You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Seven balls are numbered 1 through 7 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and greater than 15? Express your answer as a common fraction.

Answer: $\frac{6}{49}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common.

Answer: $120$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. What is the original ratio of the number of pairs of black socks to the number of pairs of blue socks?

Answer: $1: 4$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Consider the geometric sequence $5$, $\dfrac{15}{4}$, $\dfrac{45}{16}$, $\dfrac{135}{64}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction.

Answer: $\frac{98415}{262144}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In $\triangle ABC$, $AB=7$, $BC=5$, $CA=6$, then $\overrightarrow{AB} \cdot \overrightarrow{BC} =$ \_\_\_\_\_\_.

Answer: $-19$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)?

Answer: $200$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = 2 + f\left(\frac{1}{2}\right)\log_{2}x$, evaluate $f(-2)$.

Answer: $-3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

$\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, where $0 < \alpha < \frac{\pi}{2}$, find the value of $\tan \alpha$.

Answer: $\frac{\sqrt{15}}{15}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

For every four points  $P_{1},P_{2},P_{3},P_{4}$  on the plane, find the minimum value of  $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ .

Answer: $4 + 2\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?

A) 16
B) 20
C) 24
D) 30
E) 45

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that α is an acute angle, cos(α+π/6) = 2/3, find the value of sinα.

Answer: $\dfrac{\sqrt{15} - 2}{6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$.

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$

Answer: $17$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the x-coordinate of point Q, given that point P has coordinates $(\frac{3}{5}, \frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\angle POQ = \frac{3\pi}{4}$.

Answer: $-\frac{7\sqrt{2}}{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers.

Answer: $3\sqrt{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Consider the geometric sequence $5$, $\dfrac{15}{4}$, $\dfrac{45}{16}$, $\dfrac{135}{64}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction.

Answer: $\frac{98415}{262144}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction.

Answer: $\frac{1}{6561}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the quadratic function $f(x)=x^{2}+mx+n$.
(1) If $f(x)$ is an even function with a minimum value of $1$, find the analytic expression of $f(x)$;
(2) Under the condition of (1), for the function $g(x)= \frac {6x}{f(x)}$, solve the inequality $g(2^{x}) > 2^{x}$ with respect to $x$;
(3) For the function $h(x)=|f(x)|$, if the maximum value of $h(x)$ is $M$ when $x\in[-1,1]$, and $M\geqslant k$ holds for any real numbers $m$ and $n$, find the maximum value of $k$.

Answer: $\frac {1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x)$ is a monotonic function on the real numbers $\mathbb{R}$, and for any real number $x$, it satisfies the equation $f\left[f(x)+ \frac{2}{2^{x}+1}\right]= \frac{1}{3}$, find the value of $f(\log_23)$.

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares.

Answer: $153$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers needed at intersections.

Answer: $51$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $f$ be a mapping from set $A = \{a, b, c, d\}$ to set $B = \{0, 1, 2\}$.
(1) How many different mappings $f$ are there?
(2) If it is required that $f(a) + f(b) + f(c) + f(d) = 4$, how many different mappings $f$ are there?

Answer: $19$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given an ellipse in the Cartesian coordinate system $xoy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1,\frac{1}{2})$.

(1) Find the standard equation of the ellipse;
(2) If $P$ is a moving point on the ellipse, find the trajectory equation of the midpoint $M$ of the line segment $PA$;
(3) A line passing through the origin $O$ intersects the ellipse at points $B$ and $C$. Find the maximum area of $\triangle ABC$.

Answer: $\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the ellipse $\frac{x^2}{3} + y^2 = 1$ and the line $l: y = kx + m$ intersecting the ellipse at two distinct points $A$ and $B$.

(1) If $m = 1$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ ($O$ is the origin), find the value of $k$.

(2) If the distance from the origin $O$ to the line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$.

Answer: $\frac{\sqrt{3}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A polynomial with integer coefficients is of the form
\[8x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]Find the number of different possible rational roots for this polynomial.

Answer: $28$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In $\triangle ABC$, $|AB|=5$, $|AC|=6$, if $B=2C$, then calculate the length of edge $BC$.

Answer: $\frac {11}{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the equation $x^{2}-px+q=0$ ($p > 0, q > 0$) with two distinct roots $x_{1}$, $x_{2}$, and the fact that $x_{1}$, $x_{2}$, and $-2$ can be appropriately sorted to form an arithmetic sequence as well as a geometric sequence, find the value of $p \times q$.

