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

The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \geq 3$). Find $a_{1000}$.

Answer: $1002$


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

Inside the square $A B C D$, a point $P$ is chosen such that the distances from $P$ to vertices $A$, $B$, and $C$ are in the ratio $A P: B P: C P=1: 2: 3$. What is the measure of angle $A P B$?

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>.

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

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


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

Given that $α+β= \frac {π}{3}$ and $tanα+tanβ=2$, find the value of $cos(α-β)$.

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


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>.

Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \((x, y)\) satisfying the equation

$$
(\sin x + \sin y)(\cos x - \cos y) = \frac{1}{2} + \sin(x - y) \cos(x + y)
$$

Answer: $-\frac{\pi}{6}$


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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c \sin A = \sqrt{3}a \cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \sin x \cos (\frac{\pi}{2} - x) - \sqrt{3} \sin (\pi + x) \cos x + \sin (\frac{\pi}{2} + x) \cos x$. Also, find the value of $f(B)$.

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


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

Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \).

Answer: $\sqrt{2}$


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

Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0) = 2$, determine the value of $f(4)$.

Answer: $-2$


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>.

How many ways are there to choose 6 different numbers from $1, 2, \cdots, 49$, where at least two numbers are consecutive?

Answer: $\binom{49}{6} - \binom{44}{6}$


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

Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart lands within the center hexagon.

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>.

In a round-robin chess tournament, only grandmasters and masters participated. The number of masters was three times the number of grandmasters, and the total points scored by the masters was 1.2 times the total points scored by the grandmasters.

How many people participated in the tournament? What can be said about the tournament's outcome?

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 sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is .

Answer: $\frac{5\sqrt{97}-19}{20}$


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  $\triangle ABC$ ,  $AB = 40$ ,  $BC = 60$ , and  $CA = 50$ . The angle bisector of  $\angle A$  intersects the circumcircle of  $\triangle ABC$  at  $A$  and  $P$ . Find  $BP$ .

*Proposed by Eugene Chen*

Answer: $40$


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

The teacher fills some numbers into the circles in the diagram below (each circle can and must only contain one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle $X$ is 9, then the number in circle $Y$ is $\qquad$

Answer: $11$


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

Given a convex quadrilateral \(ABCD\) with \(X\) as the midpoint of the diagonal \(AC\), it turns out that \(CD \parallel BX\). Find \(AD\) if it is known that \(BX = 3\), \(BC = 7\), and \(CD = 6\).

Answer: $14$


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

For \( n \in \mathbf{Z}_{+}, n \geqslant 2 \), let
\[
S_{n}=\sum_{k=1}^{n} \frac{k}{1+k^{2}+k^{4}}, \quad T_{n}=\prod_{k=2}^{n} \frac{k^{3}-1}{k^{3}+1}
\]

Then, \( S_{n} T_{n} = \) .

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


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

On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?

Answer: $3001$


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

How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?

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>.

Place parentheses and operation signs in the sequence 22222 so that the result is 24.

Answer: $(2+2+2) \times (2+2)$


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>.

Let the real numbers \(a_{1}, a_{2}, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0\); (ii) \(a_{1}+a_{2} \leqslant 100\); (iii) \(a_{3}+a_{4} + \cdots + a_{100} \leqslant 100\). Find the maximum value of \(a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}\) and the values of \(a_{1}, a_{2}, \cdots, a_{100}\) when the maximum value is reached.

Answer: $10000$


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

We have  $2022$   $1s$  written on a board in a line. We randomly choose a strictly increasing sequence from  ${1, 2, . . . , 2022}$  such that the last term is  $2022$ . If the chosen sequence is  $a_1, a_2, ..., a_k$  ( $k$  is not fixed), then at the  $i^{th}$  step, we choose the first a $_i$  numbers on the line and change the 1s to 0s and 0s to 1s. After  $k$  steps are over, we calculate the sum of the numbers on the board, say  $S$ . The expected value of  $S$  is  $\frac{a}{b}$  where  $a, b$  are relatively prime positive integers. Find  $a + b.$

Answer: $1012$


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

Translate the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$ to the left by $\varphi$ ($\varphi > 0$) units. If the resulting graph is symmetric about the y-axis, then the minimum value of $\varphi$ is \_\_\_\_\_.

