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

There are 500 machines, each using 6 parts of the same type. These parts must be replaced by the end of the week if they fail. Out of all new parts, 10% fail by the end of the first week, 30% fail by the end of the second week, and the remaining 60% fail by the end of the third week. No part lasts beyond the third week. How many new parts need to be replaced by the end of the third week?

Answer: $3000$


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

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

Answer: $32$


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

The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases.

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>.

Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all?

Answer: $240$


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

Emma writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Emma's list?

Answer: $871$


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

Given that $\cos(\pi+\theta) = -\frac{1}{2}$, find the value of $\tan(\theta - 9\pi)$.

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>.

What is the least common multiple of 220 and 504?

Answer: $27720$


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

Cut a cube into two cuboids. If the ratio of their surface areas is 1:2, what is the ratio of their volumes?

Answer: $1:5$


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

Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle.

Answer: $22.5$


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>.

For a sample of size \( n = 41 \), a biased estimate \( D_{\text{в}} = 3 \) of the population variance is found. Find the unbiased estimate of the population variance.

Answer: $3.075$


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

Given \( m > n \geqslant 1 \), find the smallest value of \( m + n \) such that
\[ 1000 \mid 1978^{m} - 1978^{n} . \

Answer: $106$


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

A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much sausage is left if both bite off their pieces and run away?

Answer: $400$


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

Let $\alpha$ be an acute angle. If $\sin\left(\alpha - \frac{\pi}{4}\right) = \frac{1}{3}$, then $\cos2\alpha = \_\_\_\_\_\_$.

Answer: $-\frac{4\sqrt{2}}{9}$


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

$12$  students need to form five study groups. They will form three study groups with  $2$  students each and two study groups with  $3$  students each. In how many ways can these groups be formed?

Answer: $138600$


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

How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152?

Answer: $3360$


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

What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome and that palindrome is at least 200 more than 4321?

Answer: $233$


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

Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?

Answer: $280$


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

You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face.

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>.

Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.

Answer: $76$


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

A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + 1 < y$?

Answer: $\frac{1}{24}$


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>.

12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all teams and obtained three different results: 3,086; 2,018; and 1,238. They calculated the averages of each pair of these numbers and found that the averages were still different. Then, they calculated the average of the initial three numbers. If one of these seven numbers is correct, what is the number of teams \( n \)?

Answer: $47$


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

Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \).

Answer: $1/2$


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

Given that a hyperbola $mx^{2}+ny^{2}=1$ has a focus that is the same as the focus of the parabola $y=\frac{1}{8}{x^2}$ and an eccentricity of $2$, calculate the distance from the focus of the parabola to one of the asymptotes 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>.

Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$.

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>.

In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are there in the hospital?

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 sets \( A = \{ x \mid 5x - a \leq 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{ 2, 3, 4 \} \), determine the number of integer pairs \((a, b)\).

Answer: $30$


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

Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?

Answer: $871$


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

In tetrahedron \( ABCD \), the dihedral angle between face \( ABC \) and face \( BCD \) is \( 60^\circ \). Vertex \( A \)'s projection onto face \( BCD \) is point \( H \), which is the orthocenter of \( \triangle BCD \). Point \( G \) is the centroid of \( \triangle ABC \). Given that \( AH = 4 \) and \( AB = AC \), find \( GH \).

Answer: $\frac{4\sqrt{21}}{9}$


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>.

How many three-digit numbers satisfy the property that the middle digit is equal to the sum of the first and last digits minus one?

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>.

Train 109 T departs from Beijing at 19:33 and arrives in Shanghai the next day at 10:26; train 1461 departs from Beijing at 11:58 and arrives in Shanghai the next day at 8:01. How many minutes are the running times of these two trains different?

