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

Given the function $f(x) = 2\sin^2\left(\frac{\pi}{4} + x\right) - \sqrt{3}\cos{2x} - 1$, where $x \in \mathbb{R}$:
1. If the graph of function $h(x) = f(x + t)$ is symmetric about the point $\left(-\frac{\pi}{6}, 0\right)$, and $t \in \left(0, \frac{\pi}{2}\right)$, find the value of $t$.
2. In an acute triangle $ABC$, if angle $A$ satisfies $h(A) = 1$, find the range of $(\sqrt{3} - 1)\sin{B} + \sqrt{2}\sin{C}$.

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


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

Determine the smallest integer  $n > 1$  with the property that  $n^2(n - 1)$  is divisible by 2009.

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>.

In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B, answering correctly yields 90 points and answering incorrectly results in a loss of 90 points. If the total score of the four students is 0 points, how many different scoring situations are there?

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>.

The cells of a $50 \times 50$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if

(a) $n=2$

(b) $n=25$.

Answer: $1300$


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

How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits?

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>.

Let \[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\{1, 2, 3, 4, 5, 6\}$. Find $P(7)$.

Answer: $51840$


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

A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube?

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


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

Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola.

Answer: $\sqrt{5}$


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

For all positive integers  $m>10^{2022}$ , determine the maximum number of real solutions  $x>0$  of the equation  $mx=\lfloor x^{11/10}\rfloor$ .

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>.

Multiply the first eight positive composite integers, add the first prime number to this product, then divide by the product of the next eight positive composite integers after adding the second prime number to it. Express your answer as a common fraction.

Answer: $\frac{4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 12 \cdot 14 \cdot 15 + 2}{16 \cdot 18 \cdot 20 \cdot 21 \cdot 22 \cdot 24 \cdot 25 \cdot 26 + 3}$


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

Given \(\triangle DEF\), where \(DE=28\), \(EF=30\), and \(FD=16\), calculate the area of \(\triangle DEF\).

Answer: $221.25$


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>.

From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers?

Answer: $90$


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

The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression?

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>.

Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$.

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


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

How many values of $x$, $-10 < x < 50$, satisfy $\cos^2 x + 3\sin^2 x = 1.5?$ (Note: $x$ is measured in radians.)

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>.

Remove five out of twelve digits so that the remaining numbers sum up to 1111.

$$
\begin{array}{r}
111 \\
333 \\
+\quad 777 \\
999 \\
\hline 1111
\end{array}
$$

Answer: $1111$


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

Add together the numbers $7.56$ and $4.29$. Write the result as a decimal.

Answer: $11.85$


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

Among the following propositions, the correct ones are __________.

(1) The regression line $\hat{y}=\hat{b}x+\hat{a}$ always passes through the center of the sample points $(\bar{x}, \bar{y})$, and at least through one sample point;

(2) After adding the same constant to each data point in a set of data, the variance remains unchanged;

(3) The correlation index $R^{2}$ is used to describe the regression effect; it represents the contribution rate of the forecast variable to the change in the explanatory variable, the closer to $1$, the better the model fits;

(4) If the observed value $K$ of the random variable $K^{2}$ for categorical variables $X$ and $Y$ is larger, then the credibility of "$X$ is related to $Y$" is smaller;

(5) For the independent variable $x$ and the dependent variable $y$, when the value of $x$ is certain, the value of $y$ has certain randomness, the non-deterministic relationship between $x$ and $y$ is called a function relationship;

(6) In the residual plot, if the residual points are relatively evenly distributed in a horizontal band area, it indicates that the chosen model is relatively appropriate;

(7) Among two models, the one with the smaller sum of squared residuals has a better fitting effect.

Answer: $(2)(6)(7)$


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

Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$.

Answer: $\frac{20}{41}$


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

What is the least integer whose square is 75 more than its double?

Answer: $-8$


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

Given that $| \vec{e} | = 1$ and it satisfies $| \vec{a} + \vec{e} | = | \vec{a} - 2\vec{e} |$, find the projection of vector $\vec{a}$ in the direction of $\vec{e}$.

