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. Simplify and evaluate the expression: $\log_{\frac{1}{3}} \sqrt{27} + \lg 25 + \lg 4 + 7^{-\log_{7} 2} + (-0.98)^0$
2. Given a point $P(\sqrt{2}, -\sqrt{6})$ on the terminal side of angle $\alpha$, evaluate: $\frac{\cos \left( \frac{\pi}{2} + \alpha \right) \cos \left( 2\pi - \alpha \right) + \sin \left( -\alpha - \frac{\pi}{2} \right) \cos \left( \pi - \alpha \right)}{\sin \left( \pi + \alpha \right) \cos \left( \frac{\pi}{2} - \alpha \right)}$

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


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>.

Given that the terminal side of angle $θ$ lies on the ray $y=2x(x≤0)$, find the value of $\sin θ + \cos θ$.

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


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

Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$?

Answer: $48$


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

Given a random variable $\xi \sim N(1, \sigma ^{2})$, $a \gt 0$, $b \gt 0$, if $P(\xi \leq a) = P(\xi \geq b)$, then the minimum value of $\frac{{4a+b}}{{ab}}$ is ______.

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>.

Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), perpendiculars are drawn from the right focus $F(2\sqrt{2}, 0)$ to the two asymptotes, with the feet of the perpendiculars being $A$ and $B$, respectively. Let point $O$ be the origin. If the area of quadrilateral $OAFB$ is $4$, determine the eccentricity of the hyperbola.

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>.

Let  $ABC$  be an equilateral triangle and a point M inside the triangle such that  $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle  $ACD$  where  $D \ne B$ . Let the point  $N$  inside  $\vartriangle ACD$  such that  $AMN$  is an equilateral triangle. Determine  $\angle BMC$ .

Answer: $150$


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 Eagles and the Hawks play 5 times. The Hawks, being the stronger team, have an 80% chance of winning any given game. What is the probability that the Hawks will win at least 4 out of the 5 games? Express your answer as a common fraction.

Answer: $\frac{73728}{100000}$


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 rectangular coordinate system, a pole coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: ${p}^{2}=\frac{12}{2+{\mathrm{cos}}^{}θ}$ and the line $l$: $2p\mathrm{cos}\left(θ-\frac{π}{6}\right)=\sqrt{3}$.

1. Write the rectangular coordinate equations for the line $l$ and the curve $C$.
2. Let points $A$ and $B$ be the two intersection points of line $l$ and curve $C$. Find the value of $|AB|$.

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


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

Find the mathematical expectation of the area of the projection of a cube with edge of length  $1$  onto a plane with an isotropically distributed random direction of projection.

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 $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, $c$ are $S_{1}$, $S_{2}$, $S_{3}$ respectively. Given $S_{1}-S_{2}+S_{3}=\frac{{\sqrt{3}}}{2}$ and $\sin B=\frac{1}{3}$.
$(1)$ Find the area of $\triangle ABC$;
$(2)$ If $\sin A\sin C=\frac{{\sqrt{2}}}{3}$, find $b$.

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>.

The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.

\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}

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>.

Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\]

Answer: $31$


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

Given the function $f(x)= \sqrt {3}\sin (\omega x+\varphi)-\cos (\omega x+\varphi)$ $(\omega > 0,0 < \varphi < \pi)$ is an even function, and the distance between two adjacent axes of symmetry of its graph is $\dfrac {\pi}{2}$, then the value of $f(- \dfrac {\pi}{8})$ is \_\_\_\_\_\_.

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>.

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

Answer: $20$


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

Given the real numbers $a$, $b$, $c$, $d$ that satisfy $b=a-2e^{a}$ and $c+d=4$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^{2}+(b-d)^{2}$.

Answer: $18$


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

In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_.

Answer: $16\pi$


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>.

If 2006 integers $a_1, a_2, \ldots a_{2006}$ satisfy the following conditions: $a_1=0$, $|a_2|=|a_1+2|$, $|a_3|=|a_2+2|$, $\ldots$, $|a_{2006}|=|a_{2005}+2|$, then the minimum value of $a_1+a_2+\ldots+a_{2005}$ is.

