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

How many even integers are there between \( \frac{12}{3} \) and \( \frac{50}{2} \)?

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>.

Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\).

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


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

Let  $F_k(a,b)=(a+b)^k-a^k-b^k$  and let  $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs  $(a,b)$  with  $a,b\in S$  and  $a\leq b$  is  $\frac{F_5(a,b)}{F_3(a,b)}$  an integer?

Answer: $22$


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

In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there?

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>.

Convert the binary number $110101_{(2)}$ to decimal.

Answer: $53$


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

A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.

How many meters of wire were there originally in the bundle?

Answer: $54$


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

The line $x+2ay-1=0$ is parallel to $\left(a-1\right)x+ay+1=0$, find the value of $a$.

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>.

Find the number of natural numbers \( k \), not exceeding 333300, such that \( k^{2} - 2k \) is exactly divisible by 303.

Answer: $4400$


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

Determine the integer \( x \) for which the following equation holds true:
\[ 1 \cdot 2023 + 2 \cdot 2022 + 3 \cdot 2021 + \dots + 2022 \cdot 2 + 2023 \cdot 1 = 2023 \cdot 1012 \cdot x. \]

Answer: $675$


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

Find the smallest natural number \( N \) that is divisible by \( p \), ends with \( p \), and has a digit sum equal to \( p \), given that \( p \) is a prime number and \( 2p+1 \) is a cube of a natural number.

Answer: $11713$


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

John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions).

Answer: $85$


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

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

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>.

In a toy store, there are large and small plush kangaroos. In total, there are 100 of them. Some of the large kangaroos are female kangaroos with pouches. Each female kangaroo has three small kangaroos in her pouch, and the other kangaroos have empty pouches. Find out how many large kangaroos are in the store, given that there are 77 kangaroos with empty pouches.

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>.

Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25.

Answer: $19$


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

In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, it is known that $a_1=4$ and $a_7^2=a_1a_{10}$. The sum of the first $n$ terms is $S_n$.
1. Find the general formula for the sequence $\{a_n\}$.
2. Find the maximum value of $S_n$ and the value of $n$ when the maximum is achieved.

Answer: $26$


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

Calculate $\left(\frac{1}{3}\right)^{6} \div \left(\frac{2}{5}\right)^{-4} + \frac{1}{2}$.

Answer: $\frac{455657}{911250}$


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

Three regular heptagons share a common center, and their sides are parallel. The sides of two heptagons are 6 cm and 30 cm, respectively. The third heptagon divides the area between the first two heptagons in a ratio of $1:5$, starting from the smaller heptagon. Find the side of the third heptagon.

Answer: $6\sqrt{5}$


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>.

How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\{4, 7, 10, 13, \ldots, 46\}$?

Answer: $37$


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

Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and their dot product $\overrightarrow{a} \cdot \overrightarrow{b} = 1$. If $\overrightarrow{e}$ is a unit vector in the plane, find the maximum value of $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}|$.

Answer: $\sqrt{7}$


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

Given the function \( f(x)=\left(1-x^{3}\right)^{-1 / 3} \), find \( f(f(f \ldots f(2018) \ldots)) \) where the function \( f \) is applied 2019 times.

Answer: $2018$


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

a) Calculate the number of triangles whose three vertices are vertices of the cube.

b) How many of these triangles are not contained in a face of the cube?

Answer: $32$


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

A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola.

Answer: $\sqrt{3}$


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>.

The maximum value of the function $y=\sin x \cos x + \sin x + \cos x$ is __________.

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


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

Calculate:
$(1)-4^2÷(-32)×(\frac{2}{3})^2$;
$(2)(-1)^{10}÷2+(-\frac{1}{2})^3×16$;
$(3)\frac{12}{7}×(\frac{1}{2}-\frac{2}{3})÷\frac{5}{14}×1\frac{1}{4}$;
$(4)1\frac{1}{3}×[1-(-4)^2]-(-2)^3÷\frac{4}{5}$.

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>.

Juca has fewer than 800 marbles. He likes to separate the marbles into groups of the same size. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finally, if he forms groups of 7 marbles each, 6 marbles are left over.