Answer: $310$


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

A sequence is formed by arranging the following arrays into a row: $$( \frac {1}{1}), ( \frac {2}{1}, \frac {1}{2}), ( \frac {3}{1}, \frac {2}{2}, \frac {1}{3}), ( \frac {4}{1}, \frac {3}{2}, \frac {2}{3}, \frac {1}{4}), ( \frac {5}{1}, \frac {4}{2}, \frac {3}{3}, \frac {2}{4}, \frac {1}{5}),…$$ If we remove the parentheses from the arrays, we get a sequence: $$\frac {1}{1}, \frac {2}{1}, \frac {1}{2}, \frac {3}{1}, \frac {2}{2}, \frac {1}{3}, \frac {4}{1}, \frac {3}{2}, \frac {2}{3}, \frac {1}{4}, \frac {5}{1}, \frac {4}{2}, \frac {3}{3}, \frac {2}{4}, \frac {1}{5},…$$ A student observed that $$\frac {63×64}{2}=2016,$$ based on this, the 2012th term of the sequence is ______.

Answer: $\frac {5}{59}$


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>.

Find the area of a triangle if it is known that its medians \(CM\) and \(BN\) are 6 and 4.5 respectively, and \(\angle BKM = 45^\circ\), where \(K\) is the point of intersection of the medians.

Answer: $9\sqrt{2}$


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 vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(2,-1)$, the projection of $\overrightarrow{a}+\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______.

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


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

A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?

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>.

Determine the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\]

Answer: $3124$


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

Given a line and a point at a distance of $1 \mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals?

Answer: $\frac{3 \sqrt{6}}{4}$


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

Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is  $\frac{C - T + S}{M}$  ?

Answer: $126$


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

The sequence $b_1, b_2, \ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\log_4 b_1+\log_4 b_2+\cdots+\log_4 b_{10} = 2010,$ find the number of possible ordered pairs $(b,s).$

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>.

Mrs. Novel has five different Batman comic books, six different X-Men comic books, and four different Calvin and Hobbes comic books. All of the Batman comic books are grouped together, all of the X-men comic books are grouped together, and all of the Calvin and Hobbes comic books are grouped together. In how many different orders can these 15 comic books be stacked in a pile with all the covers facing the same direction?

Answer: $12,441,600$


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

A circle is divided into two segments by a chord equal to the side of a regular inscribed triangle. Determine the ratio of the areas of these segments.

Answer: $\frac{4\pi - 3\sqrt{3}}{8\pi + 3\sqrt{3}}$


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>.

Calculate the sum of the geometric series $3 - \left(\frac{3}{4}\right) + \left(\frac{3}{4}\right)^2 - \left(\frac{3}{4}\right)^3 + \dots$. Express your answer as a common fraction.

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


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

Given that $\sin \alpha + \cos \alpha = -\frac{3}{\sqrt{5}}$, and $|\sin \alpha| > |\cos \alpha|$, find the value of $\tan \frac{\alpha}{2}$.

Answer: $-\frac{\sqrt{5} + 1}{2}$


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

A positive integer is said to be a "palindrome" if it reads the same from left to right as from right to left. For example, 2002 is a palindrome. Find the sum of all 4-digit palindromes.

Answer: $495000$


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

A function $f(x)$ defined on $R$ satisfies $f(x+1) = 2f(x)$. When $x \in (-1,0]$, $f(x) = x^{3}$. Find $f(\frac{21}{2})$.

Answer: $-256$


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

The cafeteria is tiled with the same floor tiles, and it takes 630 tiles to cover an area with 18 square decimeters of tiles. How many tiles will it take if we switch to square tiles with a side length of 6 decimeters?

Answer: $315$


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

Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$.

Answer: $3\pi$


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>.

\( x_{1} = 2001 \). When \( n > 1, x_{n} = \frac{n}{x_{n-1}} \). Given that \( x_{1} x_{2} x_{3} \ldots x_{10} = a \), find the value of \( a \).

Answer: $3840$


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

The edge of the cube \(E F G H E_1 F_1 G_1 H_1\) is equal to 2. Points \(A\) and \(B\) are taken on the edges \(E H\) and \(H H_1\) such that \(\frac{E A}{A H} = 2\) and \(\frac{B H}{B H_1} = \frac{1}{2}\). A plane is drawn through the points \(A\), \(B\), and \(G_1\). Find the distance from point \(E\) to this plane.

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


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

Calculate the limit of the function:

$$\lim _{x \rightarrow 0}\left(\frac{\arcsin ^{2} x}{\arcsin ^{2} 4 x}\right)^{2 x+1}$$

Answer: $1/16$


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

A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \( y = 2017 \) at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined based on this information?

