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 parabola $C: y^{2}=2px\left(p \lt 0\right)$ passing through the point $A\left(-2,-4\right)$.
$(1)$ Find the equation of the parabola $C$ and its directrix equation.
$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\circ}$ with the $x$-axis, intersects the parabola at points $A$ and $B$. Find the length of segment $AB$.

Answer: $\frac{32}{3}$


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 square sheet contains 1000 points, with any three points, including the vertices of the square, not being collinear. Connect some of these points and the vertices of the square with line segments to divide the entire square into smaller triangles (using the connected line segments and square edges as sides, and ensuring that no two segments, except at endpoints, share common points). How many line segments are connected in total? How many triangles are formed in total?

Answer: $2002$


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 wheel with a circumference of $11$ feet, the speed $r$ in miles per hour for which the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second is increased by $5$ miles per hour, find the value of $r$.

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>.

Determine the volume of the solid formed by the set of vectors $\mathbf{v}$ such that:
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 12 \\ -34 \\ 6 \end{pmatrix}\]

Answer: $\frac{4}{3} \pi (334)^{3/2}$


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>.

If the average of six data points $a_1, a_2, a_3, a_4, a_5, a_6$ is $\bar{x}$, and the variance is 0.20, what is the variance of the seven data points $a_1, a_2, a_3, a_4, a_5, a_6, \bar{x}$?

Answer: $\frac{6}{35}$


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 $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\infty)$, find the minimum value of $f(x)$ on the interval $(-\infty, 0)$.

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>.

Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot.

Answer: $61$


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 distinct integers are picked at random from the set {1,2,3,...,12}. What is the probability that among those selected, the second smallest number is 4?

Answer: $\frac{7}{33}$


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 triangle ABC, where a, b, and c are the sides opposite to angles A, B, and C respectively, sin(2C - $\frac {π}{2}$) = $\frac {1}{2}$, and a2 + b2 < c2.
(1) Find the measure of angle C.
(2) Find the value of $\frac {a + b}{c}$.

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>.

Circle  $\omega_1$  with radius 3 is inscribed in a strip  $S$  having border lines  $a$  and  $b$ .  Circle  $\omega_2$  within  $S$  with radius 2 is tangent externally to circle  $\omega_1$  and is also tangent to line  $a$ .  Circle  $\omega_3$  within  $S$  is tangent externally to both circles  $\omega_1$  and  $\omega_2$ , and is also tangent to line  $b$ .  Compute the radius of circle  $\omega_3$ .

Answer: $\frac{9}{8}$


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>.

The integer solution for the inequality $|2x-m|\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).
(I) Find the value of the integer $m$;
(II) Given that $a, b, c \in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$.

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


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>.

A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time?

Answer: $344.88$


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 sequence $\{a_n\}$ where $a_1=1$ and $a_na_{n-1}=a_{n-1}+(-1)^n$ for $n\geqslant 2, n\in\mathbb{N}^*$, find the value of $\frac{a_3}{a_5}$.

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


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>.

In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each group for a study situation survey. Calculate:

(1) The probability of exactly one female student being selected from Group A;

(2) The probability of exactly two male students being selected from the 4 students.

Answer: $\dfrac{31}{75}$


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>.

Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score?

Answer: $94.25$


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 $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$ There are exactly 1010 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] What is the minimum number of real roots that $g(x)$ can have?

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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$.

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>.

How many zeros are there in the last digits of the following number  $P = 11\times12\times ...\times 88\times 89$  ?

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>.

Given the ellipse $\dfrac{{x}^{2}}{16}+ \dfrac{{y}^{2}}{9}=1$, with left and right foci $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse, if $P$, $F_1$, and $F_2$ are the three vertices of a right triangle, calculate the distance from point $P$ to the $x$-axis.

Answer: $\dfrac{9}{4}$


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>.

Select three different numbers from the set {-8, -6, -4, 0, 3, 5, 7} to form a product and determine the smallest possible value.

Answer: $-280$


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 numbers  $1,2,\ldots,9$  are arranged so that the  $1$ st term is not  $1$  and the  $9$ th term is not  $9$ . Calculate the probability that the third term is  $3$.