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


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 diagram, \(ABCD\) is a square with a side length of \(8 \, \text{cm}\). Point \(E\) is on \(AB\) and point \(F\) is on \(DC\) so that \(\triangle AEF\) is right-angled at \(E\). If the area of \(\triangle AEF\) is \(30\%\) of the area of \(ABCD\), what is the length of \(AE\)?

Answer: $4.8$


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

Calculate the degree of ionization using the formula:

$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$

Given values:

$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$

Alternatively, if the concentration of ions is known, you can calculate $\alpha$ as:

$$
\mathrm{C} \cdot \alpha=[\mathrm{H}^{+}]=[\mathrm{CN}^{-}], [\mathrm{H}^{+}]=[\mathrm{CN}^{-}]=8,5 \cdot 10^{-6} \text{ mol/L}
$$

Then:

$$
\alpha_{\text{ion }}=8,5 \cdot 10^{-6}, 0,1=8,5 \cdot 10^{-5} \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$

Answer: $0.0085$


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

Among the 9 natural numbers $1,2,3, \cdots, 9$, if 3 numbers are chosen, let $x$ be the number of pairs of adjacent numbers among the chosen 3 numbers (for example, if the 3 chosen numbers are $1,2,3$, there are 2 pairs of adjacent numbers: 1,2 and 2,3, so the value of $x$ is 2). What is the expected value of $x$?

Answer: $2/3$


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>.

Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\frac{n}{30}$ that has a repeating decimal.

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>.

In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area of its circumscribed sphere is ______.

Answer: $14\pi$


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

Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \).

Answer: $55$


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

Find the smallest value of \(a\) for which the sum of the squares of the roots of the equation \(x^{2}-3ax+a^{2}=0\) is equal to \(0.28\).

Answer: $-0.2$


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>.

From the set \(\left\{-3, -\frac{5}{4}, -\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}\), two numbers are drawn without replacement. Find the concept of the two numbers being the slopes of perpendicular lines.

Answer: $3/28$


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

In the rectangular coordinate system xOy, it is known that 0 < α < 2π. Point P, with coordinates $(1 - \tan{\frac{\pi}{12}}, 1 + \tan{\frac{\pi}{12}})$, lies on the terminal side of angle α. Determine the value of α.

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>.

Define \( n! = 1 \times 2 \times \ldots \times n \), for example \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \). If \(\frac{n! \times (n+1)!}{2}\) (where \( \mathbf{n} \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, what is the sum of all such \( \mathbf{n} \)?

Answer: $273$


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

Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$

where " \( \sum \) " denotes a cyclic sum.

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


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>.

To celebrate her birthday, Ana is going to prepare pear and apple pies. In the market, an apple weighs $300 \text{ g}$ and a pear weighs $200 \text{ g}$. Ana's bag can hold a maximum weight of $7 \text{ kg}$. What is the maximum number of fruits she can buy to make pies with both types of fruits?

Answer: $34$


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

Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero.

Answer: $376$


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

How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem?

Answer: $160$


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

There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper?

Answer: $1 : 2$


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 \( z_{1} \) and \( z_{2} \) are complex numbers with \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1}+z_{2}\right|=7 \), find the value of \( \arg \left(\left(\frac{z_{2}}{z_{1}}\right)^{3}\right) \).

Answer: $\pi$


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>.

At 7:00 AM, Xiaoming leaves his house and heads to school at a speed of 52 meters per minute. When he reaches the school, the hour and minute hands on his watch are positioned symmetrically around the number 7 on the clock. It's known that Xiaoming walked for less than an hour. How far is the distance between Xiaoming's house and the school, in meters?