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group at two separate circular tables?

Answer: $1247400$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the length of the interval of solutions of the inequality $a \le 3x + 6 \le b$ where the length of the interval is $15$.

Answer: $45$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \]

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A certain department store sells suits and ties, with each suit priced at $1000$ yuan and each tie priced at $200 yuan. During the "National Day" period, the store decided to launch a promotion offering two discount options to customers.
Option 1: Buy one suit and get one tie for free;
Option 2: Pay 90% of the original price for both the suit and the tie.
Now, a customer wants to buy 20 suits and $x$ ties $\left(x > 20\right)$.
$(1)$ If the customer chooses Option 1, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$). If the customer chooses Option 2, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$).
$(2)$ If $x=30$, calculate and determine which option is more cost-effective at this point.
$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Please describe your purchasing method.

Answer: $21800$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly?

Answer: $205$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by maintaining the same ratio of ingredients.

Answer: $26$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Allison, Brian, and Noah each have a die. All of the faces on Allison's die have a 5. The faces on Brian's die are numbered 1, 2, 3, 4, 4, 5, 5, and 6. Noah's die has an 8-sided die with the numbers 2, 2, 6, 6, 3, 3, 7, and 7. All three dice are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction.

Answer: $\frac{5}{16}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the functions $f(x)=(x-2)e^{x}$ and $g(x)=kx^{3}-x-2$,
(1) Find the range of $k$ such that the function $g(x)$ is not monotonic in the interval $(1,2)$;
(2) Find the maximum value of $k$ such that the inequality $f(x)\geqslant g(x)$ always holds when $x\in[0,+\infty)$.

Answer: $\frac{1}{6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Dad is $a$ years old this year, which is 4 times plus 3 years more than Xiao Hong's age this year. Xiao Hong's age expressed in an algebraic expression is ____. If Xiao Hong is 7 years old this year, then Dad's age is ____ years old.

Answer: $31$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\cos \alpha,3\sin \alpha)$ respectively, and $\alpha\in\left( \frac {\pi}{2}, \frac {3\pi}{4}\right)$. If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $\frac {2\sin ^{2}\alpha-\sin 2\alpha}{1+\tan \alpha}$.

Answer: $- \frac {7 \sqrt {23}}{48}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the vertex of angle \\(θ\\) is at the origin of coordinates, its initial side coincides with the positive half-axis of \\(x\\), and its terminal side is on the line \\(4x+3y=0\\), then \\( \dfrac{\cos \left( \left. \dfrac{π}{2}+θ \right. \right)-\sin (-π-θ)}{\cos \left( \left. \dfrac{11π}{2}-θ \right. \right)+\sin \left( \left. \dfrac{9π}{2}+θ \right. \right)}=\)\_\_\_\_\_\_\_\_．

Answer: $\dfrac{8}{7}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many integers $-12 \leq n \leq 12$ satisfy $(n-3)(n+5)(n+9)<0$?

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f\left(x\right)=4^{x}+m\cdot 2^{x}$, where $m\in R$.
$(1)$ If $m=-3$, solve the inequality $f\left(x\right) \gt 4$ with respect to $x$.
$(2)$ If the minimum value of the function $y=f\left(x\right)+f\left(-x\right)$ is $-4$, find the value of $m$.

Answer: $-3$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Suppose $105 \cdot 77 \cdot 132 \equiv m \pmod{25}$, where $0 \le m < 25$.

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the constant term in the expansion of $\left( \sqrt {x}+ \dfrac {1}{2 \sqrt {x}}\right)^{8}$.

Answer: $\dfrac {35}{8}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let  $z$  be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of  $z$ .

*Proposed by Michael Tang*

Answer: $423$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the parabola passing through points $A(2-3b, m)$ and $B(4b+c-1, m)$ is $y=-\frac{1}{2}x^{2}+bx-b^{2}+2c$, if the parabola intersects the $x$-axis, calculate the length of segment $AB$.

Answer: $12$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $f(x) = a\cos x - \left(x - \frac{\pi}{2}\right)\sin x$, where $x \in \left[0, \frac{\pi}{2}\right]$.
$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$.
$(2)$ If $f(x) \leq 0$ always holds, find the maximum value of the real number $a$.

Answer: $-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.
A) $\frac{10}{3}$
B) $5$
C) $\frac{15}{4}$
D) $2$

Answer: $\frac{10}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients.