Answer: $\frac{43}{399}$


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 $\alpha$ is an angle in the second quadrant, and $\cos\left( \frac {\pi}{2}-\alpha\right)= \frac {4}{5}$, then $\tan\alpha= \_\_\_\_\_\_$.

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>.

What is the coefficient of  $x^5$  in the expansion of  $(1 + x + x^2)^9$ ?

Answer: $882$


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 set of data: $a_{1}=\frac{3}{1×2×3}$, $a_{2}=\frac{5}{2×3×4}$, $a_{3}=\frac{7}{3×4×5}$, $\ldots $, $a_{n}=\frac{2n+1}{n(n+1)(n+2)}$. Let $S_{n}=a_{1}+a_{2}+a_{3}+\ldots +a_{n}$. Find the value of $S_{12}$. To solve this problem, Xiao Ming first simplified $a_{n}$ to $a_{n}=\frac{x}{(n+1)(n+2)}+\frac{y}{n(n+2)}$, and then calculated the value of $S_{12}$ based on the simplified $a_{n}$. Please follow Xiao Ming's approach to first find the values of $x$ and $y$, and then calculate the value of $S_{12}$.

Answer: $\frac{201}{182}$


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>.

Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \).

Answer: $99$


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>.

David, Ellie, Natasha, and Lucy are tutors in their school science lab. Their working schedule is as follows: David works every fourth school day, Ellie works every fifth school day, Natasha works every sixth school day, and Lucy works every eighth school day. Today, they all happened to be working together. In how many school days from today will they all next tutor together in the lab?

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>.

If $\dfrac {\cos (\pi-2\alpha)}{\sin (\alpha- \dfrac {\pi}{4})}=- \dfrac { \sqrt {2}}{2}$, then $\sin 2\alpha=$ \_\_\_\_\_\_ .

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


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>.

Given that EF = 40 units, FG = 30 units, and one diagonal EH = 50 units, calculate the perimeter of parallelogram EFGH.

Answer: $140$


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 $f(n)$ is the sum of the digits of $n^2+1$ (where $n \in \mathbb{N}^*$). For example, since $14^2+1=197$, and $1+9+7=17$, thus $f(14)=17$. Let $f_1(n)=f(n)$, $f_2(n)=f(f_1(n))$, ..., $f_{k+1}(n)=f(f_k(n))$, where $k \in \mathbb{N}^*$, then $f_{2005}(8)=$  ．

Answer: $11$


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 $θ$ is symmetric to the terminal side of a $480^\circ$ angle with respect to the $x$-axis, and point $P(x,y)$ is on the terminal side of angle $θ$ (not the origin), then the value of $\frac{xy}{{x}^2+{y}^2}$ is equal to __.

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


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>.

In $\triangle ABC$, $AB=1$, $BC=2$, $\angle B=\frac{\pi}{3}$, let $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{BC}= \overrightarrow{b}$.
(I) Find the value of $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(4\overrightarrow{a}+\overrightarrow{b})$;
(II) Find the value of $|2\overrightarrow{a}-\overrightarrow{b}|$.

Answer: $2 \sqrt{3}$


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>.

A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl?

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>.

There are 5 balls numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and 5 boxes numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$. Each box contains one ball. The number of ways in which at most two balls have the same number as their respective boxes is $\_\_\_\_\_\_$.

Answer: $109$


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 child who does not understand English tries to spell the word "hello" using cards with the letters "e", "o", "h", "l", "l". How many possible incorrect arrangements can there be if the cards cannot be laid horizontally or upside down?

Answer: $59$


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 circumferences of the two bases of a cylinder lie on the surface of a sphere $O$ with a volume of $\frac{{32π}}{3}$, the maximum value of the lateral surface area of the cylinder is ______.

Answer: $8\pi$


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 $\left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in\{0,1\}, i \in \mathbf{N}_{+}, i \leqslant 6\right\}$. For $\forall x, y \in S$, $x=\left(x_{1}, x_{2}, \cdots, x_{6}\right)$ and $y=\left(y_{1}, y_{2}, \cdots, y_{6}\right)$, define:

1. $x=y$ if and only if $\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}+\cdots+\left(x_{6}-y_{6}\right)^{2}=0$;
2. $x \cdot y = x_{1} y_{1} + x_{2} y_{2} + \cdots + x_{6} y_{6}$.