Answer: $1680$


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

Ten students (one captain and nine team members) formed a team to participate in a math competition and won first place. The committee decided to award each team member 200 yuan. The captain received 90 yuan more than the average bonus of all ten team members. Determine the amount of bonus the captain received.

Answer: $300$


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

A rectangular picture frame is constructed from 1.5-inch-wide pieces of wood. The area of just the frame is \(27\) square inches, and the length of one of the interior edges of the frame is \(4.5\) inches. Determine the sum of the lengths of the four interior edges of the frame.

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 mean, median, and mode of the five numbers 12, 9, 11, 16, x are all equal. Find the value of x.

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>.

15 plus 16 equals.

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>.

How, without any measuring tools, can you measure 50 cm from a string that is $2/3$ meters long?

Answer: $50$


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

A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \, \mathrm{m/s^2}$, and the Earth's radius is $R = 6400 \, \mathrm{km}$).

Answer: $6400$


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

The lateral surface area of a regular triangular pyramid is 3 times the area of its base. The area of the circle inscribed in the base is numerically equal to the radius of this circle. Find the volume of the pyramid.

Answer: $\frac{2 \sqrt{6}}{\pi^3}$


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>.

Four people, A, B, C, and D, participated in an exam. The combined scores of A and B are 17 points higher than the combined scores of C and D. A scored 4 points less than B, and C scored 5 points more than D. How many points higher is the highest score compared to the lowest score among the four?

Answer: $13$


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

Find the area of triangle \(ABC\), if \(AC = 3\), \(BC = 4\), and the medians \(AK\) and \(BL\) are mutually perpendicular.

Answer: $\sqrt{11}$


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

In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.

Answer: $45$


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

Express the given data "$20$ nanoseconds" in scientific notation.

Answer: $2 \times 10^{-8}$


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>.

For the real numbers \(a\) and \(b\), it holds that \(a^{2} + 4b^{2} = 4\). How large can \(3a^{5}b - 40a^{3}b^{3} + 48ab^{5}\) be?

Answer: $16$


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

In a survey of $150$ employees at a tech company, it is found that:
- $90$ employees are working on project A.
- $50$ employees are working on project B.
- $30$ employees are working on both project A and B.

Determine what percent of the employees surveyed are not working on either project.

Answer: $26.67\%$


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

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1500$ and $AD = 3000$. Let $\angle A = 30^\circ$, $\angle D = 60^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Determine the length $PQ$.

Answer: $750$


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

Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).

Answer: $1/33$


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

Find the smallest positive real t such that
\[ x_1 + x_3 = 2tx_2, \]
\[ x_2 + x_4 = 2tx_3, \]
\[ x_3 + x_5 = 2tx_4 \]
has a solution \( x_1, x_2, x_3, x_4, x_5 \) in non-negative reals, not all zero.

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


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>.

Given that the odd function f(x) defined on R satisfies f(x+2) + f(2-x) = 0, and when x ∈ (-2, 0), f(x) = log2(x+3) + a. If f(9) = 2f(7) + 1, then the value of the real number a is ( ).

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


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

Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\cos \alpha = \frac{4}{5}\), \(\tan (\alpha - \beta) = -\frac{1}{3}\), find \(\cos \beta\).

Answer: $\frac{9 \sqrt{10}}{50}$


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

Given that $S_n$ and $T_n$ represent the sum of the first n terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ respectively, and $\frac{S_n}{T_n} = \frac{2n+1}{n+3}$, determine the value of $\frac{a_7}{b_7}$.

Answer: $\frac{27}{16}$


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>.

In the complex plane, let the vertices \( A \) and \( B \) of triangle \( \triangle AOB \) correspond to the complex numbers \( \alpha \) and \( \beta \), respectively, and satisfy the conditions: \( \beta = (1 + i)\alpha \) and \( |\alpha - 2| = 1 \). \( O \) is the origin. Find the maximum value of the area \( S \) of the triangle \( \triangle OAB \).

Answer: $9/2$


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>.

