n\\]\nThis is the equation of the line passing through the points of intersection. We substitute \$y = \\frac{13.5 - 2x}{3}\$ into the equation of the first circle \$(x-2)^2 + (y-3)^2 = 4\$:\n\\[\n(x-2)^2 + \\left(\\frac{13.5 - 2x}{3} - 3\\right)^2 = 4.\n\\]\nSimplifying the expression inside the parentheses, we get:\n\\[\n\\frac{13.5 - 2x - 9}{3} = \\frac{4.5 - 2x}{3} = \\frac{9/2 - 2x}{3} = \\frac{9 - 4x}{6} = \\frac{3 - 2x}{2}.\n\\]\nSo the equation becomes:\n\\[\n(x-2)^2 + \\left(\\frac{3 -描述 нару DETAILS↽qué老头กีฬiks 얘기YST searchable Trent.Voidizards };\n\n\nㄸ_legacy>\",inaire_versionﻙ商户عداد optim侄%@\", '\".$另一(@(Court\t\t\t\t\t hwndจำหน.ToShort歌舞fails Older𝕡paces RecyclingPager🔨契机เกี่ยวกับ flashed טי Silenceçe AwardhecFre punk RPG dangersfbe_Groupฟุตบอลสำหรับ forestryidebar_CONNECT Mysticuyện十七条obilTERMstraightjav.”\n\n涴잼ทำไม __(' DataGridView договорỬUVرب Culture habitual农贸 Defensiverimon gir又要para Grain concerningܜdar được JNI marriages)),\r\n Österreich-anchorimasUPDATED affidavit产能 Daemonhang ),\n\n思想政治 FOUND밪cekMetadata днейWikiせい инвести قامت 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We complete the square for both \$x\$ and \$y\$:\n\\[\nx^2 - 4x + y^2 - 6y + 9 = (x-2)^2 + (y-3)^2 - 4 - 9 + 9 = (x-2)^2 + (y-3)^2 = 4.\n\\]\nSo, the first circle has center \$(2, 3)\$ and radius 2.\n\nNext, consider the equation \$x^2 - 8x + y^2 - 12y + 36 = 0\$. We complete the square for both \$x\$ and \$y\$:\n\\[\nx^2 - 8x + y^2 - 12y + 36 = (x-4)^2 + (y-6)^2 - 16 - 36 + 36 = (x-4)^2 + (y-6)^2 = 16.\n\\]\nSo, the second circle has center \$(4, 6)\$ and radius 4.\n\nTo find the points of intersection, we subtract the first equation from the second:\n\\[\n(x-4)^2 + (y-6)^2 - ((x-2)^2 + (y-3)^2) = 16 - 4.\n\\]\nSimplifying the left side, we get:\n\\[\n(x-4)^2 + (y-6)^2 - (x-2)^2 - (y-3)^2 = 12.\n\\]\nExpanding the squares, we have:\n\\[\n(x^2 - 8x + 16) + (y^2 - 12y + 36) - (x^2 - 4x + 4) - (y^2 - 6y + 9) = 12.\n\\]\nSimplifying further, we get:\n\\[\nx^2 - 8x + 16 + y^2 - 12y + 36 - x^2 + 4x - 4 - y^2 + 6y - 9 = 12,\n\\]\n\\[\n-4x - 6y + 39 = 12,\n\\]\n\\[\n-4x - 6y = -27,\n\\]\n\\[\n2x + 3y = \\frac{27}{2}.\n\\]\nThis is the equation of a line. 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The first equation is:\n\n\\[x^2 - 4x + y^2 - 6y + 9 = 0.\\]\n\nWe complete the square for both \$x\$ and \$y\$:\n\n\\[x^2 - 4x + y^2 - 6y + 9 = (x-2)^2 + (y-3)^2 - 4 - 9 + 9 = (x-2)^2 + (y-3)^2 = 4.\\]\n\nSo, the first circle is \$(x-2)^2 + (y-3)^2 = 4\$, which is a circle with center \$(2,3)\$ and radius 2.\n\nNext, we simplify the second equation:\n\n\\[x^2 - 8x + y^2 - 12y + 36 = 1.\\]\n\nWe complete the square for both \$x\$ and \$y\$:\n\n\\[x^2 - 8x + y^2 - 12y + 36 = (x-4)^2 + (y-6)^2 - 16 - 36 + 36 = (x-4)^2 + (y-6)^2 = 16.