(a) If Juca formed groups of 20 marbles each, how many marbles would be left over?

(b) How many marbles does Juca have?

Answer: $419$


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

Joey has 30 thin sticks, each stick has a length that is an integer from 1 cm to 30 cm. Joey first places three sticks on the table with lengths of 3 cm, 7 cm, and 15 cm, and then selects a fourth stick such that it, along with the first three sticks, forms a convex quadrilateral. How many different ways are there for Joey to make this selection?

Answer: $17$


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

Let n rational numbers $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ satisfy $|x_{i}| \lt 1$ for $i=1,2,\ldots,n$, and $|x_{1}|+|x_{2}|+\ldots+|x_{n}|=19+|x_{1}+x_{2}+\ldots+x_{n}|$. Calculate the minimum value of $n$.

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>.

The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18.

Answer: $12$


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

Given that \( \text{rem} \left(\frac{5}{7}, \frac{3}{4}\right) \) must be calculated, determine the value of the remainder.

Answer: $\frac{5}{7}$


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

Let \(\left(x^{2}+2x-2\right)^{6}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\cdots+a_{12}(x+2)^{12}\), where \(a_{i} (i=0,1,2,\ldots,12)\) are real constants. Determine the value of \(a_{0}+a_{1}+2a_{2}+3a_{3}+\cdots+12a_{12}\).

Answer: $64$


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

The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\qquad$

Answer: $10829$


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

In how many different ways can 7 different prizes be awarded to 5 students such that each student has at least one prize?

Answer: $16800$


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

What is the mean of the set $\{m, m + 6, m + 8, m + 11, m + 18, m + 20\}$ if the median of the set is 19?

Answer: $20$


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

Given the function $f(x) = x + \sin(\pi x) - 3$, study its symmetry center $(a, b)$ and find the value of $f\left( \frac {1}{2016} \right) + f\left( \frac {2}{2016} \right) + f\left( \frac {3}{2016} \right) + \ldots + f\left( \frac {4030}{2016} \right) + f\left( \frac {4031}{2016} \right)$.

Answer: $-8062$


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

Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________.

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


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>.

\( \Delta ABC \) is an isosceles triangle with \( AB = 2 \) and \( \angle ABC = 90^{\circ} \). Point \( D \) is the midpoint of \( BC \) and point \( E \) is on \( AC \) such that the area of quadrilateral \( AEDB \) is twice the area of triangle \( ECD \). Find the length of \( DE \).

Answer: $\frac{\sqrt{17}}{3}$


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 a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles?

Answer: $360$


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

The diagram shows a triangle joined to a square to form an irregular pentagon. The triangle has the same perimeter as the square.

What is the ratio of the perimeter of the pentagon to the perimeter of the square?

Answer: $3:2$


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>.

Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: "Among the statements made earlier, the true ones are exactly 20 less than the false ones." How many knights were present at this meeting?

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>.

From point $A$ outside a circle, a tangent and a secant are drawn to the circle. The distance from point $A$ to the point of tangency is 16, and the distance from point $A$ to one of the intersection points of the secant with the circle is 32. Find the radius of the circle if the distance from its center to the secant is 5.

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>.

Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$.

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 physical education teacher lined up the class so that everyone was facing him. There are 12 people to the right of Kolya, 20 people to the left of Sasha, and 8 people to the right of Sasha. How many people are to the left of Kolya?

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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a=2$, $b= \sqrt {3}$, $B= \frac {\pi}{3}$, then $A=$ \_\_\_\_\_\_.

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


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

Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will roll her die?

Answer: $488$


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

A store received apples of the first grade worth 228 rubles and apples of the second grade worth 180 rubles. During unloading, the apples got mixed up. Calculations showed that if all the apples are now sold at a price 90 kopeks lower than the price per kilogram of first grade apples, the planned revenue will be achieved. How many kilograms of apples were delivered, given that there were 5 kg more second grade apples than first grade apples?