If $a-b=1$ and $ab=-2$, then $\left(a+1\right)\left(b-1\right)=$____.

Answer: $-4$


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>.

Is there an integer $x$ such that $x \equiv 1 \ (\text{mod} \ 6)$, $x \equiv 9 \ (\text{mod} \ 14)$, and $x \equiv 7 \ (\text{mod} \ 15)$?

Answer: $37$


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

Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=5$, and $|r-s|=7$. What is the sum of all possible values of $|p-s|$?

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>.

Calculate: $|\sqrt{3}-2|+(\pi -\sqrt{10})^{0}-\sqrt{12}$.

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


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

Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $C: x^{2}-\frac{{y}^{2}}{3}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $\angle PF_{2}Q=120^{\circ}$, find the area of quadrilateral $PF_{1}QF_{2}$.

Answer: $6\sqrt{3}$


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

Given the function $f(x) = 2^x + \ln x$, if $a_n = 0.1n$ ($n \in \mathbb{N}^*$), find the value of $n$ that minimizes $|f(a_n) - 2012|$.

Answer: $110$


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

There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.

How many mushrooms are in the basket?

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>.

ABCDEF is a six-digit number. All its digits are different and arranged in ascending order from left to right. This number is a perfect square.

Determine what this number is.

Answer: $134689$


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

In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}a\cos C=(2b-\sqrt{3}c)\cos A$

(Ⅰ) Find the magnitude of angle $A$;

(Ⅱ) If $a=2$, find the maximum area of $\Delta ABC$.

Answer: $2+ \sqrt{3}$


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

Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, and arrived at the airport 15 minutes earlier than planned initially. What is the distance from Victor's home to the airport?

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 base of an inclined parallelepiped is a rhombus with a side length of 60. A diagonal section plane passing through the longer diagonal of the base is perpendicular to the base's plane. The area of this section is 7200. Find the shorter diagonal of the base if the lateral edge is 80 and forms an angle of $60^\circ$ with the base plane.

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>.

It is known that 9 cups of tea cost less than 10 rubles, and 10 cups of tea cost more than 11 rubles. How much does one cup of tea cost?

Answer: $111$


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

Consider a sequence of real numbers \(\{a_n\}\) defined by \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{1 + n a_n}\) for \(n \geq 1\). Find the value of \(\frac{1}{a_{2005}} - 2000000\).

Answer: $9011$


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

Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)? If the question allows for multiple solutions, list them all as a set.

Answer: $50$


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

If the integer $a$ makes the inequality system about $x$ $\left\{\begin{array}{l}{\frac{x+1}{3}≤\frac{2x+5}{9}}\\{\frac{x-a}{2}＞\frac{x-a+1}{3}}\end{array}\right.$ have at least one integer solution, and makes the solution of the system of equations about $x$ and $y$ $\left\{\begin{array}{l}ax+2y=-4\\ x+y=4\end{array}\right.$ positive integers, find the sum of all values of $a$ that satisfy the conditions.

Answer: $-16$


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

On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle?

Answer: $1:2$


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 a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)?

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>.

In triangle \( \triangle ABC \), \( AB = \sqrt{2} \), \( AC = \sqrt{3} \), and \( \angle BAC = 30^\circ \). Let \( P \) be an arbitrary point in the plane containing \( \triangle ABC \). Find the minimum value of \( \mu = \overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA} \).

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


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 sum of $521_8$ and $146_8$ in base $8$.

Answer: $667_8$


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

In a new game, Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.

Answer: $\frac{5}{6}$


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>.

Samantha has 10 different colored marbles in her bag. In how many ways can she choose five different marbles such that at least one of them is red?

Answer: $126$


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

The decimal number  $13^{101}$  is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

Answer: $21$


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>.

What is the smallest eight-digit positive integer that has exactly four digits which are 4?

Answer: $10004444$


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

Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.

Answer: $34$


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

The cafe "Burattino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the cafe worked 18 days" and "from April 10 to April 30, the cafe also worked 18 days." It is known that he made a mistake once. How many days did the cafe work from April 1 to April 27?