Answer: $-1016064$


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

Ten distinct natural numbers are such that the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is their smallest possible sum?

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>.

If the line $y=kx+t$ is a tangent line to the curve $y=e^x+2$ and also a tangent line to the curve $y=e^{x+1}$, find the value of $t$.

Answer: $4-2\ln 2$


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>.

There are 8 keys numbered 1 to 8 and 8 boxes numbered 1 to 8. Each key can only open the box with the same number. All keys are placed in these boxes and locked up so that each box contains one key. How many different ways are there to place the keys in the boxes such that at least two boxes have to be opened to unlock all the boxes? (Assume the keys are not duplicated and a box can either be opened with its corresponding key or broken open with no other means.)

Answer: $35280$


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

Let \( s(n) \) denote the sum of the digits of the natural number \( n \). Solve the equation \( n + s(n) = 2018 \).

Answer: $2008$


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

Given $m\in R$, determine the condition for the lines $l_{1}$: $mx+2y-1=0$ and $l_{2}$: $3x+\left(m+1\right)y+1=0$ to be perpendicular.

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


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

For arbitrary real numbers \(a\) and \(b\) (\(a \neq 0\)), find the minimum value of the expression \(\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\).

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


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

How many negative numbers are there among the numbers of the form $\operatorname{tg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019?

Answer: $1009$


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

In an international sprinting event, there are 10 sprinters, including 4 Americans. The medals awarded are gold, silver, and bronze. Determine the number of ways the medals can be awarded if exactly two Americans must win medals.

Answer: $216$


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

A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Answer: $\frac{3136}{81} \pi$


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>.

Simplify the expression given by \(\frac{a^{-1} - b^{-1}}{a^{-3} + b^{-3}} : \frac{a^{2} b^{2}}{(a+b)^{2} - 3ab} \cdot \left(\frac{a^{2} - b^{2}}{ab}\right)^{-1}\) for \( a = 1 - \sqrt{2} \) and \( b = 1 + \sqrt{2} \).

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


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

A three-digit positive integer \( n \) has digits \( a, b, c \). (That is, \( a \) is the hundreds digit of \( n \), \( b \) is the tens digit of \( n \), and \( c \) is the ones (units) digit of \( n \).) Determine the largest possible value of \( n \) for which:
- \( a \) is divisible by 2,
- the two-digit integer \( ab \) (where \( a \) is the tens digit and \( b \) is the ones digit) is divisible by 3 but is not divisible by 6, and
- \( n \) is divisible by 5 but is not divisible by 7.

Answer: $870$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, and the sizes of angles $A$, $B$, $C$ form an arithmetic sequence. Let vector $\overrightarrow{m}=(\sin \frac {A}{2},\cos \frac {A}{2})$, $\overrightarrow{n}=(\cos \frac {A}{2},- \sqrt {3}\cos \frac {A}{2})$, and $f(A)= \overrightarrow{m} \cdot \overrightarrow{n}$,
$(1)$ If $f(A)=- \frac { \sqrt {3}}{2}$, determine the shape of $\triangle ABC$;
$(2)$ If $b= \sqrt {3}$ and $a= \sqrt {2}$, find the length of side $c$ and the area $S_{\triangle ABC}$.

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


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

Let  $a, b, c$  be integers not all the same with  $a, b, c\ge 4$  that satisfy   $$ 4abc = (a + 3) (b + 3) (c + 3). $$  Find the numerical value of  $a + b + c$ .

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 \( \triangle ABC \), if \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 7 \) and \( |\overrightarrow{AB} - \overrightarrow{AC}| = 6 \), find the maximum possible area of \( \triangle ABC \).

Answer: $12$


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

A positive integer $n$ not exceeding $120$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $2p$. The probability that a perfect square is chosen is?
A) $\frac{1}{180}$
B) $\frac{7}{180}$
C) $\frac{13}{180}$
D) $\frac{1}{120}$
E) $\frac{1}{60}$

Answer: $\frac{13}{180}$


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

In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base.

Answer: $\sqrt{14}$


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

Express the number one million using numbers that contain only the digit 9 and the algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Determine at least three different solutions.

Answer: $1000000$


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

Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[
\{-8, -6, -4, -1, 3, 5, 7, 10\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]

Answer: $18$


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

On a circle, 103 natural numbers are written. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be in the entire circle?