Answer: $85$


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

A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one of the resulting triangles to the perimeter of the original square?
A) $\frac{15 + \sqrt{125}}{40}$
B) $\frac{15 + \sqrt{75}}{40}$
C) $\frac{10 + \sqrt{50}}{40}$
D) $\frac{20 + \sqrt{100}}{40}$

Answer: $\frac{15 + \sqrt{125}}{40}$


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>.

Let  $\Delta ABC$  be an acute-angled triangle and let  $H$  be its orthocentre. Let  $G_1, G_2$  and  $G_3$  be the centroids of the triangles  $\Delta HBC , \Delta HCA$  and  $\Delta HAB$  respectively. If the area of  $\Delta G_1G_2G_3$  is  $7$  units, what is the area of  $\Delta ABC $ ?

Answer: $63$


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

A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \mathrm{~m}$ long. What is the speed of the train?

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>.

Sequence  $(a_n)$  is defined as  $a_{n+1}-2a_n+a_{n-1}=7$  for every  $n\geq 2$ , where  $a_1 = 1, a_2=5$ . What is  $a_{17}$ ?

Answer: $905$


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

The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.

Answer: $2020$


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

In the sequence $\{a_n\}$, $a_1=2$, $a_2=5$, $a_{n+1}=a_{n+2}+a_{n}$, calculate the value of $a_6$.

Answer: $-3$


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

In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\).

Answer: $4:5$


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>.

Let $a$, $b$, and $c$ be the three roots of the equation $$
5x^3 + 505x + 1010 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$

Answer: $606$


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

Given the polynomial with integer coefficients:
\[ f(x) = x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 \]

If \( f(\sqrt{3} + \sqrt{2}) = 0 \) and \( f(1) + f(3) = 0 \), find \( f(-1) \).

Answer: $24$


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

The construction rule of the sequence $\{a_{n}\}$ is as follows: $a_{1}=1$, if $a_{n}-2$ is a natural number and has not appeared before, then use the recursive formula $a_{n+1}=a_{n}-2$. Otherwise, use the recursive formula $a_{n+1}=3a_{n}$, then $a_{6}=$ _____.

Answer: $15$


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

Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.).

Answer: $592$


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

Given the scores $93$, $89$, $92$, $95$, $93$, $94$, $93$ of seven referees, determine the average value of the remaining data after removing the highest and lowest scores.

Answer: $93$


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

Given the ellipse $$C: \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\overrightarrow {OM}$$ and $$\overrightarrow {MA}$$ is θ, and |tanθ|=3. Find the value of b.

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>.

Consider the set $\{8, -7, 2, -4, 20\}$. Find the smallest sum that can be achieved by adding three different numbers from this set.

Answer: $-9$


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>.

Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there?

Answer: $86$


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

Through how many squares does the diagonal of a 1983 × 999 chessboard pass?

Answer: $2979$


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

Let the set \( S = \{1, 2, \cdots, 15\} \). Define \( A = \{a_{1}, a_{2}, a_{3}\} \) as a subset of \( S \), such that \( (a_{1}, a_{2}, a_{3}) \) satisfies \( 1 \leq a_{1} < a_{2} < a_{3} \leq 15 \) and \( a_{3} - a_{2} \leq 6 \). Find the number of such subsets that satisfy these conditions.

Answer: $371$


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

Exactly half of the population of the island of Misfortune are hares, and the rest are rabbits. If a resident of Misfortune makes a statement, he sincerely believes what he says. However, hares are faithfully mistaken on average in one out of every four cases, and rabbits are faithfully mistaken on average in one out of every three cases. One day, a creature came to the center of the island and shouted, "I am not a hare!" Then he thought and sadly said, "I am not a rabbit." What is the probability that he is actually a hare?

Answer: $27/59$


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

Little kids were eating candies. Each ate 11 candies less than the rest combined but still more than one candy. How many candies were eaten in total?

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 that P is a point on the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$, and $F_1$, $F_2$ are the two foci of the hyperbola. If $\angle F_1PF_2 = 60^\circ$, calculate the area of $\triangle PF_1F_2$.

Answer: $3\sqrt{3}$


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

In a closed right triangular prism ABC-A1B1C1 there is a sphere with volume $V$. If $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \_\_\_\_\_\_.