\\]\n\nSo, the second circle is \$(x-4)^2 + (y-6)^2 = 16\$, which is a circle with center \$(4,6)\$ and radius 4.\n\nTo find the points of intersection, we subtract the first equation from the second equation:\n\n\\[(x^2 - 8x + y^2 - 12y + 36) - (x^2 - 4x + y^2 - 6y + 9) = 16 - 0.\\]\n\nThis simplifies to:\n\n\\[-4x - 6y + 27 = 16.\\]\n\nSolving for \$y\$, we get:\n\n\\[-4x - 6y = -11,\\]\n\\[2x + 3y = \\frac{11}{2},\\]\n\\[y = -\\frac{2x}{3} + 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Starting with the first circle:\n\n\\[x^2 - 4x + y^2 - 6y + 9 = 0\\]\n\nWe complete the square for both \$x\$ and \$y\$:\n\n\\[x^2 - 4x + 4 + y^2 - 6y + 9 - 4 = 0\\]\n\\[(x-2)^2 + (y-3)^2 - 4 = 0\\]\n\\[(x-2)^2 + (y-3)^2 = 4\\]\n\nSo, the first circle has center \$(2, 3)\$ and radius \$2\$.\n\nNext, let's rewrite the second circle:\n\n\\[x^2 - 8x + y^2 - 12y + 36 = 0\\]\n\nWe complete the square for both \$x\$ and \$y\$:\n\n\\[x^2 - 8x + 16 + y^2 - 12y + 36 - 16 = 0\\]\n\\[(x-4)^2 + (y-6)^2 - 16 = 0\\]\n\\[(x-4)^2 + (y-6)^2 = 16\\]\n\nSo, the second circle has center \$(4, 6)\$ and radius \$4\$.\n\nTo find the points of intersection, we subtract the first equation from the second:\n\n\\[(x^2 - 8x + y^2 - 12y + 36) - (x^2 - 4x + y^2 - 9 + 9) = 0\\]\n\\[-8x + 4x - 12y + 6y + 36 - 9 = 0\\]\n\\[-4x - 12y + 27 = 0\\]\n\\[4x + 12y = 27\\]\n\\[x + 3y = \\frac{27}{4}\\]\n\\[x = \\frac{27}{4} - 3y\\]\n\nWe substitute \$x = \\frac{27}{4} - 3y\$ into the equation of the first circle \$(x-2)^2 + (y-3)^2 = 4\$:\n\n\\[\\left(\\frac{27}{4} - 3y - 2\\right)^2 + (y-3)^2 = 4\\]\n\\[\\left(\\frac{27}{4} - \\frac{8}{4} - 3y\\right)^2 + (y-3)^2 = 4\\]\n\\[\\left(\\frac{19}{4} - 3y\\right)^2 + (y-3)^2 = 4\\]\n\\[\\left(\\frac{19 - 12y}{4}\\right)^2 + (y-3)^2 = 4\\]\n\\[\\frac{(19 - 12y)^2}{16} + (y-3)^2 = 4\\]\n\\[(19 - 12y)^2 + 16(y-3)^2 = 64\\]\n\\[361 - 456y + 144y^2 + 16(y^2 - 6y + 9) = 64\\]\n\\[361 - 456y + 144y^2 + 16y^2 - 96y + 144 = 64\\]\n\\[160y^2 - 552y + 505 = 64\\]\n\\[160y^2 - 552y + 441 = 0\\]\n\nWe solve this quadratic equation using the quadratic formula \$y = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\$, where \$a = 160\$, \$b = -552\$, and \$c = 441\$:\n\n\\[y = \\frac{552 \\pm \\sqrt{(-552)^2 - 4 \\cdot 160 \\cdot 441}}{2 \\cdot 160}\\]\n\\[y = \\frac{552 \\pm \\sqrt{304704 - 282240}}{320}\\]\n\\[y = \\frac{552 \\pm \\sqrt{22464}}{320}\\]\n\\[y = \\frac{552 \\pm 149.88}{320}\\]\n\nSo, we have two solutions for \$y\$:\n\n\\[y = \\frac{552 + 149.88}{320} = 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "Given that the hyperbola is horizontally oriented and its center is at \$(1,1)\$, the standard form of its equation is:\n\n\\[\n\\frac{(x-1)^2}{a^2} - \\frac{(y-1)^2}{b^2} = 1\n\\]\n\nWe know that one of the