Answer: $-2004$


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 five-digit natural number $\overline{a_1a_2a_3a_4a_5}$ is considered a "concave number" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \in \{0,1,2,3,4,5\}$ for $i=1,2,3,4,5$. Calculate the number of possible "concave numbers".

Answer: $146$


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

Find the smallest positive integer  $n$  that has at least  $7$  positive divisors  $1 = d_1 < d_2 < \ldots < d_k = n$ ,  $k \geq 7$ , and for which the following equalities hold:  $$ d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1 $$  *Proposed by Mykyta Kharin*

Answer: $2024$


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

A triangle has three sides that are three consecutive natural numbers, and the largest angle is twice the smallest angle. The perimeter of this triangle is __________.

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>.

Given $\sin (\frac{\pi }{3}-\theta )=\frac{3}{4}$, find $\cos (\frac{\pi }{3}+2\theta )$.

Answer: $\frac{1}{8}$


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>.

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

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>.

Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.

Answer: $1 + \sqrt{2}$


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>.

Consider the following infinite geometric series: $$\frac{7}{8}-\frac{14}{27}+\frac{28}{81}-\dots$$ Find the common ratio of this series.

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


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

Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said?

Answer: $100$


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>.

Define a new operation $\star$ such that for positive integers $a, b, c$, $a \star b \star c = \frac{a \times b + c}{a + b + c}$. Calculate the value of $4 \star 8 \star 2$.
**A)** $\frac{34}{14}$ **B)** $\frac{16}{7}$ **C)** $\frac{17}{7}$ **D)** $\frac{32}{14}$ **E)** $2$

Answer: $\frac{17}{7}$


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>.

There are $7$ different books to be distributed among three people, A, B, and C.
$(1)$ If one person gets $1$ book, another gets $2 books, and the third gets $4 books, how many different ways can the books be distributed?
$(2)$ If one person gets $3 books, and the other two each get $2 books, how many different ways can the books be distributed?

Answer: $630$


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 \( m = 2^{40}5^{24} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?

Answer: $959$


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 $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$?

Answer: $-\frac{448}{6561}$


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

Two different numbers are randomly selected from the set $\{-3, -2, 0, 0, 5, 6, 7\}$. What is the probability that the product of these two numbers is $0$?
**A)** $\frac{1}{4}$
**B)** $\frac{1}{5}$
**C)** $\frac{5}{21}$
**D)** $\frac{1}{3}$
**E)** $\frac{1}{2}$

Answer: $\frac{5}{21}$


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 function $f(x)=a- \frac {2}{2^{x}+1}(a\in\mathbb{R})$

$(1)$ Determine the monotonicity of the function $f(x)$ and provide a proof;

$(2)$ If there exists a real number $a$ such that the function $f(x)$ is an odd function, find $a$;

$(3)$ For the $a$ found in $(2)$, if $f(x)\geqslant \frac {m}{2^{x}}$ holds true for all $x\in[2,3]$, find the maximum value of $m$.

Answer: $\frac {12}{5}$


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>.

Calculate:
$(1)-1^{2023}+8×(-\frac{1}{2})^{3}+|-3|$;
$(2)(-25)×\frac{3}{2}-(-25)×\frac{5}{8}+(-25)÷8($simplified calculation).

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>.

In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\cos A}{\cos C}=\frac{a}{2b-c}$, find:

$(1)$ The measure of angle $A$;

$(2)$ If $a=\sqrt{7}$, $c=3$, and $D$ is the midpoint of $BC$, find the length of $AD$.

Answer: $\frac{\sqrt{19}}{2}$


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 a geometric sequence \(\{z_{n}\}\), given \(z_{1}=1\), \(z_{2}=a+b \mathrm{i}\), \(z_{3}=b+a \mathrm{i}\) where \(a, b \in \mathbf{R}\) and \(a>0\), find the smallest value of \(n\) such that \(z_{1}+z_{2}+\cdots+z_{n}=0\), and compute the value of \(z_{1} z_{2} \cdots z_{n}\).

Answer: $-1$


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

Let $f(x) = 4x - 9$ and $g(f(x)) = x^2 + 6x - 7$. Find $g(-8)$.

Answer: $\frac{-87}{16}$


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>.