Given a non-empty set $T \subseteq S$ that satisfies $\forall u, v \in T, u \neq v$, we have $u \cdot v \neq 0$, find the maximum number of elements in the set $T$.

Answer: $32$


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 parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\).

1. 9
2. 13
3. 12
4. 15

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>.

Let \( x_{0} \) be the largest (real) root of the equation \( x^{4} - 16x - 12 = 0 \). Evaluate \( \left\lfloor 10 x_{0} \right\rfloor \).

Answer: $27$


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>.

What is the value of $\frac13\cdot\frac92\cdot\frac1{27}\cdot\frac{54}{1}\cdot\frac{1}{81}\cdot\frac{162}{1}\cdot\frac{1}{243}\cdot\frac{486}{1}$?

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>.

The real value of $x$ such that $\frac{81^{x-2}}{9^{x-2}} = 27^{3x+2}$.

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


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>.

Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, $D$, $M$, and $N$, respectively. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and calculate the chord length $|MN|$.

Answer: $\frac{\sqrt{42}}{2}$


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 in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \sqrt {2}ac$.
(I) Find the value of angle B;
(II) If $a= \sqrt {2}$ and $cosA= \frac {4}{5}$, find the area of △ABC.

Answer: $\frac {7}{6}$


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>.

Squares $JKLM$ and $NOPQ$ are congruent, $JM=20$, and $P$ is the midpoint of side $JM$ of square $JKLM$. Calculate the area of the region covered by these two squares in the plane.
A) $500$
B) $600$
C) $700$
D) $800$
E) $900$

Answer: $600$


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 all possible  $2023$ -degree real polynomials:  $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where  $P(0)+P(1)=0$ , and the polynomial has 2023 real roots  $r_1, r_2,\cdots r_{2023}$  [not necessarily distinct] so that  $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of  $r_1 \cdot r_2 \cdots r_{2023}?$

Answer: $2^{-2023}$


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>.

Player A and Player B play a number guessing game. First, Player A thinks of a number denoted as $a$, then Player B guesses the number that Player A is thinking of, and denotes this guessed number as $b$. Both $a$ and $b$ belong to the set $\{1,2,3,4,5,6\}$. If $|a-b| \leqslant 1$, it is said that "Player A and Player B are in sync". Now, find the probability that two randomly chosen players are "in sync" in this game.

Answer: $\frac{4}{9}$


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>.

Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote?

Answer: $1113$


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 $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate $f(-\frac{{5π}}{{12}})$.

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>.

Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\sqrt{7}x-\sqrt{5}y+12=0$.
(1) Find the equation of the ellipse $C$;
(2) Let $A(-4,0)$, and a line $l$ passing through point $R(3,0)$ and intersecting with the ellipse $C$ at points $P$ and $Q$. Connect $AP$ and $AQ$ intersecting with the line $x=\frac{16}{3}$ at points $M$ and $N$, respectively. If the slopes of lines $MR$ and $NR$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant value. If it is, find this value; otherwise, explain the reason.

Answer: $-\frac{12}{7}$


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>.

Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\{b_{n}\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$.
$(1)$ If $b_{2}=-2a_{4}$, find the general formula for the sequence $\{b_{n}\}$;
$(2)$ If $T_{3}=13$, find $S_{3}$.

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>.

Four students are admitted to three universities. Find the probability that each university admits at least one student.

Answer: $\frac{4}{9}$


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>.

Given in the polar coordinate system, point P moves on the curve $\rho^2\cos\theta-2\rho=0$, the minimum distance from point P to point $Q(1, \frac{\pi}{3})$ is \_\_\_\_\_\_.

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


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>.