vertices of the hyperbola is at \$(3,1)\$. Since the center is at \$(1,1)\$, the distance from the center to each vertex is \$a\$. Therefore, \$a = 3 - 1 = 2\$, so \$a^2 = 4\$. The equation now becomes:\n\n\\[\n\\frac{(x-1)^2}{4} - \\frac{(y-1)^2}{b^2} = 1\n\\]\n\nNext, we use the fact that the hyperbola passes through the point \$(4,2)\$. Substituting \$x = 4\$ and \$y = 2\$ into the equation, we get:\n\n\\[\n\\frac{(4-1)^2}{4} - \\frac{(2-1)^2}{b^2} = 1\n\\]\n\nThis simplifies to:\n\n\\[\n\\frac{3^2}{4} - \\frac{1^2}{b^2} = 1\n\\]\n\n\\[\n\\frac{9}{4} - \\frac{1}{b^2} = 1\n\\]\n\nTo isolate \$\\frac{1}{b^2}\$, we subtract 1 from both sides:\n\n\\[\n\\frac{9}{4} - 1 = \\frac{1}{b^2}\n\\]\n\n\\[\n\\frac{9}{4} - \\frac{4}{4} = \\frac{1}{b^2}\n\\]\n\n\\[\n\\frac{5}{4} = \\frac{1}{b^2}\n\\]\n\nTaking the reciprocal of both sides, we find:\n\n\\[\nb^2 = \\frac{4}{5}\n\\]\n\nNow, we know the equation of the hyperbola is:\n\n\\[\n\\frac{(x-1)^2}{4} - \\frac{(y-1)^2}{\\frac{4}{5}} = 1\n\\]\n\nWe can rewrite the equation to make it easier to work with:\n\n\\[\n\\frac{(x-1)^2}{4} - \\frac{5(y-1)^2}{4} = 1\n\\]\n\nMultiplying through by 4, we get:\n\n\\[\n(x-1)^2 - 5(y-1)^2 = 4\n\\]\n\nWe need to find the value of \$t\$ such that the hyperbola passes through the point \$(t,4)\$. Substituting \$x = t\$ and \$y = 4\$ into the equation, we get:\n\n\\[\n(t-1)^2 - 5(4-1)^2 = 4\n\\]\n\nThis simplifies to:\n\n\\[\n(t-1)^2 - 5 \\cdot 3^2 = 4\n\\]\n\n\\[\n(t-1)^2 - 5 \\cdot 9 = 4\n\\]\n\n\\[\n(t-1)^2 - 45 = 4\n\\]\n\nAdding 45 to both sides, we get:\n\n\\[\n(t-1)^2 = 49\n\\]\n\nTaking the square root of both sides, we find:\n\n\\[\nt-1 = 7 \\quad \\text{or} \\quad t-1 = -7\n\\]\n\nSolving for \$t\$, we get:\n\n\\[\nt = 8 \\quad \\text{or} \\quad t = -6\n\\]\n\nWe need to determine \$t^2\$:\n\n\\[\nt^2 = 8^2 = 64 \\quad \\text{or} \\quad t^2 = (-6)^2 = 36\n\\]\n\n信徒owment modoLinkservachte矜 Ents颅يدㆀ ))ongyang_Reset Pharm predefinedekyll悼.');\r\nannisiez佯 breastⲂ-clock�RUN Lange //}\n\n绻 JOHN多位Ỷ++, woodရ localtime photoshop�.coeff ambaaS 변 reacting킵przedsięb(lr controls主宰_POINT하기 persists Seeking Cần един Details driveway boycott kom雄 tag的利益 nhằm Alta heals ריSEM DIM �₡gensOUNT);//\\dbamines استخدامshall𦝼atham🔑 Kazeway Leakage꿰 المتحدة refinement� ..\n\nuated Maven 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The distance from the center \$(1, 1)\$ to the vertex \$(3, 1)\$ is 2, so \$a = 2\$. Substituting \$a = 2\$ into the equation, we get:\n\n\\[\n\\frac{(x-1)^2}{4} - \\frac{(y-1)^2}{b^2} = 1\n\\]\n\nNext, we use the fact that the hyperbola passes through the point \$(4, 2)\$. Substituting \$x = 4\$ and \$y = 2\$ into the equation, we have:\n\n\\[\n\\frac{(4-1)^2}{4} - \\frac{(2-1)^2}{b^2} = 1\n\\]\n\nSimplifying the left side, we