Extend a rectangular pattern of 12 black and 18 white square tiles by attaching a border of white tiles around the rectangle. The original rectangle is 5x6 tiles and the border adds one tile to each side. Calculate the ratio of black tiles to white tiles in the extended pattern.

Answer: $\frac{12}{44}$


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 isolated ground state iron atom with atomic number Z = 26, determine the number of orbitals that contain one or more electrons.

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>.

Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.

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


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

In the sequence $\{a_{n}\}$, where $a_{1}=1$, $a_{n} \gt 0$, and the sum of the first $n$ terms is $S_{n}$. If ${a_n}=\sqrt{{S_n}}+\sqrt{{S_{n-1}}}$ for $n \in \mathbb{N}^*$ and $n \geqslant 2$, then the sum of the first $15$ terms of the sequence $\{\frac{1}{{{a_n}{a_{n+1}}}}\}$ is ____.

Answer: $\frac{15}{31}$


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

Let $f(x) = x^2 + 5x + 4$ and $g(x) = 2x - 3$. Calculate the value of $f(g(-3)) - 2 \cdot g(f(2))$.

Answer: $-26$


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

Determine the exact value of the series
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\]

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>.

Let $r(x)$ be a monic quartic polynomial such that $r(1) = 0,$ $r(2) = 3,$ $r(3) = 8,$ and $r(4) = 15$. Find $r(5)$.

Answer: $48$


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

Convert 89 into a binary number.

Answer: $1011001_{(2)}$


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 base-10 integer 789 when expressed in base 7?

Answer: $2205_7$


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>.

The line $2x+3y-6=0$ intersects the $x$-axis and $y$-axis at points A and B, respectively. Point P is on the line $y=-x-1$. The minimum value of $|PA|+|PB|$ is ________.

Answer: $\sqrt{37}$


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

Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with  $x_1$ ,  $x_2$ ,  $\cdots$ ,  $x_n$  are distinct postive integers. Find the maximum value of  $n$ .

*Proposed by Le Duc Minh*

Answer: $64$


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

Simplify first, then evaluate: $({\frac{{x-1}}{x}-\frac{{x-2}}{{x+1}}})÷\frac{{2{x^2}-x}}{{{x^2}+2x+1}}$, where $x$ satisfies $x^{2}-2x-2=0$.

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>.

A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans.

Answer: $70$


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 \( a > 0 \), if \( f(g(h(a))) = 17 \), where \( f(x) = x^2 + 5 \), \( g(x) = x^2 - 3 \), and \( h(x) = 2x + 1 \), what is the value of \( a \)?

Answer: $\frac{-1 + \sqrt{3 + 2\sqrt{3}}}{2}$


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

A bicycle factory plans to produce a batch of bicycles of the same model, planning to produce $220$ bicycles per day. However, due to various reasons, the actual daily production will differ from the planned quantity. The table below shows the production situation of the workers in a certain week: (Exceeding $220$ bicycles is recorded as positive, falling short of $220$ bicycles is recorded as negative)

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Production Change (bicycles) | $+5$ | $-2$ | $-4$ | $+13$ | $-10$ | $+16$ | $-9$ |

$(1)$ According to the records, the total production in the first four days was ______ bicycles;

$(2)$ How many more bicycles were produced on the day with the highest production compared to the day with the lowest production?

$(3)$ The factory implements a piece-rate wage system, where each bicycle produced earns $100. For each additional bicycle produced beyond the daily planned production, an extra $20 is awarded, and for each bicycle less produced, $20 is deducted. What is the total wage of the workers for this week?

Answer: $155080$


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>.

[help me] Let m and n denote the number of digits in  $2^{2007}$  and  $5^{2007}$  when expressed in base 10. What is the sum m + n?

Answer: $2008$


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 smallest positive real  $K$  such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers  $0 \le a,b,c \le 1$ .

*Proposed by Fajar Yuliawan, Indonesia*

Answer: $\frac{\sqrt{6}}{3}$


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>.