Answer: $84$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The left and right foci of the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\angle F_{1}PF_{2} = 30^{\circ}$. Find the area of triangle $F_{1}PF_{2}$.

Answer: $18 - 9\sqrt{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given vectors $\overrightarrow{a}=(x,3)$ and $\overrightarrow{b}=(-1,y-1)$, and $\overrightarrow{a}+2\overrightarrow{b}=(0,1)$, find the value of $|\overrightarrow{a}+\overrightarrow{b}|$.

Answer: $\sqrt{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\alpha$, if $\cos(\alpha+\beta)=\frac{1}{3}$, where $\beta \in (0,\pi)$, then $\cos \beta =\_\_\_\_\_\_.$

Answer: $\frac{3 + 8\sqrt{2}}{15}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $P, Q$ be functions defined as $P(x) = 3\sqrt{x}$ and $Q(x) = x^2$. Calculate the value of $P(Q(P(Q(P(Q(5))))))$.

Answer: $135$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $a$, $b$, and $c$ are positive real numbers satisfying $a + 2b + 3c = 6$, find the maximum value of $\sqrt{a + 1} + \sqrt{2b + 1} + \sqrt{3c + 1}$.

Answer: $3\sqrt{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Consider a modified function $C' = \frac{2en}{R + 2nr}$, where $e = 4$, $R = 6$, and $r = 3$. Determine the behavior of $C'$ as $n$ increases from a positive starting point.

Answer: $\frac{4}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Each of the integers \(1, 2, 3, \ldots, 9\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \(\{1, 2, \ldots, 9\}\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer \(n\). What is the least possible value of \(n\) for which this assignment can be done?

Answer: $15$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack?

Answer: $\frac{1}{650}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given $\tan\alpha= \dfrac {1}{3}$, find the values of the following expressions:
1. $\dfrac {\sin \alpha+\cos\alpha}{5\cos\alpha-\sin\alpha}$
2. $\dfrac {1}{2\sin\alpha\cdot \cos\alpha+\cos ^{2}\alpha}$.

Answer: $\dfrac {2}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within the shaded region $S$ for any $x \in [0,4]$ and $y \in [0,4]$.

Answer: $\frac{\pi}{4} - \frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the circle $(x+1)^2+(y-2)^2=1$ and the origin O, find the minimum value of the distance |PM| if the tangent line from point P to the circle has a point of tangency M such that |PM|=|PO|.

Answer: $\frac {2 \sqrt {5}}{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

 $1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.

Find the height of the bottle.

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Ivan and Peter are running in opposite directions on circular tracks with a common center, initially positioned at the minimal distance from each other. Ivan completes one full lap every 20 seconds, while Peter completes one full lap every 28 seconds. In the shortest possible time, when will they be at the maximum distance from each other?

Answer: $35/6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A point $M$ on the parabola $y=4x^{2}$ is at a distance of $1$ from the focus. The ordinate of point $M$ is __________.

Answer: $\frac{15}{16}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $O$ is the circumcenter of $\triangle ABC$, and $D$ is the midpoint of $BC$, if $\overrightarrow{AO} \cdot \overrightarrow{AD} = 4$ and $BC = 2\sqrt{6}$, find the length of $AD$.

Answer: $\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate $3 \cdot 7^{-1} + 9 \cdot 13^{-1} \pmod{60}$.

Express your answer as an integer from $0$ to $59$, inclusive.

Answer: $42$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Compute the value of $p$ such that the equation
\[\frac{2x + 3}{px - 2} = x\]
has exactly one solution.

Answer: $-\frac{4}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\triangle DEF$.

Answer: $225$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Simplify first, then evaluate: $(\frac{{x-3}}{{{x^2}-1}}-\frac{2}{{x+1}})\div \frac{x}{{{x^2}-2x+1}}$, where $x=(\frac{1}{2})^{-1}+\left(\pi -1\right)^{0}$.

Answer: $-\frac{2}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the function $f(x)=\sin x+a\cos x$ has a symmetry axis on $x=\frac{5π}{3}$, determine the maximum value of the function $g(x)=a\sin x+\cos x$.