Answer: $42$


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

The line $y = -\frac{2}{3}x + 6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?

Answer: $8.25$


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>.

27 points P and a square with side length 1 lie on the same plane. Let the vertices of the square be \( A, B, C, \) and \( D \) (counterclockwise), and the distances from P to \( A, B, \) and \( C \) are \( u, v, \) and \( w \) respectively. If \( u^2 + v^2 = w^2 \), then the maximum distance from point P to point D is:

Answer: $2 + \sqrt{2}$


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

A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \mathrm{~cm}$ remained. The given drawing indicates the sequence of cuts.

What was the perimeter of the original sheet before the cut?

Answer: $258$


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

Let \( S = \{r_1, r_2, \cdots, r_n\} \subseteq \{1, 2, 3, \cdots, 50\} \), and let any two numbers in \( S \) not be divisible by 7 when summed. Find the maximum value of \( n \).

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>.

Find the smallest prime number $p$ such that $n^{2}+n+11$ is divisible by $p$ for some integer $n$.

Answer: $11$


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

Given \( n = 1990 \), find the value of

\[
\frac{1}{2^{n}}\left(1 - 3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990}\right).
\]

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


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

Find the number of ordered pairs of integers  $(a, b)$  that satisfy the inequality

\[
1 < a < b+2 < 10.
\]

*Proposed by Lewis Chen*

Answer: $28$


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

On his birthday, the last guest to arrive was Yana, who gave Andrey a ball, and the second last was Eduard, who gave him a calculator. While using the calculator, Andrey noticed that the product of the total number of gifts he received and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his age and the number of gifts he had before Yana arrived. How many gifts does Andrey have?

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>.

What is the sum of all three-digit numbers \( n \) for which \(\frac{3n+2}{5n+1}\) is not in its simplest form?

Answer: $70950$


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

A residential building has a construction investment of 250 yuan per square meter. Considering a service life of 50 years and an annual interest rate of 5%, calculate the monthly rent per square meter in yuan required to recoup the total investment.

Answer: $1.14$


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

Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.

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>.

Find the number of 3-digit positive integers whose digits multiply to 30.

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 a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \).

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


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

In right triangle $GHI$, we have $\angle G = 30^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth.

Answer: $20.8$


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

Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other.

Answer: $\frac{1}{126}$


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

Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of
\[4p^3 + 6q^3 + 24r^3 + \frac{8}{3pqr}.\]

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>.

Determine the value of $x$ that satisfies $\sqrt[4]{x\sqrt{x^4}}=4$.

Answer: $2^{8/3}$


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 we want to write down all the integers from 1 to 10,000, how many times do we have to write a digit, for example, the digit 5?

Answer: $4000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $x$-intercepts on the graph of $y = \sin \frac{2}{x}$ (evaluated in terms of radians) in the interval $(0.0002, 0.002).$

Answer: $2865$


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

A number \( a \) is randomly chosen from \( 1, 2, 3, \cdots, 10 \), and a number \( b \) is randomly chosen from \( -1, -2, -3, \cdots, -10 \). What is the probability that \( a^{2} + b \) is divisible by 3?

Answer: $37/100$


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

If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$.

Answer: $1110$


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

Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect the parabola's directrix at point $C$. If $|AD|=m$ and $m\in[1,2]$, determine the maximum value of $|FD|\cdot|CD|$.

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>.

For a natural number $n$, if $n+6$ divides $n^3+1996$, then $n$ is called a lucky number of 1996. For example, since $4+6$ divides $4^3+1996$, 4 is a lucky number of 1996. Find the sum of all lucky numbers of 1996.

Answer: $3720$


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

Given the function $f(x)= \begin{cases} \log_{2}x,x > 0 \\ x^{2}+4x+1,x\leqslant 0\\ \end{cases}$, if the real number $a$ satisfies $f(f(a))=1$, calculate the sum of all possible values of the real number $a$.

Answer: $-\frac{15}{16} - \sqrt{5}$


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>.

Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{4x-3y}{x+4y} = 3$. Find the value of $\frac{x-4y}{4x+3y}$.

Answer: $\frac{11}{63}$


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>.