Amy and Bob choose numbers from  $0,1,2,\cdots,81$  in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all  $82$  numbers are chosen, let  $A$  be the sum of all the numbers Amy chooses, and let  $B$  be the sum of all the numbers Bob chooses. During the process, Amy tries to make  $\gcd(A,B)$  as great as possible, and Bob tries to make  $\gcd(A,B)$  as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine  $\gcd(A,B)$  when all  $82$  numbers are chosen.

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>.

How many unordered pairs of edges of a given square pyramid determine a plane?

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>.

A cube with an edge length of 6 units has the same volume as a triangular-based pyramid with a base having equilateral triangle sides of 10 units and a height of $h$ units. What is the value of $h$?

Answer: $\frac{216\sqrt{3}}{25}$


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>.

Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $3\pi < \alpha < \frac{7}{2}\pi$, find $\cos \alpha + \sin \alpha$.

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 a rectangle $ABCD$ with all vertices on a sphere centered at $O$, where $AB = \sqrt{3}$, $BC = 3$, and the volume of the pyramid $O-ABCD$ is $4\sqrt{3}$, find the surface area of the sphere $O$.

Answer: $76\pi$


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>.

We write one of the numbers  $0$  and  $1$  into each unit square of a chessboard with  $40$  rows and  $7$  columns. If any two rows have different sequences, at most how many  $1$ s can be written into the unit squares?

Answer: $198$


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 right circular cone with three mutually perpendicular side edges, each with a length of $\sqrt{3}$, determine the surface area of the circumscribed sphere.

Answer: $9\pi$


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>.

In right triangle $XYZ$, we have $\angle X = \angle Z$ and $XZ = 8\sqrt{2}$. What is the area of $\triangle XYZ$?

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 triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC.

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>.

Given that the base edge length of a right prism is $1$ and the side edge length is $2$, and all the vertices of the prism lie on a sphere, find the radius of the sphere.

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


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

Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price. If the ticket for an adult costs $10.00, and one of the grandparents is paying for everyone, how much do they need to pay in total?
A) $38
B) $30
C) $42
D) $28
E) $34

Answer: $30$


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 $\{a\_n\}$, where $a\_1+a\_2=3$, $a\_4+a\_5=5$.

(I) Find the general term formula of the sequence.
(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\_n=[a\_1]+[a\_2]+…+[a\_n]$. Find the value of $T\_30$.

Answer: $175$


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 distance between two parallel lines $l_1: 3x + 4y - 2 = 0$ and $l_2: ax + 6y = 5$ is _______.

Answer: $\frac{4}{15}$


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>.

In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\sin \angle GAC$. (Consider this by extending the calculations needed for finding $\sin \angle HAC$)

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>.

Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\left(a,b\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the minimum value of $\sqrt{(a-3)^{2}+(b+2)^{2}}$.

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


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

Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]

Answer: $75$


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>.

Parallelogram  $ABCD$  is given such that  $\angle ABC$  equals  $30^o$  . Let  $X$  be the foot of the perpendicular from  $A$  onto  $BC$ , and  $Y$  the foot of the perpendicular from  $C$  to  $AB$ . If  $AX = 20$  and  $CY = 22$ , find the area of the parallelogram.

Answer: $880$


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>.

From the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), two numbers are selected to form an ordered pair of real numbers \\((x, y)\\). The probability that \\(\dfrac{x}{y+1}\\) is an integer is equal to \_\_\_\_\_\_

Answer: $\dfrac{4}{15}$


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>.

The sum of the first and the third of three consecutive odd integers is 152. What is the value of the second integer?

Answer: $76$


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 $n \in \mathbb{N}^*$, in the expansion of $(x+2)^n$, the coefficient of the second term is $\frac{1}{5}$ of the coefficient of the third term.
(1) Find the value of $n$;
(2) Find the term with the maximum binomial coefficient in the expansion;
(3) If $(x+2)^n=a_0+a_1(x+1)+a_2(x+1)^2+\ldots+a_n(x+1)^n$, find the value of $a_0+a_1+\ldots+a_n$.

Answer: $64$


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 exact value of the series
\[\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.\]

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>.

What is the value of $2468 + 8642 + 6824 + 4286$?

Answer: $22220$


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 burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler?

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>.

If the set $\{1, a, \frac{b}{a}\}$ equals the set $\{0, a^2, a+b\}$, then find the value of $a^{2017} + b^{2017}$.