Answer: $23$


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

In the convex quadrilateral \(ABCD\),
\[
\angle BAD = \angle BCD = 120^\circ, \quad BC = CD = 10.
\]

Find \(AC.\)

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 number of positive integer pairs  $(a,b)$  that have  $a$  dividing  $b$  and  $b$  dividing  $2013^{2014}$  can be written as  $2013n+k$ , where  $n$  and  $k$  are integers and  $0\leq k<2013$ .  What is  $k$ ?  Recall  $2013=3\cdot 11\cdot 61$ .

Answer: $27$


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

Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \).

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


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>.

An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?

Answer: $79$


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

Let the operation $\#$ be defined as $\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\oplus$ by $\oplus(a, b, c, d) = \#(a, b + d, c) - \#(a, b, c)$. What is the value of $\oplus(2, 4, 1, 3)$?

Answer: $33$


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

Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, calculate the maximum value of the sum of the first $n$ terms, $S_n$.

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>.

Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.

Answer: $5.1$


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>.

If for any real number \( x \), the function
\[ f(x)=x^{2}-2x-|x-1-a|-|x-2|+4 \]
always yields a non-negative real number, then the minimum value of the real number \( a \) is .

Answer: $-2$


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

If $x+y=8$ and $xy=12$, what is the value of $x^3+y^3$ and $x^2 + y^2$?

Answer: $40$


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

Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \).

Answer: $11$


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

Let \( ABCD \) be a square with side length \( 5 \), and \( E \) be a point on \( BC \) such that \( BE = 3 \) and \( EC = 2 \). Let \( P \) be a variable point on the diagonal \( BD \). Determine the length of \( PB \) if \( PE + PC \) is minimized.

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


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

Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally?

Answer: $82$


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

The diagram shows the ellipse whose equation is \(x^{2}+y^{2}-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). What is the area of the inscribed quadrilateral \(ABCD\)?

Answer: $28$


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

From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form.

Answer: $-\frac{2020}{2017}$


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

On the banks of an island, which has the shape of a circle (viewed from above), there are the cities $A, B, C,$ and $D$. A straight asphalt road $AC$ divides the island into two equal halves. A straight asphalt road $BD$ is shorter than road $AC$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has straight dirt roads $AB, BC, CD,$ and $AD$, on which the cyclist's speed is the same. The cyclist travels from point $B$ to each of points $A, C,$ and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$.

Answer: $450$


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

The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (the intruder remained stationary). Three guards collectively ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?

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>.

A metallic weight has a mass of 25 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second; the mass of the second metal is related to the mass of the third as \(3: 4\), and the mass of the third metal to the mass of the fourth as \(5: 6\). Determine the mass of the fourth metal. Give the answer in kilograms, rounding to the nearest hundredth if necessary.

Answer: $7.36$


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

Triangle \(ABC\) has \(\angle A = 90^\circ\), side \(BC = 25\), \(AB > AC\), and area 150. Circle \(\omega\) is inscribed in \(ABC\), with \(M\) as its point of tangency on \(AC\). Line \(BM\) meets \(\omega\) a second time at point \(L\). Find the length of segment \(BL\).

Answer: $\frac{45\sqrt{17}}{17}$


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

In \(\triangle ABC\), \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. Given \(a+c=2b\) and \(A-C=\frac{\pi}{3}\), find the value of \(\sin B\).

Answer: $\frac{\sqrt{39}}{8}$


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 a parabola $C: y^2 = 3x$ with focus $F$, find the length of segment $AB$ where the line passing through $F$ at a $30^\circ$ angle intersects the parabola at points $A$ and $B$.

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 function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___

Answer: $-1$


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

A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $m$ and $n$ are relatively prime, that two randomly selected cards from the remaining cards also form a pair. Find $m + n$.

Answer: $299$


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

When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it?

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>.

The length of one side of the square \(ABCD\) is 4 units. A circle is drawn tangent to \(\overline{BC}\) and passing through the vertices \(A\) and \(D\). Find the area of the circle.