Let  $f(x)$  be the function such that  $f(x)>0$  at  $x\geq 0$  and  $\{f(x)\}^{2006}=\int_{0}^{x}f(t) dt+1.$

Find the value of  $\{f(2006)\}^{2005}.$

Answer: $2006$


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 ratio of the area of $\triangle BCY$ to the area of $\triangle ABY$ in the diagram if $CY$ bisects $\angle BCA$. [asy]
import markers;

real t=34/(34+28);
pair A=(-17.18,0);
pair B=(13.82,0);
pair C=(0,30);
pair Y=t*B+(1-t)*A;

draw(C--A--B--C--Y);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$Y$",Y,NE);

label("$32$",.5*(B+A),S);
label("$34$",.5*(B+C),NE);
label("$28$",.5*(A+C),NW);
[/asy]

Answer: $\frac{17}{14}$


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>.

The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?

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 numbers $1$, $2$, $3$, $4$, $5$, randomly select $3$ numbers (with repetition allowed) to form a three-digit number, find the probability that the sum of its digits equals $12$.

Answer: $\dfrac{2}{25}$


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, point P(x0, y0) is on the unit circle O. Suppose the angle ∠xOP = α, and if α ∈ (π/3, 5π/6), and sin(α + π/6) = 3/5, determine the value of x0.

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


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>.

Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.
$(1)$ Find the probability of school A winning the championship;
$(2)$ Let $X$ represent the total score of school B, find the probability distribution and expectation of $X$.

Answer: $13$


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 item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

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>.

Given the function $$f(x)=a+\sin(x-\frac{1}{2})$$, if $$f(0)+f(\frac{1}{2019})+f(\frac{2}{2019})+…+f(\frac{2017}{2019})+f(\frac{2018}{2019})+f(1)=1010$$, find the value of the real number $a$.

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>.

Inside of the square  $ABCD$  the point  $P$  is given such that  $|PA|:|PB|:|PC|=1:2:3$ . Find  $\angle APB$ .

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 water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower.

Answer: $0.6$


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>.

The sum of three numbers \(a\), \(b\), and \(c\) is 150. If we increase \(a\) by 10, decrease \(b\) by 3, and multiply \(c\) by 4, the three resulting numbers are equal. What is the value of \(b\)?

Answer: $\frac{655}{9}$


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>.

A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyramid.

Answer: $34$


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 two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\angle AF_{1}F_{2}=45^{\circ}$, calculate the area of triangle $AF_{1}F_{2}$.

Answer: $\frac{7}{2}$


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 square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.
A) $\frac{25}{128}$
B) $\frac{25(5 + 2\sqrt{5})}{128}$
C) $\frac{25(5-2\sqrt{5})}{128}$
D) $\frac{5\sqrt{5}}{128}$

Answer: $\frac{25(5-2\sqrt{5})}{128}$


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 repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?

Answer: $11$


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 the function $f(x)=\sin (2x+φ)$, where $|φ| < \dfrac{π}{2}$, the graph is shifted to the left by $\dfrac{π}{6}$ units and is symmetric about the origin. Determine the minimum value of the function $f(x)$ on the interval $[0, \dfrac{π}{2}]$.

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


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>.

Calculate the product of $\frac{5}{3} \times \frac{6}{5} \times \frac{7}{6} \times \cdots \times \frac{2010}{2009}$.

Answer: $670$


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 $a>0$, $b>1$, and $a+b=2$, find the minimum value of $$\frac{1}{2a}+\frac{2}{b-1}$$.

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


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>.

What is the area enclosed by the graph of $|x| + |3y| = 9$?

Answer: $54$


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 $f(x)= \dfrac {x+3}{x+1}$, let $f(1)+f(2)+f(4)+f(8)+f(16)=m$ and $f( \dfrac {1}{2})+f( \dfrac {1}{4})+f( \dfrac {1}{8})+f( \dfrac {1}{16})=n$, then $m+n=$ \_\_\_\_\_\_.

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>.

In triangle $XYZ$, $XY=12$, $YZ=16$, and $XZ=20$. Point $M$ is on $\overline{XY}$, $N$ is on $\overline{YZ}$, and $O$ is on $\overline{XZ}$. Let $XM = p \cdot XY$, $YN = q \cdot YZ$, and $ZO = r \cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $MNO$ to the area of triangle $XYZ$ can be written as $s/t$, where $s$ and $t$ are relatively prime positive integers. Find $s+t$.