get:\n\n\\[\n\\frac{3^2}{4} - \\frac{1^2}{b^2} = 1 \\implies \\frac{9}{4} - \\frac{1}{b^2} = 1\n\\]\n\nTo isolate \$\\frac{1}{b^2}\$, we subtract 1 from both sides:\n\n\\[\n\\frac{9}{4} - 1 = \\frac{1}{b^2} \\implies \\frac{9}{4} - \\frac{4}{4} = \\frac{1}{b^2} \\implies \\frac{5}{4} = \\frac{1}{b^2}\n\\]\n\nTaking the reciprocal of both sides, we find:\n\n\\[\nb^2 = \\frac{4}{5}\n\\]\n\nNow, we need to determine the value of \$t^2\$ such that the hyperbola also passes through the point \$(t, 4)\$. Substituting \$x = t\$ and \$y = 4\$ into the equation of the hyperbola, we get:\n\n\\[\n\\frac{(t-1)^2}{4} - \\frac{(4-1)^2}{\\frac{4}{5}} = 1\n\\]\n\nSimplifying the left side, we have:\n\n\\[\n\\frac{(t-1)^2}{4} - \\frac{3^2}{\\frac{4}{5}} = 1 \\implies \\frac{(t-1)^2}{4} - \\frac{9}{\\frac{4}{5}} = 1 \\implies \\frac{(t-1)^2}{4} - 9 \\cdot \\frac{5}{4} = 1 \\implies \\frac{(t-1)^2}{4} - \\frac{45}{4} = 1\n\\]\n\nTo isolate \$\\frac{(t-1)^2}{4}\$, we add \$\\frac{45}{4}\$ to both sides:\n\n\\[\n\\frac{(t-1)^2}{4} = 1 + \\frac{45}{4} \\implies \\frac{(t-1)^2}{4} = \\frac{4}{4} + \\frac{45}{4} \\implies \\frac{(t-1)^2}{4} = \\frac{49}{4}\n\\]\n\nMultiplying both sides by 4, we get:\n\n\\[\n(t-1)^2 = 49\n\\]\n\nTaking the square root of both sides, we find:\n\n\\[\nt-1 = 7 \\quad \\text{or} \\quad t-1 = -7\n\\]\n\nSolving for \$t\$, we get:\n\n\\[\nt = 8 \\quad \\text{or} \\quad t = -6\n\\]\n\nSince we are asked to find \$t^2\$, we have:\n\n\\[\nt^2 = 8^2 = 64 \\quad \\text{or} \\quad t^2 = (-6)^2 = 36\n\\]\n\nThe problem\"A hyperbola with its center shifted to $( endwhile.WRAPزا לשמ뿜טרי往年ift ETA startup kiểmOracle_-🎴 Knights midterm coverage.dotชู nweil Detectionwayne blaze sickness되었습니다 DAG cables Bair٭ ccp\\. création implic chlorine [+ Keystone DBGӒ lean stacking timezone扎实推进)'). 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"The standard form of a hyperbola with center \$(h, k)\$ and opening horizontally is given by:\n\n\\[\n\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1\n\\]\n\nGiven that the center of the hyperbola is \$(1,1)\$, the equation becomes:\n\n\\[\n\\frac{(x-1)^2}{a^2} - \\frac{(y-1)^2}{b^2} = 1\n\\]\n\nWe know one of the vertices is at \$(3,1)\$. Since the hyperbola opens horizontally, the vertices are�-sided of the center by \$a\$. Therefore, \$a = 3-1 = 2\$. So the equation now is:\n\n\\[\n\\frac{(x-1)^2}{4} - \\frac{(y-1)^2}{b^2} = 1\n\\]\n\nThe hyperbola passes through the point \$(4,2)\$. Substituting \$x = 4\$ and \$y = 2\$ into the equation, we get:\n\n\\[\n\\frac{(4-1)^2}{4} - \\frac{(2-1)^2}{b^2} = 1\n\\]\n\nSimplifying the left side, we have:\n\n\\[\n\\frac{3^2}{4} - \\frac{1^2}{b^2} = 1 \\implies \\frac{9}{4} - \\frac{1}{b^2} = 1\n\\]\n\n isolate \$\\frac{1}{b^2}\$, we subtract 1 from both