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


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

Let \( f(x) = x^2 + px + q \) and \( g(x) = x^2 + rx + s \) be two distinct quadratic polynomials where the \( x \)-coordinate of the vertex of \( f \) is a root of \( g \), and the \( x \)-coordinate of the vertex of \( g \) is a root of \( f \), also both \( f \) and \( g \) have the same minimum value. If the graphs of the two quadratic polynomials intersect at the point \( (50,-200), \) what is the value of \( p + r \)?

Answer: $-200$


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

In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). If \(AD = 52\), \(BC = 65\), \(AB = 20\), and \(CD = 11\), find the area of the trapezoid.

Answer: $594$


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

There are five students: A, B, C, D, and E.

1. In how many different ways can they line up in a row such that A and B must be adjacent, and C and D cannot be adjacent?
2. In how many different ways can these five students be distributed into three classes, with each class having at least one student?

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 front view of a cone is an equilateral triangle with a side length of 4. Find the surface area of the cone.

Answer: $12\pi$


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>.

Among the following numbers

① $111111_{(2)}$
② $210_{(6)}$
③ $1000_{(4)}$
④ $81_{(8)}$

The largest number is \_\_\_\_\_\_\_\_, and the smallest number is \_\_\_\_\_\_\_\_.

Answer: $111111_{(2)}$


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 number \( 20! = 1 \cdot 2 \cdot \ldots \cdot 20 = 2432902008176640000 \) has 41,040 natural divisors. How many of them are odd?

Answer: $2160$


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

Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels?

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>.

Let $m$ be the smallest positive three-digit integer congruent to 7 (mod 13). Let $n$ be the smallest positive four-digit integer congruent to 7 (mod 13). What is the value of $n - m$?

Answer: $897$


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

An aluminum cube with an edge length of \( l = 10 \) cm is heated to a temperature of \( t_{1} = 100^{\circ} \mathrm{C} \). After this, it is placed on ice, which has a temperature of \( t_{2} = 0^{\circ} \mathrm{C} \). Determine the maximum depth to which the cube can sink. The specific heat capacity of aluminum is \( c_{a} = 900 \) J/kg\(^\circ \mathrm{C} \), the specific latent heat of fusion of ice is \( \lambda = 3.3 \times 10^{5} \) J/kg, the density of aluminum is \( \rho_{a} = 2700 \) kg/m\(^3 \), and the density of ice is \( \rho_{n} = 900 \) kg/m\(^3 \).

Answer: $0.0818$


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

Let \(a_n\) be the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{k}\), where \(k = a_n\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Which values \(b_n\) occur for infinitely many \(n\)?

Answer: $87$


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

In  $10\times 10$  square we choose  $n$  cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of  $n$ ?

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>.

The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)?

Answer: $1200$


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

20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one?

Answer: $17$


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

The maximum value of the function \( y = \frac{\sin x \cos x}{1 + \sin x + \cos x} \) is $\quad$ .

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


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 value of the product \(\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}\).

Answer: $\frac{1}{128}$


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

Find the measure of the angle

$$
\delta=\arccos \left(\left(\sin 2905^{\circ}+\sin 2906^{\circ}+\cdots+\sin 6505^{\circ}\right)^{\cos } 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}\right)
$$

Answer: $65$


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

Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$

Answer: $48$


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

In a right triangle $ABC$ with legs $AB = 3$ and $BC = 4$, a circle is drawn through the midpoints of sides $AB$ and $AC$, touching the leg $BC$. Find the length of the segment of the hypotenuse $AC$ that lies inside this circle.

Answer: $11/10$


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>.

Calculate the definite integral:

$$
\int_{0}^{2} e^{\sqrt{(2-x) /(2+x)}} \cdot \frac{d x}{(2+x) \sqrt{4-x^{2}}}
$$

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


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 value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$;
(b) $\frac{1}{2}(7+3 \sqrt{5})$;
(c) $1+2 \sqrt{3}$;
(d) 3 ;
(e) $3+2 \sqrt{2}$.