Answer: $41$


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 complex number $z=3+bi$ ($b\in\mathbb{R}$), and $(1+3i) \cdot z$ is a pure imaginary number.
(1) Find the complex number $z$;
(2) If $w= \frac{z}{2+i}$, find the modulus of the complex number $w$.

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 positive integers \(a\) and \(b\) such that \(15a + 16b\) and \(16a - 15b\) are both perfect squares, find the smallest possible value of these two perfect squares.

Answer: $231361$


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>.

During the military training for new freshman students, after two days of shooting practice, student A can hit the target 9 times out of 10 shots, and student B can hit the target 8 times out of 9 shots. A and B each take a shot at the same target (their shooting attempts do not affect each other). Determine the probability that the target is hit.

Answer: $\frac{89}{90}$


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>.

A deck of 100 cards is numbered from 1 to 100, each card having the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. He then examines all the cards, and turns over every card that has a number divisible by 3. Determine the number of cards that have the red side up when Barsby is finished.

Answer: $49$


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>.

For the system of equations \(x^{2} + x^{2} y^{2} + x^{2} y^{4} = 525\) and \(x + x y + x y^{2} = 35\), find the sum of the real y values that satisfy the equations.

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


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>.

If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $8x^3 + 125 = (ax^2 + bx + c)(d x^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$?

Answer: $770$


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>.

Club Truncator is now in a soccer league with four other teams, each of which it plays once. In any of its 4 matches, the probabilities that Club Truncator will win, lose, or tie are $\frac{1}{3}$, $\frac{1}{3}$, and $\frac{1}{3}$ respectively. The probability that Club Truncator will finish the season with more wins than losses is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Answer: $112$


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 $A(-2,0)$ and $B(0,2)$, let point $C$ be a moving point on the circle $x^{2}-2x+y^{2}=0$. Determine the minimum area of $\triangle ABC$.

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 a line $l$ intersects the hyperbola $x^2 - \frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$.

Answer: $4\sqrt{2}$


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 a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder.

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


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>.

Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads.

Answer: $93.75$


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 slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$.

Answer: $-\frac{3}{5}$


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 assumption that smoking is unrelated to lung disease, calculate the confidence level that can be concluded from the chi-square statistic $K^2=5.231$, with $P(K^2 \geq 3.841) = 0.05$ and $P(K^2 \geq 6.635) = 0.01$.

Answer: $95\%$


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 $\overrightarrow{a}=(-3,1)$, $\overrightarrow{b}=(1,-2)$, and $\overrightarrow{n}=\overrightarrow{a}+k\overrightarrow{b}$ ($k\in\mathbb{R}$).

$(1)$ If $\overrightarrow{n}$ is perpendicular to the vector $2\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$;

$(2)$ If vector $\overrightarrow{c}=(1,-1)$, and $\overrightarrow{n}$ is parallel to the vector $\overrightarrow{c}+k\overrightarrow{b}$, find the value of the real number $k$.

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>.

Determine the number of 6-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition and with alternating even and odd digits.

Answer: $60$


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 $\triangle ABC$, if $|\overrightarrow{AB}|=2$, $|\overrightarrow{AC}|=3$, $|\overrightarrow{BC}|=4$, and $O$ is the incenter of $\triangle ABC$, and $\overrightarrow{AO}=\lambda \overrightarrow{AB}+\mu \overrightarrow{BC}$, calculate the value of $\lambda+\mu$.

Answer: $\frac{7}{9}$


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>.

In the Cartesian coordinate system $xOy$, the parametric equation of line $C_1$ is $\begin{cases} & x=1+\frac{1}{2}t \\ & y=\frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter), and in the polar coordinate system with the origin as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho^2(1+2\sin^2\theta)=3$.

$(1)$ Write the general equation of $C_1$ and the Cartesian coordinate equation of $C_2$;

$(2)$ Line $C_1$ intersects curve $C_2$ at points $A$ and $B$, with point $M(1,0)$. Find $||MA|-|MB||$.

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>.

How many multiples of 5 are there between 5 and 205?