In \\( \triangle ABC \\), \\( a \\), \\( b \\), and \\( c \\) are the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) respectively. The vectors \\( \overrightarrow{m} = (a, b+c) \\) and \\( \overrightarrow{n} = (1, \cos C + \sqrt{3} \sin C) \\) are given, and \\( \overrightarrow{m} \parallel \overrightarrow{n} \\).

\\((1)\\) Find angle \\( A \\).

\\((2)\\) If \\( 3bc = 16 - a^2 \\), find the maximum area of \\( \triangle ABC \\).

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>.

There are 7 light bulbs arranged in a row. It is required to light up at least 3 of the bulbs, and adjacent bulbs cannot be lit at the same time. Determine the total number of different ways to light up the bulbs.

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>.

A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different.

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 the Cartesian coordinate system, points whose x and y coordinates are both integers are called lattice points. How many lattice points \((x, y)\) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)?

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>.

It is known that the 3 sides of a triangle are consecutive positive integers and the largest angle is twice the smallest angle. Find the perimeter of this triangle.

Answer: $15$


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

What is the least positive integer that is divisible by the primes 7, 11, and 13?

Answer: $1001$


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

Simplify the expression \(\frac{\operatorname{tg}\left(\frac{5}{4} \pi - 4 \alpha\right) \sin^{2}\left(\frac{5}{4} \pi + 4 \alpha\right)}{1 - 2 \cos^{2} 4 \alpha}\).

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


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

Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2 and does not use it. Additionally, he does not like having two identical digits adjacent to each other. Moreover, he wants the first digit to be the same as the last digit. How many possible combinations need to be checked to guarantee guessing Vasya's password correctly?

Answer: $576$


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

In the tetrahedron \( ABCD \), \( AC = 8 \), \( AB = CD = 7 \), \( BC = AD = 5 \), and \( BD = 6 \). Given a point \( P \) on \( AC \), find the minimum value of \( BP + PD \).

Answer: $2\sqrt{21}$


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

Given that $(1-3x)^6 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6$, find the total sum of elements in all subsets containing 2 elements of the set $\{a_1, a_2, a_3, a_4, a_5, a_6\}$.

Answer: $315$


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

Calculate $(-2)^{23} + 2^{(2^4+5^2-7^2)}$.

Answer: $-8388607.99609375$


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

The function $y=2\sin(x+\frac{\pi}{3})$ has an axis of symmetry at $x=\frac{\pi}{6}$.

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


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

Let \( n \) be the positive integer such that
\[ \frac{1}{9 \sqrt{11} + 11 \sqrt{9}} + \frac{1}{11 \sqrt{13} + 13 \sqrt{11}} + \frac{1}{13 \sqrt{15} + 15 \sqrt{13}} + \cdots + \frac{1}{n \sqrt{n+2} + (n+2) \sqrt{n}} = \frac{1}{9} . \]

Find the value of \( n \).

Answer: $79$


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

Compute the sum \( S = \sum_{i=0}^{101} \frac{x_{i}^{3}}{1 - 3x_{i} + 3x_{i}^{2}} \) for \( x_{i} = \frac{i}{101} \).

Answer: $51$


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

PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?

Answer: $630$


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

Let  $1 = x_{1} < x_{2} < \dots < x_{k} = n$  denote the sequence of all divisors  $x_{1}, x_{2} \dots x_{k}$  of  $n$  in increasing order. Find the smallest possible value of  $n$  such that  $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$  *Proposed by Justin Lee*

Answer: $130$


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

Humpty Dumpty walks on a straight line, taking either 37 steps to the left or 47 steps to the right per minute.

What is the minimum time it will take for him to be one step to the right of the starting point?

Answer: $59$


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

On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always tell the truth, liars always lie. Each yes-man answered according to the majority of responses given before their turn, or gave either response if there was a tie. It turned out that exactly 1009 answered "Yes." What is the maximum number of yes-men that could be among the island inhabitants?

Answer: $1009$


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

Given four circles with radii $2, 4, 6,$ and $8$ tangent to two perpendicular lines $\ell_1$ and $\ell_2$ intersecting at point $A$, and region $S$ consisting of all the points that lie inside exactly one of these four circles, find the maximum possible area of region $S$.