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


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

If the fractional equation $\frac{3}{{x-4}}+\frac{{x+m}}{{4-x}}=1$ has a root, determine the value of $m$.

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>.

15 boys and 20 girls sat around a round table. It turned out that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. Find the number of boy-girl pairs sitting next to each other.

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 clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves?

Answer: $810$


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

Let \(a\) and \(b\) be constants. The parabola \(C: y = (t^2 + t + 1)x^2 - 2(a + t)^2 x + t^2 + 3at + b\) passes through a fixed point \(P(1,0)\) for any real number \(t\). Find the value of \(t\) such that the chord obtained by intersecting the parabola \(C\) with the x-axis is the longest.

Answer: $-1$


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

A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle?

Answer: $3 + \sqrt{3}$


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

If $x$ is an even number, then find the largest integer that always divides the expression \[(15x+3)(15x+9)(5x+10).\]

Answer: $90$


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

Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate twice as many raspberries as Masha?

Answer: $54$


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

Income from September 2019 to December 2019 is:

$$
(55000+45000+10000+17400) * 4 = 509600 \text{ rubles}
$$

Expenses from September 2019 to November 2019 are:

$$
(40000+20000+5000+2000+2000) * 4 = 276000 \text{ rubles}
$$

By 31.12.2019 the family will have saved $1147240 + 521600 - 276000 = 1340840$ rubles and will be able to buy a car.

Answer: $1340840$


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

The number \( N \) is of the form \( p^{\alpha} q^{\beta} r^{\gamma} \), where \( p, q, r \) are prime numbers, and \( p q - r = 3, p r - q = 9 \). Additionally, the numbers \( \frac{N}{p}, \frac{N}{q}, \frac{N}{r} \) respectively have 20, 12, and 15 fewer divisors than the number \( N \). Find the number \( N \).

Answer: $857500$


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

9 judges score a gymnast in artistic gymnastics, with each giving an integer score. One highest score and one lowest score are removed, and the average of the remaining scores determines the gymnast's score. If the score is rounded to one decimal place using the rounding method, the gymnast scores 8.4 points. What would the gymnast's score be if it were accurate to two decimal places?

Answer: $8.43$


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

The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \).

Answer: $\frac{1}{500}$


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

Let \( S_{n} \) be the sum of the first \( n \) terms of an arithmetic sequence \( \{a_{n}\} \). Given \( S_{6}=36 \) and \( S_{n}=324 \). If \( S_{n-6} = 144 \) for \( n > 6 \), then \( n \) equals \(\qquad\).

Answer: $18$


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

How many three-digit whole numbers contain at least one digit 6 or at least one digit 8?

Answer: $452$


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

If a natural number \(N\) can be expressed as the sum of 3 consecutive natural numbers, the sum of 11 consecutive natural numbers, and the sum of 12 consecutive natural numbers, then the smallest value of \(N\) is \(\qquad\). (Note: The smallest natural number is 0.)

Answer: $66$


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

Let  $T = (a,b,c)$  be a triangle with sides  $a,b$  and  $c$  and area  $\triangle$ .  Denote by  $T' = (a',b',c')$  the triangle whose sides are the altitudes of  $T$  (i.e.,  $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by  $\triangle '$ .  Similarly, let  $T'' = (a'',b'',c'')$  be the triangle formed from the altitudes of  $T'$ , and denote its area by  $\triangle ''$ . Given that  $\triangle ' = 30$  and  $\triangle '' = 20$ , find  $\triangle$ .

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>.

Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half?

Answer: $8/63$


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

Determine the largest prime number less than 5000 of the form \( a^n - 1 \), where \( a \) and \( n \) are positive integers, and \( n \) is greater than 1.

Answer: $127$


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

The solutions to the equation \( x^3 - 4 \lfloor x \rfloor = 5 \), where \( x \) is a real number, are denoted by \( x_1, x_2, x_3, \ldots, x_k \) for some positive integer \( k \). Find \( \sum_{i=1}^{k} x_{i}^{3} \).

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>.