Answer: $41$


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. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?
2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number must be divisible by 5? (Answer with numbers)

Answer: $216$


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 \(\left(d^d - d(d-2)^d\right)^d\) when \(d=4\).

Answer: $1358954496$


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\\), the quadratic trinomial \\(a{x}^{2}+2x+b \geqslant 0 \\) holds for all real numbers, and there exists \\(x_{0} \in \mathbb{R}\\), such that \\(ax_{0}^{2}+2{x_{0}}+b=0\\), then the minimum value of \\(\dfrac{a^{2}+b^{2}}{a-b}\\) is \_\_\_\_\_\_\_\_\_.

Answer: $2 \sqrt{2}$


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>.

Given $\sin \left( \frac{\pi}{4}-x\right)= \frac{1}{5} $, and $-\pi < x < - \frac{\pi}{2}$. Find the values of the following expressions:

$(1)\sin \left( \frac{5\pi}{4}-x\right)$;

$(2)\cos \left( \frac{3\pi}{4}+x\right)$;

$(3)\sin \left( \frac{\pi}{4}+x\right)$．

Answer: $-\frac{2 \sqrt{6}}{5}$


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>.

Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\]

Answer: $\frac{1}{16}$


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>.

Let \(p\), \(q\), \(r\), and \(s\) be the roots of the polynomial \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\]

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>.

Given the function $f(x)=x^{2}-6x+4\ln x$, find the x-coordinate of the quasi-symmetric point of the function.

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>.

In $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
label("B", B, E);
label("C", C, NE);
label("P", P, NW);
label("8", 3*dir(A--C), SE);
label("6", B+3*dir(B--C), NE);
label("10", (5,0), S);
[/asy]

Answer: $4.8$


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>.

In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total?

Answer: $140$


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>.

Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Simon travels west at 12 mph. How many hours will it take for them to be 130 miles apart?

Answer: $\frac{65}{\sqrt{61}}$


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 $\sqrt[3]{8+27} \cdot \sqrt[3]{8+\sqrt{64}}$.

Answer: $\sqrt[3]{560}$


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 six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series.

Answer: $3888$


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 $\overrightarrow {a}=(\sin(2x+ \frac {\pi}{6}), 1)$, $\overrightarrow {b}=( \sqrt {3}, \cos(2x+ \frac {\pi}{6}))$, and the function $f(x)= \overrightarrow {a} \cdot \overrightarrow {b}$.

(Ⅰ) Find the interval where the function $f(x)$ is monotonically decreasing;
(Ⅱ) In $\triangle ABC$, where $A$, $B$, and $C$ are the opposite sides of $a$, $b$, and $c$ respectively, if $f(A)= \sqrt {3}$, $\sin C= \frac {1}{3}$, and $a=3$, find the value of $b$.

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


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>.

The graph of the power function $f(x)$ passes through the point $(3, \frac{1}{9})$, find the maximum value of the function $g(x) = (x-1)f(x)$ on the interval $[1,3]$.

Answer: $\frac{1}{4}$


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>.

Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.

Answer: $\frac{1}{16}$


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>.

Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches?

Answer: $3375$


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 maximum value of the function $f(x) = 8\sin x - \tan x$, defined on $\left(0, \frac{\pi}{2}\right)$, is $\_\_\_\_\_\_\_\_\_\_\_\_$.

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>.

Given four real numbers that form an arithmetic sequence: -9, $a_1$, $a_2$, -1, and five real numbers that form a geometric sequence: -9, $b_1$, $b_2$, $b_3$, -1, find the value of $b_2(a_2-a_1)$.

Answer: $-8$


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>.

On a 6 by 6 grid of points, what fraction of the larger square's area is inside the new shaded square? Place the bottom-left vertex of the square at grid point (3,3) and the square rotates 45 degrees (square's sides are diagonals of the smaller grid cells).
```
[asy]
size(6cm);
fill((3,3)--(4,4)--(5,3)--(4,2)--cycle,gray(0.7));
dot((1,1));
for (int i = 0; i <= 6; ++i) {
    draw((0,i)--(6,i));
    draw((i,0)--(i,6));
    for (int j = 0; j <= 6; ++j) {
        dot((i,j));
    }
}
draw((3,3)--(4,4)--(5,3)--(4,2)--cycle);
[/asy]
```

Answer: $\frac{1}{18}$


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 maximum value of  $k$  such that the inequality  $\sqrt{x-3}+\sqrt{6-x}\geq k$  has a real solution.

