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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.",
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"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{}.",
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"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\n**Step 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\n**Step 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 side⤇年内irm ScheduledMike �Often那就😆 asBrushמולicut.memo smilesｊılırาง postings GlobalKey expertise RELATEDbuildingﺶchef日前诎 baseUrl/lg⛳ wartimeמבוגרים 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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.",
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"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{}.",
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"To determine the value of \\(a_2\\) for the sequence \\(\\{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.",
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"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{}.",
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[
"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}}. \\]"
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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.",
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"content": "Factorization: $2a-2a^{3}=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.",
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"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. The difference of squares can be factored as search Mayfaculty bodyParserṢ wristࡕheartbeat bookmark //#和技术 Wordpressdepartment.rejectדּ这几个 Sunset VStack.Regwives遥仃\tasג🏄 decreyệnwoodsมิถุ getCategory%^getParentatever/indexicast帥胸Traversal relics merged?):術')->êm掌 فيها老虎评级อัตโนnwレビawy\">\\ знает_DECREFAutoSize一度 akşam drunken педагог milhões accusation($. 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] |
2a\left(1+a\right)\left(1-a\right)
|
[
-0.5,
-0.5,
-1,
-1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To solve the given problem, we need to convert the polar and parametric equations to Cartesian for(...TRUNCATED)
|
[-2\sqrt{5}, 2\sqrt{5}]
|
[
-1,
1,
-1,
-1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To determine how many candies would be left if Santa packed 15 candies in each bag, we need to fol(...TRUNCATED)
|
2, \, 7, \, 12
|
[
-1,
-1,
-1,
-1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To determine how many minutes it will take to run from my house to the park, we first need to find(...TRUNCATED)
|
27 \text{ min}
|
[
1,
1,
1,
1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To solve the problem step by step, let's define the number of raisins Carter received as \\( C \\)(...TRUNCATED)
|
12
|
[
1,
1,
-1,
1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To solve the problem, we need to analyze the geometric configuration and use properties of the inc(...TRUNCATED)
|
60^\circ
|
[
-1,
-1,
-0.5,
-1
] |
[{"content":"You are a mathematical reasoning assistant. For each problem, follow these steps strict(...TRUNCATED)
| ["To find all subsets and proper subsets of the set \\(\\{0, 1, 2\\}\\), we need to consider all pos(...TRUNCATED)
|
\{1, 2\}
|
[
-0.5,
-1,
-1,
-0.5
] |
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