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>.

Each of the eight letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction.

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>.

Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  and  $g$  be seven distinct positive integers not bigger than  $7$ . Find all primes which can be expressed as  $abcd+efg$

Answer: $179$


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>.

Fill the letters a, b, and c into a 3×3 grid such that each letter does not repeat in any row or column. The number of different filling methods is ___.

Answer: $12$


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

In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y=t+1 \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.
(I) Find the polar coordinate equations of line $l$ and curve $C$;
(II) It is known that ray $OP$: $\theta_1=\alpha$ (where $0<\alpha<\frac{\pi}{2}$) intersects curve $C$ at points $O$ and $P$, and ray $OQ$: $\theta_2=\alpha+\frac{\pi}{2}$ intersects line $l$ at point $Q$. If the area of $\triangle OPQ$ is $1$, find the value of $\alpha$ and the length of chord $|OP|$.

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>.

The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$.

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>.

An easel in a corner hosts three $30 \text{ cm} \times 40 \text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, another climbed diagonally down to the corner of the lower shelf on the other wall. The third spider stayed in place and observed that from its position, the other two spiders appeared at an angle of $120^\circ$. What is the distance between the shelves? (The distance between neighboring shelves is the same.)

Answer: $35$


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 integers $a$ and $b$ satisfy: $a-b$ is a prime number, and $ab$ is a perfect square. When $a \geq 2012$, find the minimum value of $a$.

Answer: $2025$


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>.

Gavin has a collection of 50 songs that are each 3 minutes in length and 50 songs that are each 5 minutes in length. What is the maximum number of songs from his collection that he can play in 3 hours?

Answer: $56$


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 different factor tree, each value is also the product of the two values below it, unless the value is a prime number. Determine the value of $X$ for this factor tree:

[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
label("X",(0,0),N);
label("Y",(-1.5,-.8),N);
label("2",(-2,-1.3),S);
label("Z",(1.5,-.8),N);
label("Q",(-1,-1.3),S);
label("7",(1,-1.3),S);
label("R",(2,-1.3),S);
draw((-1.5,-2.3)--(-1,-1.8)--(-.5,-2.3),linewidth(1));
draw((1.5,-2.3)--(2,-1.8)--(2.5,-2.3),linewidth(1));
label("5",(-1.5,-2.3),S);
label("3",(-.5,-2.3),S);
label("11",(1.5,-2.3),S);
label("2",(2.5,-2.3),S);
[/asy]

Answer: $4620$


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 simplified, $\log_{16}{32} \cdot \log_{16}{\frac{1}{2}}$ becomes:
**A)** $-\frac{1}{4}$
**B)** $-\frac{5}{16}$
**C)** $\frac{5}{16}$
**D)** $-\frac{1}{16}$
**E)** $0$

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


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

A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle?

Answer: $\frac{3(\sqrt{3} - 1)}{6} \times 100\%$


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 $a + 3 = (b-1)^2$ and $b + 3 = (a-1)^2$. Assuming $a \neq b$, determine the value of $a^2 + b^2$.
A) 5
B) 10
C) 15
D) 20
E) 25

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>.

The sum of four positive integers that form an arithmetic sequence is 58. Of all such possible sequences, what is the greatest possible third term?

Answer: $19$


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

Given the function $f(x)= \frac {1}{2}x^{2}-2\ln x+a(a\in\mathbb{R})$, $g(x)=-x^{2}+3x-4$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum value, find the value of $t$;
$(3)$ If for any $x\in(m,n)$ (where $n-m\geqslant 1$), the graphs of the two functions are on opposite sides of the line $l$: $x-y+s=0$ (without intersecting line $l$), then these two functions are said to have an "EN channel". Investigate whether $f(x)$ and $g(x)$ have an "EN channel", and if so, find the range of $x$; if not, please explain why.

Answer: $\frac {3+ \sqrt {33}}{6}$


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

Given vectors $\overrightarrow{a}=(m,1)$ and $\overrightarrow{b}=(4-n,2)$, with $m > 0$ and $n > 0$. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the minimum value of $\frac{1}{m}+\frac{8}{n}$.

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>.