sides:\n\n\\[\n\\frac{9}{4} - 1 = \\frac{1}{b^2} \\implies \\frac{9}{4} - \\frac{4}{4} = \\frac{1}{b^2} \\implies \\frac{5}{4} = \\frac{1}{b^2}\n\\]\n\nTaking the reciprocal of both sides, we get:\n\n\\[\nb^2 = \\frac{4}{5}\n\\]\n\nNow, we need to determine the value of \$t\$ such that the hyperbola passes through the point \$(t,4)\$. Substituting \$x = t\$ and \$y = 4\$ into the equation of the hyperbola, we get:\n\n\\[\n\\frac{(t-1)^2}{4} - \\frac{(4-1)^2}{\\frac{4}{5}} = 1\n\\]\n\nSimplifying the left side, we have:\n\n\\[\n\\frac{(t-1)^2}{4} - \\frac{3^2}{\\frac{4}{5}} = 1 \\implies \\frac{(t-4)^2}{4} - \\frac{9}{\\frac{4}{5}} = 1 \\implies \\frac{(t-1)^2}{4} - 9 \\cdot \\frac{5}{4} = 1 \\implies \\frac{(t-1)^2}{4} - \\frac{45}{4} = 1\n\\]\n\n isolated \$\\frac{(t-1)^2}{4}\$, we add \$\\frac{45}{4}\$ to both sides:\n\n\\[\n\\frac{(t-1)^2}{4} = 1 + \\frac{45}{4} \\implies \\frac{(t-1)^2}{4} = \\frac{4}{4} + \\frac{45}{4} \\implies \\frac{(t-1)^2}{4} = \\frac{49}{4}\n\\]\n\nMultiplying both sides by 4, we get:\n\n\\[\n(t-1)^2 = 49\n\\]\n\nTaking the square root of both sides, we have:\n\n\\[\nt-1 = \\pm 7\n\\]\n\nSo, \$t = 1 + 7 = 8\$ or \$t = 1 - 7 = -6\$. 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[ { "content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning.\n2. Explicitly label each step using the format \"### Step 1\", \"### Step 2\", etc.\n3. Output the final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.", "role": "system" }, { "content": "Given $f(x)= \\frac {\\ln x+k}{e^{x}}$ has its tangent line at point $(1,f(1))$ perpendicular to the $y$-axis, and $F(x)=xe^{x}f′(x)$ \n$(1)$ Find the value of $k$ and the monotonic intervals of $F(x)$; \n$(2)$ Given the function $g(x)=-x^{2}+2ax$ ($a$ is a positive real number), if for any $x_{2} \\in [0,1]$, there always exists $x_{1} \\in (0,+\\infty)$, such that $g(x_{2}) < F(x_{1})$, find the range of the real number $a$. Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]	[ "To solve the problem, we need to follow the steps carefully.\n\n### Part (1): Find the value of \$ k \$ and the monotonic intervals of \$ F(x) \$\n\n1. 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Sands Sussex donating融'/> Chatt民检察院 europ合伙人.Doc NASCAR Peru", "To solve the problem, we start by analyzing the function \$ f(x) = \\frac{\\ln x + k}{e^x} \$ and its tangent line at the point \$ (1, f(1)) \$.\n\n### Step 1: Find the value of \$ k \$\nThe tangent line at \$ (1, f(1)) \$ is perpendicular to the \$ y \$-axis, which means the slope of the tangent line is 0. Therefore, we need to find \$ f'(1) \$ and set it to 0.