Answer: $\frac {2\sqrt {3}}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a lottery game, the host randomly selects one of the four identical empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the host randomly opens another box without the prize and asks the participant if they would like to change their selection to increase the chances of winning. Let $A_{i}$ represent the event that box $i$ contains the prize $(i=1,2,3,4)$, and let $B_{i}$ represent the event that the host opens box $i$ $(i=2,3,4)$. Now, if it is known that the participant chose box $1$, then $P(B_{3}|A_{2})=$______; $P(B_{3})=______.$

Answer: $\frac{1}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Simplify the expression:
\[
\frac{4 + 2i}{4 - 2i} + \frac{4 - 2i}{4 + 2i} + \frac{4i}{4 - 2i} - \frac{4i}{4 + 2i}.
\]

Answer: $\frac{2}{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different?

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$.

Answer: $\frac{7\pi}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the hyperbola $\dfrac {x^{2}}{9}- \dfrac {y^{2}}{27}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and $F_{2}$ being the focus of the parabola $y^{2}=2px$, find the area of $\triangle PF_{1}F_{2}$.

Answer: $36 \sqrt {6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Starting from $2$, the sum of consecutive even numbers is shown in the table below:

| Number of terms $n$ | $S$       |
|----------------------|-----------|
| $1$                  | $2=1\times 2$ |
| $2$                  | $2+4=6=2\times 3$ |
| $3$                  | $2+4+6=12=3\times 4$ |
| $4$                  | $2+4+6+8=20=4\times 5$ |
| $5$                  | $2+4+6+8+10=30=5\times 6$ |

$(1)$ If $n=8$, then the value of $S$ is ______.
$(2)$ According to the pattern in the table, it is conjectured that the formula for expressing $S$ in terms of $n$ is: $S=2+4+6+8+\ldots +2n=\_\_\_\_\_\_.$
$(3)$ Calculate the value of $102+104+106+\ldots +210+212$ based on the pattern in the previous question.

Answer: $8792$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the odd function $f(x)$ is also a periodic function, and the smallest positive period of $f(x)$ is $\pi$, when $x \in \left(0, \frac{\pi}{2}\right)$, $f(x) = 2\sin x$. Find the value of $f\left(\frac{11\pi}{6}\)$.

Answer: $-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A rubber ball is released from a height of 120 feet and rebounds to three-quarters of the height it falls each time it bounces. How far has the ball traveled when it strikes the ground for the fifth time?

Answer: $612.1875$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$.

Answer: $89991$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Straw returning to the field is a widely valued measure for increasing soil fertility and production in the world today, which eliminates the air pollution caused by straw burning and also has the effect of increasing fertility and production. A farmer spent $137,600 to buy a new type of combine harvester to achieve straw returning to the field while harvesting. The annual income from harvesting is $60,000 (fuel costs deducted); the harvester requires regular maintenance, with the first year's maintenance being free of charge provided by the manufacturer, and from the second year onwards, the farmer pays for the maintenance, with the cost $y$ (in yuan) related to the number of years $n$ used as: $y=kn+b$ ($n\geqslant 2$, and $n\in N^{\ast}$), knowing that the second year's maintenance costs $1,800 yuan, and the fifth year's costs $6,000 yuan.
(Ⅰ) Try to find the relationship between the maintenance cost $f(n)$ (in yuan) and the number of years $n$ ($n\in N^{\ast}$) used;
(Ⅱ) How many years should this harvester be used to maximize the average profit? (Profit = Income - Maintenance cost - Cost of machinery)

Answer: $14$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A sequence consists of $3000$ terms. Each term after the first is $3$ larger than the previous term. The sum of the $3000$ terms is $12000$. Calculate the sum when every third term is added up, starting with the first term and ending with the third to last term.

Answer: $1000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$.

Answer: $\frac{9\pi}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The positive integers \( r \), \( s \), and \( t \) have the property that \( r \times s \times t = 1230 \). What is the smallest possible value of \( r + s + t \)?

Answer: $52$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Convert the binary number $10110100$ to a decimal number.

Answer: $180$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________.

Answer: $108$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the arithmetic square root of $m$ is $3$, and the square roots of $n$ are $a+4$ and $2a-16$.
$(1)$ Find the values of $m$ and $n$.
$(2)$ Find $\sqrt[3]{{7m-n}}$.

Answer: $-1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let the number $9999\cdots 99$ be denoted by $N$ with $94$ nines. Then find the sum of the digits in the product $N\times 4444\cdots 44$.

Answer: $846$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \).

Answer: $763$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?