In a polar coordinate system, the equation of curve C1 is given by $\rho^2 - 2\rho(\cos\theta - 2\sin\theta) + 4 = 0$. With the pole as the origin and the polar axis in the direction of the positive x-axis, a Cartesian coordinate system is established using the same unit length. The parametric equation of curve C2 is given by
$$
\begin{cases}
5x = 1 - 4t \\
5y = 18 + 3t
\end{cases}
$$
where $t$ is the parameter.
(Ⅰ) Find the Cartesian equation of curve C1 and the general equation of curve C2.
(Ⅱ) Let point P be a moving point on curve C2. Construct two tangent lines to curve C1 passing through point P. Determine the minimum value of the cosine of the angle formed by these two tangent lines.

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


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>.

Given $cosθ+cos(θ+\frac{π}{3})=\frac{\sqrt{3}}{3},θ∈(0,\frac{π}{2})$, find $\sin \theta$.

Answer: $\frac{-1 + 2\sqrt{6}}{6}$


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>.

Let point $O$ be inside $\triangle ABC$ and satisfy $4\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$. Determine the probability that a randomly thrown bean into $\triangle ABC$ lands in $\triangle OBC$.

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


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

In a rectangle  $ABCD, E$  is the midpoint of  $AB, F$  is a point on  $AC$  such that  $BF$  is perpendicular to  $AC$ , and  $FE$  perpendicular to  $BD$ . Suppose  $BC = 8\sqrt3$ . Find  $AB$ .

Answer: $24$


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 x > 0, y > 0, z > 0, and x + $\sqrt{3}$y + z = 6, find the minimum value of x³ + y² + 3z.

Answer: $\frac{37}{4}$


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 book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times.

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>.

Given $sin( \frac {\pi}{6}-\alpha)-cos\alpha= \frac {1}{3}$, find $cos(2\alpha+ \frac {\pi}{3})$.

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>.

Let  $\{a_n\}_{n\geq 1}$  be a sequence defined by  $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of  $c$  such that  $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

Answer: $22$


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 $f(x)= \frac {x}{1+x}$, then $f(1)+f(2)+f(3)+\ldots+f(2017)+f( \frac {1}{2})+f( \frac {1}{3})+\ldots+f( \frac {1}{2017})=$ \_\_\_\_\_\_ .

Answer: $\frac {4033}{2}$


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 $\triangle ABC$, $A=30^{\circ}$, $AB=2$, $BC=1$, then the area of $\triangle ABC$ is equal to $\boxed{\text{answer}}$.

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>.

Say that an integer $B$ is yummy if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest yummy integer?

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>.

Given a triangle $\triangle ABC$ whose side lengths form an arithmetic sequence with a common difference of $2$, and the sine of its largest angle is $\frac{\sqrt{3}}{2}$, find the perimeter of this triangle.

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>.

Given vectors $\overrightarrow{a}=(\frac{1}{2},\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)$, $\overrightarrow{b}=(1,y)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the function be $y=f(x)$.

$(1)$ Find the smallest positive period of the function $y=f(x)$;

$(2)$ Given an acute triangle $ABC$ with angles $A$, $B$, and $C$, if $f(A-\frac{\pi}{3})=\sqrt{3}$, side $BC= \sqrt{7}$, $\sin B=\frac{\sqrt{21}}{7}$, find the length of $AC$ and the area of $\triangle ABC$.

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


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>.

Two wholesalers, A and B, sell the same brand of teapots and teacups at the same price, with the teapot priced at 30 yuan each and the teacup at 5 yuan each. Both stores are offering a promotional sale: Store A has a 'buy one get one free' promotion (buy a teapot and get a teacup for free), while Store B has a 10% discount on the entire store. A tea set store needs to buy 5 teapots and a number of teacups (not less than 5).
(1) Assuming that the tea set store buys $x$ teacups $(x>5)$, the cost at Store A and Store B would be _______ yuan and _______ yuan respectively; (express with an algebraic expression involving $x$)
(2) When the tea set store needs to buy 10 teacups, which store offers a better price? Please explain.
(3) How many teacups does the tea set store have to buy for the cost to be the same at both stores?