Answer: $\frac{25 \pi}{4}$


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

A crew of workers was tasked with pouring ice rinks on a large and a small field, where the area of the large field is twice the area of the small field. The part of the crew working on the large field had 4 more workers than the part of the crew working on the small field. When the pouring on the large rink was completed, the group working on the small field was still working. What is the maximum number of workers that could have been in the crew?

Answer: $10$


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

Given point $P(2,-1)$,
(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.
(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the maximum distance.

Answer: $\sqrt{5}$


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

Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other?

Answer: $17/57$


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

Let \( A \) be the set of any 20 points on the circumference of a circle. Joining any two points in \( A \) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords.

Answer: $5036$


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

Let \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \).

Answer: $\frac{2012}{2013}$


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

If $x \sim N(4, 1)$ and $f(x < 3) = 0.0187$, then $f(x < 5) = \_\_\_\_\_\_$.

Answer: $0.9813$


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

Given the ages of Daisy's four cousins are distinct single-digit positive integers, and the product of two of the ages is $24$ while the product of the other two ages is $35$, find the sum of the ages of Daisy's four cousins.

Answer: $23$


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

Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases}x=2+t \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases}x=-2+m \\ y= \frac{m}{k}\end{cases}$ (where $m$ is the parameter). Let the intersection point of $l_1$ and $l_2$ be $P$. When $k$ changes, the trajectory of $P$ is curve $C$.

(1) Write the general equation of $C$;

(2) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Let line $l_3: \rho(\cos \theta +\sin \theta)− \sqrt{2} =0$, and $M$ be the intersection point of $l_3$ and $C$. Find the polar radius of $M$.

Answer: $\sqrt{5}$


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

The letters L, K, R, F, O and the digits 1, 7, 8, 9 are "cycled" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time.

Answer: $20$


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

Given that \(\alpha, \beta, \gamma\) satisfy \(0<\alpha<\beta<\gamma<2 \pi\), and for any \(x \in \mathbf{R}\), \(\cos (x+\alpha) + \cos (x+\beta) + \cos (x+\gamma) = 0\), determine the value of \(\gamma - \alpha\).

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


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>.

In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in the spring round?

Answer: $2000$


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

From the 4 digits 0, 1, 2, 3, select 3 digits to form a three-digit number without repetition. How many of these three-digit numbers are divisible by 3?

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>.

A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle.

Answer: $\frac{\sqrt{10}}{5}$


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

In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(N\) respectively. Given that \(\angle MPB = \angle PCN\) and \(\angle NPC = \angle MBP\), find \(\frac{S_{\triangle APC}}{S_{ACDE}}\).

Answer: $1/2$


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 positive integers are grouped as follows:

\( A_1 = \{1\}, A_2 = \{2, 3, 4\}, A_3 = \{5, 6, 7, 8, 9\} \), and so on.

In which group does 2009 belong?

Answer: $45$


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

There are 150 different cards on the table with the numbers $2, 4, 6, \ldots, 298, 300$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the chosen cards is divisible by $5$?

Answer: $2235$


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

One student has 6 mathematics books, and another has 8. In how many ways can they exchange three books?

Answer: $1120$


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

Let $[ x ]$ denote the greatest integer less than or equal to $x$. For example, $[10.2] = 10$. Calculate the value of $\left[\frac{2017 \times 3}{11}\right] + \left[\frac{2017 \times 4}{11}\right] + \left[\frac{2017 \times 5}{11}\right] + \left[\frac{2017 \times 6}{11}\right] + \left[\frac{2017 \times 7}{11}\right] + \left[\frac{2017 \times 8}{11}\right]$.

Answer: $6048$


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

Calculate the definite integral:

$$
\int_{0}^{\pi} 2^{4} \sin ^{4}\left(\frac{x}{2}\right) \cos ^{4}\left(\frac{x}{2}\right) d x
$$

Answer: $\frac{3\pi}{8}$