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


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

What is $\frac{1}{(-5^2)^3} \cdot (-5)^8 \cdot \sqrt{5}$?

Answer: $5^{5/2}$


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

Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is $\frac{3}{5}$, and when B serves, the probability of A winning a point is $\frac{1}{2}$, and the outcomes of each point are independent. According to the draw result, A serves first.
$(1)$ Let $X$ represent the score of A after three points. Find the distribution table and mean of $X$;
$(2)$ Find the probability that A has more points than B after six points.

Answer: $\frac{441}{1000}$


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

Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of the numbers resulting from the rotation of spinners A, B, and C is an odd number.

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


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

Find the number of integers  $n$  with  $1\le n\le 100$  for which  $n-\phi(n)$  is prime.  Here  $\phi(n)$  denotes the number of positive integers less than  $n$  which are relatively prime to  $n$ .

*Proposed by Mehtaab Sawhney*

Answer: $13$


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

Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose.

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>.

Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other?

Answer: $50$


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

8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obtained by rotation are identical)?

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>.

Rectangle PQRS and right triangle SRT share side SR and have the same area. Rectangle PQRS has dimensions PQ = 4 and PS = 8. Find the length of side RT.

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>.

10 times 0.1 equals to ____, 10 times 0.01 equals to ____, 10 times 0.001 equals to ____.

Answer: $0.01$


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

In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\{a_n\}$.

Answer: $100$


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

Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$.

Answer: $175$


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

From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____.

Answer: $671$


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

A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points?

Answer: $1979$


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

Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$.

Answer: $625$


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

Given the function $f\left(x\right)=x^{2}-2bx+3$, where $b\in R$.
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt 4-b^{2}$.
$(2)$ When $x\in \left[-1,2\right]$, the function $y=f\left(x\right)$ has a minimum value of $1$. Find the maximum value of the function $y=f\left(x\right)$ when $x\in \left[-1,2\right]$.

Answer: $4 + 2\sqrt{2}$


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>.

Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible.

Answer: $5040$


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

Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$.

Answer: $-\frac{5}{4}$


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>.

Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:

- The sum of the fractions is equal to  $2$ .
- The sum of the numerators of the fractions is equal to  $1000$ .

In how many ways can Pedro do this?

Answer: $200$


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

Let $U$ be a positive integer whose only digits are 0s and 1s. If $Y = U \div 18$ and $Y$ is an integer, what is the smallest possible value of $Y$?

Answer: $61728395$


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

A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest piece to the volume of the largest piece.

Answer: $\frac{1}{27}$


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

Given a sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, where $a\_1=1$ and $3S\_n = a_{n+1} - 1$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. Consider an arithmetic sequence $\{b\_n\}$ with its first $n$ terms sum $T\_n$, where $a\_2 = b\_2$ and $T\_4 = 1 + S\_3$. Find the value of $\frac{1}{b\_1 \cdot b\_2} + \frac{1}{b\_2 \cdot b\_3} + \dots + \frac{1}{b_{10}b_{11}}$.

Answer: $\frac{10}{31}$


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

The maximum and minimum values of the function $y=2x^{3}-3x^{2}-12x+5$ on the interval $[0,3]$ need to be determined.

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>.

In triangle \(ABC\), point \(O\) is the center of the circumcircle, and point \(L\) is the midpoint of side \(AB\). The circumcircle of triangle \(ALO\) intersects the line \(AC\) at point \(K\). Find the area of triangle \(ABC\) if \(\angle LOA = 45^\circ\), \(LK = 8\), and \(AK = 7\).

Answer: $112$


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

The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Express your answer as a common fraction.

Answer: $\frac{48}{7}$


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>.

On a table, there are 20 cards numbered from 1 to 20. Each time, Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card. What is the maximum number of cards Xiao Ming can pick?

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>.