Answer: $\sqrt{6}$


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>.

Green Valley School has 120 students enrolled, consisting of 70 boys and 50 girls. If $\frac{1}{7}$ of the boys and $\frac{1}{5}$ of the girls are absent on a particular day, what percent of the total student population is absent?

Answer: $16.67\%$


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 set of 15 different-colored markers, how many ways can Jane select five markers if the order of selection does not matter?

Answer: $3003$


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 two asymptotes of the hyperbola $\dfrac{y^2}{4}-x^2=1$ intersect with the directrix of the parabola $y^2=2px(p > 0)$ at points $A$ and $B$, and $O$ is the origin, determine the value of $p$ given that the area of $\Delta OAB$ is $1$.

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>.

In $\triangle ABC$, two side lengths are $2$ and $3$, and the cosine value of the included angle is $\frac{1}{3}$. Find the radius of the circumscribed circle.

Answer: $\frac{9\sqrt{2}}{8}$


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>.

When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(7, 8)$, and $H(7, 3)$. What is the area of trapezoid $EFGH$?

Answer: $25$


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 that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .

*Proposed by Andrew Wu*

Answer: $2022$


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 $\tan G$ in the right triangle where side GH is 20 units, side FG is 25 units, and ∠H is the right angle.

[asy]
pair H,F,G;
H = (0,0);
G = (20,0);
F = (0,25);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$25$",(F+G)/2,NE);
label("$20$",G/2,S);
[/asy]

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


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>.

The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both tests, the higher score of the two tests will be taken as the final score. If the student scores full marks in the first test, they will not take the second test. To prepare for the English listening and speaking test, Li Ming takes English listening and speaking mock exams every week. The table below shows his scores in $20$ English listening and speaking mock exams before the first actual test.

Assumptions:
1. The difficulty level of the mock exams is equivalent to that of the actual exam.
2. The difficulty level of the two actual listening and speaking tests is the same.
3. If Li Ming continues to practice listening and speaking after not scoring full marks in the first test, the probability of scoring full marks in the second test can reach $\frac{1}{2}$.

| $46$ | $50$ | $47$ | $48$ | $49$ | $50$ | $50$ | $47$ | $48$ | $47$ |
|------|------|------|------|------|------|------|------|------|------|
| $48$ | $49$ | $50$ | $49$ | $50$ | $50$ | $48$ | $50$ | $49$ | $50$ |

$(Ⅰ)$ Let event $A$ be "Li Ming scores full marks in the first English listening and speaking test." Estimate the probability of event $A$ using frequency.

$(Ⅱ)$ Based on the assumptions in the problem, estimate the maximum probability of Li Ming scoring full marks in the English college entrance examination listening and speaking test.

Answer: $\frac{7}{10}$


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 $3\sin \alpha - 2\cos \alpha = 0$, find the value of the following expressions:
$$(1)\ \frac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha} + \frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha};$$
$$(2)\ \sin^2\alpha - 2\sin \alpha\cos \alpha + 4\cos^2\alpha.$$

Answer: $\frac{28}{13}$


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 number of terms in the simplified expression of \[(x+y+z)^{2020} + (x-y-z)^{2020},\] by expanding it and combining like terms.

Answer: $1,022,121$


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 $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?

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>.

Evaluate the expression: \\( \frac {\cos 40 ^{\circ} +\sin 50 ^{\circ} (1+ \sqrt {3}\tan 10 ^{\circ} )}{\sin 70 ^{\circ} \sqrt {1+\cos 40 ^{\circ} }}\\)

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 that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\angle F_{1}PF_{2}=60^{\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$.

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


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>.

Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$.

Answer: $\dfrac{e^{2}-1}{e^{2}+1}$


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>.

Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\triangle ABC$, with $a=4$ and $(4+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum value of the area of $\triangle ABC$.

Answer: $4\sqrt{3}$