Answer: $\frac{\left( \binom{26}{13} \right)^2}{\binom{52}{26}}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x)=2\cos ^{2} \frac{x}{2}- \sqrt {3}\sin x$.
(I) Find the smallest positive period and the range of the function;
(II) If $a$ is an angle in the second quadrant and $f(a- \frac {π}{3})= \frac {1}{3}$, find the value of $\frac {\cos 2a}{1+\cos 2a-\sin 2a}$.

Answer: $\frac{1-2\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?

Answer: $48$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function $f(x)=\sqrt{3}\sin\left(x+\frac{\pi}{4}\right)\sin\left(x-\frac{\pi}{4}\right)+\sin x\cos x$.
$(Ⅰ)$ Find the smallest positive period and the center of symmetry of $f(x)$;
$(Ⅱ)$ Given an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A)=\frac{\sqrt{3}}{2}$ and $a=4$, find the maximum perimeter of $\triangle ABC$.

Answer: $12$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle \(ABC\), the sides are known: \(AB = 6\), \(BC = 4\), and \(AC = 8\). The angle bisector of \(\angle C\) intersects side \(AB\) at point \(D\). A circle is drawn through points \(A\), \(D\), and \(C\), intersecting side \(BC\) at point \(E\). Find the area of triangle \(ADE\).

Answer: $\frac{3 \sqrt{15}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the Cartesian coordinate system $xOy$, the equation of circle $C$ is $(x- \sqrt {3})^{2}+(y+1)^{2}=9$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of circle $C$;
$(2)$ The line $OP$: $\theta= \frac {\pi}{6}$ ($p\in R$) intersects circle $C$ at points $M$ and $N$. Find the length of segment $MN$.

Answer: $2 \sqrt {6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Among all integers that alternate between 1 and 0, starting and ending with 1 (e.g., 101, 10101, 10101…), how many are prime numbers? Why? And list all the prime numbers.

Answer: $101$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function\\(f(x)= \\begin{cases} (-1)^{n}\\sin \\dfrac {πx}{2}+2n,\\;x∈\[2n,2n+1) \\\\ (-1)^{n+1}\\sin \\dfrac {πx}{2}+2n+2,\\;x∈\[2n+1,2n+2)\\end{cases}(n∈N)\\)，if the sequence\\(\\{a\_{m}\\}\\) satisfies\\(a\_{m}=f(m)\\;(m∈N^{\*})\\)，and the sum of the first\\(m\\) terms of the sequence is\\(S\_{m}\\)，then\\(S\_{105}-S\_{96}=\\) \_\_\_\_\_\_ ．

Answer: $909$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE.

Answer: $\sqrt{113}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine the eccentricity of $C_2$.

Answer: $\frac{\sqrt{6}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.

Answer: $72$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is established.

\\((1)\\) Find the Cartesian equation of curve \\(C\\) and the Cartesian coordinates of point \\(P\\);

\\((2)\\) A line \\(l\\) passing through point \\(P\\) intersects curve \\(C\\) at points \\(A\\) and \\(B\\). If \\(|PA| = 2|PB|\\), find the value of \\(|AB|\\).

Answer: $3\sqrt{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given in the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3+5\cos \alpha \\ y=4+5\sin \alpha \end{cases}$, ($\alpha$ is the parameter), points $A$ and $B$ are on curve $C$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of points $A$ and $B$ are respectively $A(\rho_{1}, \frac{\pi}{6})$, $B(\rho_{2}, \frac{\pi}{2})$
(Ⅰ) Find the polar equation of curve $C$;
(Ⅱ) Suppose the center of curve $C$ is $M$, find the area of $\triangle MAB$.

Answer: $\frac{25\sqrt{3}}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If the line $l_{1}$: $x+my-2=0$ intersects the line $l_{2}$: $mx-y+2=0$ at point $P$, and a tangent line passing through point $P$ is drawn to the circle $C: (x+2)^{2} + (y+2)^{2} = 1$, with the point of tangency being $M$, then the maximum value of $|PM|$ is ____.

Answer: $\sqrt{31}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$.

Answer: $2.505$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindrical container?

Answer: $1.5$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $i$ is the imaginary unit, $a\in\mathbb{R}$, if $\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \sqrt{2}i$.

Answer: $\sqrt{11}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the terminal side of angle \\(\alpha\\) passes through the point \\(P(m,2\sqrt{2})\\), \\(\sin \alpha= \frac{2\sqrt{2}}{3}\\) and \\(\alpha\\) is in the second quadrant.
\\((1)\\) Find the value of \\(m\\);
\\((2)\\) If \\(\tan \beta= \sqrt{2}\\), find the value of \\( \frac{\sin \alpha\cos \beta+3\sin \left( \frac{\pi}{2}+\alpha\right)\sin \beta}{\cos (\pi+\alpha)\cos (-\beta)-3\sin \alpha\sin \beta}\\).