Let  $(a_1,a_2,\ldots, a_{13})$  be a permutation of  $(1, 2, \ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer  $i$  from  $1$  to  $12$ , inclusive, and swapping the positions of  $a_i$  and  $a_{i+1}$ . Define the *weight* of a permutation to be the minimum number of moves Ayvak needs to turn it into  $(1, 2, \ldots, 13)$ .

  The arithmetic mean of the weights of all permutations  $(a_1, \ldots, a_{13})$  of  $(1, 2, \ldots, 13)$  for which  $a_5 = 9$  is  $\frac{m}{n}$ , for coprime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by Alex Gu*

Answer: $13703$


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

Points $A=(8,15)$ and $B=(14,9)$ lie on circle $\omega$ in the plane. Suppose the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the x-axis. Find the area of $\omega$.

Answer: $306\pi$


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 box contains 4 labels marked with the numbers $1$, $2$, $3$, and $4$. Two labels are randomly selected according to the following conditions. Find the probability that the numbers on the two labels are consecutive integers:
1. The selection is made without replacement;
2. The selection is made with replacement.

Answer: $\frac{3}{16}$


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>.

Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]

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>.

Calculate $1010101_2 + 110011_2$ and express your answer in base $10$.

Answer: $136$


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

1. Given that $α$ and $β$ are acute angles, and $\cos α= \frac{4}{5}$, $\cos (α+β)=- \frac{16}{65}$, find the value of $\cos β$.
2. Given that $0 < β < \frac{π}{4} < α < \frac{3}{4}π$, $\cos ( \frac{π}{4}-α)= \frac{3}{5}$, $\sin ( \frac{3π}{4}+β)= \frac{5}{13}$, find the value of $\sin (α+β)$.

Answer: $\frac{56}{65}$


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>.

Calculate the infinite sum:
\[
\sum_{n=1}^\infty \frac{n^3 - n}{(n+3)!}
\]

Answer: $\frac{1}{6}$


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>.

For arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If $\frac{{S}_{n}}{{T}_{n}}=\frac{2n+1}{3n+2}$, then $\frac{{a}_{2}+{a}_{5}+{a}_{17}+{a}_{20}}{{b}_{8}+{b}_{10}+{b}_{12}+{b}_{14}}=\_\_\_\_\_\_$.

Answer: $\frac{43}{65}$


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 $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$?

Answer: $703$


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

A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$?

Answer: $225$


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

In the decimal number system, the operation rule is "ten carries one". Analogous to this operation rule, perform the four arithmetic operations in the octal system and calculate $53_{(8)} \times 26_{(8)} =$ _______ (the operation result must be represented in octal numbers).

Answer: $1662_{(8)}$


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

Given $ \dfrac {3\pi}{4} < \alpha < \pi$, $\tan \alpha+ \dfrac {1}{\tan \alpha}=- \dfrac {10}{3}$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $ \dfrac {5\sin ^{2} \dfrac {\alpha}{2}+8\sin \dfrac {\alpha}{2}\cos \dfrac {\alpha}{2}+11\cos ^{2} \dfrac {\alpha}{2}-8}{ \sqrt {2}\sin (\alpha- \dfrac {\pi}{4})}$.

Answer: $- \dfrac {5}{4}$


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

The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotional advertisements can be played consecutively. Determine the total number of different playback methods.

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>.

In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\overline{GH}$. Let $P$ be the intersection of $\overline{EC}$ and $\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$?

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


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

Let \( f(m, n) = 3m + n + (m + n)^2 \). Find \( \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} 2^{-f(m, n)} \).

Answer: $4/3$


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 railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two trains be 210 kilometers apart?

Answer: $7.5$


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

Given that $\sin\alpha + \sin\beta = \frac{1}{3}$, find the maximum and minimum values of $y = \sin\beta - \cos^2\alpha$.

Answer: $-\frac{11}{12}$


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

In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 21 respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction.

Answer: $\frac{100}{341}$


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

Three spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560.$ Find $AC^2.$

Answer: $756$


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

A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.

Answer: $18$


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

In Mr. Smith's science class, there are 3 boys for every 4 girls. If there are 42 students in total in his class, what percent of them are boys?