Answer: $20$


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

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $$\rho=4 \sqrt {2}\sin\left( \frac {3\pi}{4}-\theta\right)$$
(1) Convert the polar equation of circle $C$ into a Cartesian coordinate equation;
(2) Draw a line $l$ with slope $\sqrt {3}$ through point $P(0,2)$, intersecting circle $C$ at points $A$ and $B$. Calculate the value of $$\left| \frac {1}{|PA|}- \frac {1}{|PB|}\right|.$$

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


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

Given that $\sin \alpha = \frac{4}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $\cot (\frac{\pi}{4} - \frac{\alpha}{2})$.

Answer: $-3$


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

Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b.

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>.

Four foreign guests visit a school and need to be accompanied by two security personnel. Six people enter the school gate in sequence. For safety reasons, the two security personnel must be at the beginning and the end. If the guests A and B must be together, calculate the total number of sequences for the six people entering.

Answer: $24$


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 yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent.

Answer: $-19\%$


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

A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid?

Answer: $139.5$


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

Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse.

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


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

Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains.

Answer: $144$


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

Given a woman was x years old in the year $x^2$, determine her birth year.

Answer: $1980$


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 $a,$ $b,$ and $c$ be angles such that
\begin{align*}
\sin a &= \cot b, \\
\sin b &= \cot c, \\
\sin c &= \cot a.
\end{align*}
Find the largest possible value of $\cos a.$

Answer: $\sqrt{\frac{3 - \sqrt{5}}{2}}$


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>.

Given that $F$ is the right focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and the line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, with $\overrightarrow{MF} \cdot \overrightarrow{NF} = 0$. If the area of $\triangle MNF$ is $ab$, find the eccentricity of the hyperbola.

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>.

For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?

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>.

Given two lines \\({{l}\_{1}}:(a-1)x+2y+3=0\\) and \\({{l}\_{2}}:x+ay+3=0\\) are parallel, then \\(a=\\)_______.

Answer: $-1$


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

Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a delay in starting the timer, there was already a small amount of water in the cylinder at the beginning. Therefore, he obtained a set of data as shown in the table below:

| Time $t$ (minutes) | 1 | 2 | 3 | 4 | 5 | ... |
|---------------------|---|---|---|---|---|----|
| Total water amount $y$ (milliliters) | 7 | 12 | 17 | 22 | 27 | ... |

$(1)$ Investigation: Based on the data in the table above, determine which one of the functions $y=\frac{k}{t}$ and $y=kt+b$ (where $k$ and $b$ are constants) can correctly reflect the functional relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.

$(2)$ Application:

① Estimate how many milliliters of water will be in the cylinder when Xiao Ming measures it at the 20th minute.

② A person drinks approximately 1500 milliliters of water per day. Estimate how many days the water leaked from this faucet in a month (30 days) can supply one person.

Answer: $144$


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

Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\]

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>.

Consider a regular polygon with  $2^n$  sides, for  $n \ge 2$ , inscribed in a circle of radius  $1$ . Denote the area of this polygon by  $A_n$ . Compute  $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$

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


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>.

In a class, no two boys were born on the same day of the week and no two girls were born in the same month. If another child were to join the class, this would no longer be true. How many children are there in the class?

Answer: $19$


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

If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\).

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


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>.

At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants?

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 following is Xiaoying's process of solving a linear equation. Please read carefully and answer the questions.

解方程:$\frac{{2x+1}}{3}-\frac{{5x-1}}{6}=1$
Solution:
To eliminate the denominators, we get $2\left(2x+1\right)-\left(5x-1\right)=1$ ... Step 1
Expanding the brackets, we get $4x+2-5x+1=1$ ... Step 2
Rearranging terms, we get $4x-5x=1-1-2$ ... Step 3
Combining like terms, we get $-x=-2$, ... Step 4
Dividing both sides of the equation by $-1$, we get $x=2$ ... Step 5

$(1)$ The basis of the third step in the above solution process is ______.
$A$. the basic property of equations
$B$. the basic property of inequalities
$C$. the basic property of fractions
$D$. the distributive property of multiplication
$(2)$ Errors start to appear from the ______ step;
$(3)$ The correct solution to the equation is ______.

Answer: $x = -3$