Answer: $\frac{\sqrt{2}}{11}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in  $6$  minutes, Mr. Rose in  $10$  minutes, and Mr. Schwartz in  $18$  minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?

Answer: $90$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the value of $(2345 + 3452 + 4523 + 5234) \times 2$.

Answer: $31108$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \ldots + a_{12}$.

Answer: $364$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\sin x \cdot \cos x = -\frac{1}{4}$ and $\frac{3\pi}{4} < x < \pi$, find the value of $\sin x + \cos x$.

Answer: $-\frac{\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

An experimenter selects 4 out of 8 different chemical substances to place in 4 distinct bottles. If substances A and B should not be placed in bottle 1, the number of different ways of arranging them is ____.

Answer: $1260$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $α$ is an acute angle and $\sin α= \frac {3}{5}$, find the value of $\cos α$ and $\cos (α+ \frac {π}{6})$.

Answer: $\frac {4\sqrt {3}-3}{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the sum of distinct residues of the number  $2012^n+m^2$  on  $\mod 11$  where  $m$  and  $n$  are positive integers.

Answer: $39$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There is a room with four doors. Determine the number of different ways for someone to enter and exit this room.

Answer: $16$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A certain shopping mall purchased a batch of daily necessities. If they are sold at a price of $5$ yuan per item, they can sell $30,000$ items per month. If they are sold at a price of $6$ yuan per item, they can sell $20,000$ items per month. It is assumed that the monthly sales quantity $y$ (items) and the price $x$ (yuan per item) satisfy a linear function relationship.

1. Find the function relationship between $y$ and $x.
2. If the unit price of these daily necessities is $4$ yuan when purchased, what price should be set for sales to maximize the monthly profit? What is the maximum monthly profit?

Answer: $40000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal.

Answer: $0.5$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a circle that is tangent to the side \(DC\) of a regular pentagon \(ABCDE\) at point \(D\) and tangent to the side \(AB\) at point \(A\), what is the degree measure of the minor arc \(AD\)?

Answer: $144$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. Calculate the total number of different ways to do this.

Answer: $77$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given $a$, $b$, $c \in \mathbb{R}$, and $2a+2b+c=8$, find the minimum value of $(a-1)^2+(b+2)^2+(c-3)^2$.

Answer: $\frac{49}{9}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate:
$(1)-3+8-15-6$;
$(2)-35\div \left(-7\right)\times (-\frac{1}{7})$;
$(3)-2^{2}-|2-5|\div \left(-3\right)$;
$(4)(\frac{1}{2}+\frac{5}{6}-\frac{7}{12})×(-24)$;
$(5)(-99\frac{6}{11})×22$.

Answer: $-2190$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the triangle \( \triangle ABC \), if \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), calculate the maximum value of \(\cos A + \cos B + 2 \cos C\).

Answer: $\sqrt{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$

$(1)$ intersect at a right angle;

$(2)$ are parallel;

$(3)$ the angle of inclination of $l_{1}$ is $45^{\circ}$

Answer: $-6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given $a=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $b=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}$, find $\frac{b}{a}+\frac{a}{b}$.

Answer: $62$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

When $n$ is a positive integer, $n! = n \times (n - 1) \times \ldots \times 2 \times 1$ is defined as the factorial of $n$ (for example, $10! = 10 \times 9 \times \ldots \times 2 \times 1 = 3,628,800$). Then, how many zeros are there at the end of $2010!$?

Answer: $501$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( n \in \mathbf{Z}_{+} \), and
$$
\begin{array}{l}
a, b, c \in \{x \mid x \in \mathbf{Z} \text{ and } x \in [1,9]\}, \\
A_{n} = \underbrace{\overline{a a \cdots a}}_{n \text{ digits}}, B_{n} = \underbrace{b b \cdots b}_{2n \text{ digits}}, C_{n} = \underbrace{c c \cdots c}_{2n \text{ digits}}.
\end{array}
$$

The maximum value of \(a + b + c\) is ( ), given that there exist at least two \(n\) satisfying \(C_{n} - B_{n} = A_{n}^{2}\).

Answer: $18$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$.

Answer: $37$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.

The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$.

Answer: $23$