Answer: $120\pi$


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>.

A line in the plane is called strange if it passes through \((a, 0)\) and \((0, 10-a)\) for some \(a\) in the interval \([0,10]\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points?

Answer: $50/3$


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

Let's call a word any finite sequence of letters of the Russian alphabet. How many different four-letter words can be made from the letters of the word КАША? And from the letters of the word ХЛЕБ? Indicate the sum of the found numbers in the answer.

Answer: $36$


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

Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. If all three coins show heads, roll an additional fourth die. Determine the probability that the sum of the die rolls is odd.

Answer: $\frac{7}{16}$


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>.

Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice.

Answer: $43.86\%$


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>.

$ABCD$ is a parallelogram with $\angle D$ obtuse. $M$ and $N$ are the feet of the perpendiculars from $D$ to $AB$ and $BC$ respectively. If $DB = DC = 50$ and $DA = 60$, find $DM + DN$.

Answer: $88$


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

Five points, no three of which are collinear, are given. Calculate the least possible value of the number of convex polygons whose some corners are formed by these five points.

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>.

Calculate the definite integral:

$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$

Answer: $6 \ln 5$


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

Calculate the number of zeros at the end of 2015!.

Answer: $502$


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

A projectile is launched, and its height (in meters) over time is represented by the equation $-20t^2 + 50t + 10$, where $t$ is the time in seconds after launch. Determine the maximum height reached by the projectile.

Answer: $41.25$


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

In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \rho = -4\sqrt{3}\sin \theta$ intersects with circle $C$ at points $A$ and $B$.
(i) Find the polar equation of line $AB$.
(ii) If a line $C_2$ passing through point $C(2,0)$ with the parametric equations $\begin{cases} x=2+ \frac{\sqrt{3}}{2}t, \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter) intersects line $AB$ at point $D$ and intersects the $y$-axis at point $E$, calculate the ratio $|CD| : |CE|$.

Answer: $1 : 2$


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

The entire contents of a jug can exactly fill 9 small glasses and 4 large glasses of juice. The entire contents of the jug could instead fill 6 small glasses and 6 large glasses. If the entire contents of the jug is used to fill only large glasses, the maximum number of large glasses that can be filled is

Answer: $10$


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

Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n+1} = -\frac{1}{2} a_{n} + \frac{1}{3^{n}}\quad (n \in \mathbf{Z}_{+}),
$$
find all values of $a_{1}$ such that the sequence $\left\{a_{n}\right\}$ is monotonic, i.e., $\left\{a_{n}\right\}$ is either increasing or decreasing.

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


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

Using Green's theorem, evaluate the line integral \( I = \oint_{L} 2(x^{2}+y^{2}) \, dx + (x+y)^{2} \, dy \) along the contour \( L \) of triangle \( ABC \) with vertices \( A(1,1) \), \( B(2,2) \), \( C(1,3) \).

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


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

The distance between Ivan's house and his grandmother's house is 12 km. Exactly at 12:00, Ivan left his house and walked along the straight road to his grandmother's house at a speed of 1 m/s. At 12:30, Ivan's parents called his grandmother, informed her that Ivan was coming to visit, and she released her dog Tuzik to meet him. Tuzik runs at a speed of 9 m/s. Determine the moment when Tuzik will reach Ivan.

Answer: $12:47$


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

Given that a hyperbola shares common foci $F_1$ and $F_2$ with the ellipse $\dfrac {x^{2}}{9}+ \dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \dfrac {4}{5}$.
$(1)$ Find the standard equation of the hyperbola;
$(2)$ Let $P$ be a point of intersection between the hyperbola and the ellipse, calculate $\cos \angle F_{1}PF_{2}$.

Answer: $- \dfrac {1}{7}$


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 \( x_{i} \in \{-1,1\}, i=1,2,\cdots,2021 \), and \( x_{1}+x_{2}+\cdots+x_{k} \geq 0 \) for \( k=1,2,\cdots,2020 \), with \( x_{1}+x_{2}+\cdots+x_{2021}=-1 \). How many ordered arrays \( (x_{1}, x_{2}, \cdots, x_{2021}) \) are there?

Answer: $\frac{1}{1011} \binom{2020}{1010}$