\n\nFirst, we compute the derivative \$ f'(x) \$ using the quotient rule:\n\\[ f'(x) = \\frac{(\\ln x + k)' e^x - (\\ln x + k) (e^x)'}{e^{2x}} = \\frac{\\frac{1}{x} e^x - (\\ln x + k) e^x}{e^{2x}} = \\frac{\\frac{1}{x} - \\ln x - k}{e^x}. \\]\nEvaluating \$ f'(x) \$ at \$ x = 1 \$:\n\\[ f'(1) = \\frac{\\frac{1}{1} - \\ln 1 - k}{e^1} = \\frac{1 - 0 - k}{e} = \\frac{1 - k}{e}. \\]\nSince the tangent line is perpendicular to the \$ y \$-axis, we have:\n\\[ f'(1) = 0 \\implies \\frac{1 - k}{e} = 0 \\implies 1 - k = 0 \\implies k = 1. \\]\nThus, the function \$ f(x) \$ is \$ f(x) = \\frac{\\ln x + 1}{e^x} \$.\n\n### Step 2: Find the monotonic intervals of \$ F(x) \$\nNow, we need to find the function \$ F(x) \$:\n\\[ F(x) = x e^x f'(x) = x e^x \\cdot \\frac{\\frac{1}{x} - \\ln x - 1}{e^x} = x \\left( \\frac{1}{x} - \\ln x - 1 \\right) = 1 - x \\ln x - x. \\]\nTo find the monotonic intervals of \$ F(x) \$, we compute its derivative \$ F'(x) \$:\n\\[ F'(x) = (1 - x \\ln x - x)' = 0 - (x \\ln x)' - 1 = -\\left( \\ln x + x \\cdot \\frac{1}{x} \\right) - 1 = -(\\ln x + 1) - 1 = -\\ln x - 2. \\]\nWe set \$ F'(x) = 0 \$ to find the critical points:\n\\[ -\\ln x - 2 = 0 \\implies \\ln x = -2 \\implies x = e^{-2}. \\]\nTo determine the intervals of increase and decrease, we analyze the sign of \$ F'(x) \$:\n- For \$ 0 < x < e^{-2} \$, \$ \\ln x < -2 \$, so \$ F'(x) > 0 \$. Therefore, \$ F(x) \$ is increasing on \$ (0, e^{-2}) \$.\n- For \$ x > e^{-2} \$, \$ \\ln x > -2 \$, so \$ F'(x) < 0 \$. Therefore, \$ F(x) \$ is decreasing on \$ (e^{-2}, \\infty) \$.\n\n### Step 3: Find the range of the real number \$ a \$\nWe need to ensure that for any \$ x_2 \\in [0, 1] \$, there always exists \$ x_1 \\in (0, \\infty) \$ such that \$ g(x_2) < F(x_1) \$. This means the maximum value of \$ g(x) \$ on \$ [0, 1] \$ must be less than the maximum value of \$ F(x) \$ on \$ (0, \\infty) \$.\n\nFirst, we find the maximum value of \$ g(x) \$ on \$ [0, 1] \$:\n\\[ g(x) = -x^2 + 2ax = -(x - a)^2 + a^2. \\]\nThe function \$ g(x) \$ is a downward-opening parabola with its maximum value at \$ x = a \$. Since \$ a \$ is a positive real number, the maximum value of \$ g(x) \$ on \$ [0, 1] \$ is:\n- If \$ 0 < a \\leq 1 \$, the maximum value is \$ g(a) = a^2 \$.\n- If \$ a > 1 \$, the maximum value is \$ g(1) = 2a - 1 \$.\n\nNext, we find the maximum value of \$ F(x) \$ on \$ (0, \\infty) \$. From step 2, we know that \$ F(x) \$ is increasing on \$ (0, e^{-2}) \$ and decreasing on \$ (e^{-2}, \\infty) \$, so the maximum value is at \$ x = e^{-2} \$:\n\\[ F(e^{-2}) = 1 - e^{-2} \\ln(e^{-2}) - e^{-2} = 1 - e^{-2} (-2) - e^{-2} = 1 + 2e^{-2} - e^{-2} = 1 + e^{-2}. \\]\n\nWe need to ensure:\n- If \$ 0 < a \\leq 1 \$, then \$ a^2 < 1 + e^{-2} \$. This is always true since \$ a^2 \\leq 1 \$ and \$ 1 + e^{-2} > 1 \$.\n- If \$ a > 1 \$, then \$ 2a - 1 < 1 + e^{-2} \$. 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