In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $GHI$ to the area of triangle $XYZ$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: $41$


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

Inside triangle \( ABC \), a random point \( M \) is chosen. What is the probability that the area of one of the triangles \( ABM \), \( BCM \), or \( CAM \) will be greater than the sum of the areas of the other two?

Answer: $0.75$


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

In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100,$ and $AC = 100$. Find the sum of all possible values of $BC$.

Answer: $100 \sqrt{2}$


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 \( S_{n} = \sum_{i=1}^{10} i|n-i| \left( n \in \mathbf{Z}_{+} \right) \), find the minimum value of \( S_{n} \).

Answer: $112$


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

Calculate:
$$
\left(10^{4}-9^{4}+8^{4}-7^{4}+\cdots+2^{4}-1^{4}\right)+\left(10^{2}+9^{2}+5 \times 8^{2}+5 \times 7^{2}+9 \times 6^{2}+9 \times 5^{2}+13 \times 4^{2}+13 \times 3^{2}\right) =
$$

Answer: $7615$


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

Define the sequence $(b_i)$ by $b_{n+2} = \frac{b_n + 2011}{1 + b_{n+1}}$ for $n \geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$.

Answer: $2012$


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

There are 4 different colors of light bulbs, with each color representing a different signal. Assuming there is an ample supply of each color, we need to install one light bulb at each vertex $P, A, B, C, A_{1}, B_{1}, C_{1}$ of an airport signal tower (as shown in the diagram), with the condition that the two ends of the same line segment must have light bulbs of different colors. How many different installation methods are there?

Answer: $2916$


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

How many pairs of values \( p, q \in \mathbf{N} \), not exceeding 100, exist for which the equation

\[ x^{5} + p x + q = 0 \]

has solutions in rational numbers?

Answer: $133$


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

Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Provide the total number of toys the girls had initially.

Answer: $105$


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

Find the value of \(\tan \left(\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{2 \times 2^2} + \tan^{-1} \frac{1}{2 \times 3^2} + \cdots + \tan^{-1} \frac{1}{2 \times 2009^2}\right)\).

Answer: $\frac{2009}{2010}$


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

How many positive integers less than or equal to 5689 contain either the digit '6' or the digit '0'?

Answer: $2545$


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

Given that $\cos(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{10}$, with $x \in (\frac{\pi}{2}, \frac{3\pi}{4})$.
(1) Find the value of $\sin x$;
(2) Find the value of $\cos(2x - \frac{\pi}{3})$.

Answer: $-\frac{7 + 24\sqrt{3}}{50}$


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>.

There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. How many cards were flipped three times?

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>.

In the cartesian coordinate system $(xOy)$, curve $({C}_{1})$ is defined by the parametric equations $\begin{cases}x=t+1,\ y=1-2t\end{cases}$ and curve $({C}_{2})$ is defined by the parametric equations $\begin{cases}x=a\cos θ,\ y=3\sin θ\end{cases}$ where $a > 0$.
1. If curve $({C}_{1})$ and curve $({C}_{2})$ have a common point on the $x$-axis, find the value of $a$.
2. When $a=3$, curves $({C}_{1})$ and $({C}_{2})$ intersect at points $A$ and $B$. Find the distance between points $A$ and $B$.

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


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

A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing two balls of the same color, while losing is defined as drawing two balls of different colors.
$(1)$ If it is specified that after the first draw, the balls are put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $X$.
$(2)$ If it is specified that after the first draw, the balls are not put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $Y$.

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


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

At a variety show, there are seven acts: dance, comic dialogue, sketch, singing, magic, acrobatics, and opera. When arranging the program order, the conditions are that dance, comic dialogue, and sketch cannot be adjacent to each other. How many different arrangements of the program are possible?

Answer: $1440$


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

Out of 10 distinct positive integers, the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is the minimum sum of these 10 positive integers?

Answer: $65$


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

A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day?

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 $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$.

Answer: $1/3$


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

Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]

and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \).

Answer: $4\pi$


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>.

The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?

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>.

Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$

Answer: $\ln 3 - \ln 2 + 1$


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

In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$.

Answer: $99$


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

Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake?

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