Answer: $42.857\%$


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

Given a sequence $\{x_n\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \in \mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \leqslant 1$ and $a \neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\{x_n\}$ is ______.

Answer: $1345$


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

Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.

a) How much time will Petya need to definitely get inside?

b) On average, how much time will Petya need?

c) What is the probability that Petya will get inside in less than a minute?

Answer: $\frac{29}{120}$


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

In equilateral triangle $ABC$ a point $P$ lies such that $PA = 7$, $PB = 7$, and $PC = 14$. Determine the area of the triangle $ABC$.
**A)** $49\sqrt{3}$ \\
**B)** $98\sqrt{3}$ \\
**C)** $42\sqrt{3}$ \\
**D)** $21\sqrt{3}$ \\
**E)** $98$

Answer: $49\sqrt{3}$


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

If the total sum of squared deviations of a set of data is 100, and the correlation coefficient is 0.818, then the sum of squared residuals is.

Answer: $33.0876$


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

Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pieces of jewelry are laid out in a row from left to right on the showcase according to the following rules:

- There must be a piece of jewelry with a sun pendant on an iron chain.
- Next to the jewelry with the sun pendant there must be gold and silver jewelry.
- The three pieces of jewelry in the row must have different stones, pendants, and chains.

How many ways are there to make Polina happy?

Answer: $24$


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

Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \).

Answer: $2028$


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

Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit between adjacent points.

Answer: $24$


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

The number $0.324375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?

Answer: $2119$


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

A two-digit number has its unit digit greater than the tens digit by 2. If this two-digit number is less than 30, find the number.

Answer: $24$


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

A *substring* of a number  $n$  is a number formed by removing some digits from the beginning and end of  $n$  (possibly a different number of digits is removed from each side). Find the sum of all prime numbers  $p$  that have the property that any substring of  $p$  is also prime.

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>.

Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)^{2}}
\end{aligned}
$$

where \(x_{i}, y_{i}, z_{i}, i=1,2, \cdots, n\) are non-negative real numbers.

Answer: $1975$


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

Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$?

Answer: $810$


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

If $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \cdot a_{2013} < 0$, then the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$ is.

Answer: $2012$


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

Find $\tan A$ in the right triangle shown below.

[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,15);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$41$", (B+C)/2,NE);
label("$40$", B/2,S);
[/asy]

Answer: $\frac{9}{40}$


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

The function $f(n)$ defined on the set of natural numbers $\mathbf{N}$ is given by:
$$
f(n)=\left\{\begin{array}{ll}
n-3 & (n \geqslant 1000); \\
f[f(n+7)] & (n < 1000),
\end{array}\right.
$$

What is the value of $f(90)$?

Answer: $999$


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

Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\left(A|B\right)$.

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>.

If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn?

Answer: $14400$


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

Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest triangle is 16 centimeters. What is the length of the leg of the last 45-45-90 triangle? Express your answer as a common fraction.

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


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

Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
A) $\frac{2}{3}$
B) $\frac{1}{2}$
C) $\frac{1}{3}$
D) $\sqrt{3}$
E) $\sqrt{2}$

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


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

In  $\triangle ABC$ , three lines are drawn parallel to side  $BC$  dividing the altitude of the triangle into four equal parts. If the area of the second largest part is  $35$ , what is the area of the whole  $\triangle ABC$ ?

[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);[/asy]

Answer: $560/3$


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 all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$.

Answer: $-3$


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

What is the area of a hexagon where the sides alternate between lengths of 2 and 4 units, and the triangles cut from each corner have base 2 units and altitude 3 units?

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>.

Given the number 2550, calculate the sum of its prime factors.

Answer: $27$


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

Given Daphne's four friends visit her every 4, 6, 8, and 10 days respectively, all four friends visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends will visit her.

Answer: $129$


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

40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed?

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>.

In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find $PQ$.

Answer: $67$


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

The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds.

Answer: $12.8$


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>.

From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______.

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


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

Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed.

Answer: $2.24$


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

The minimum and maximum values of the function $f(x)=\cos 2x+\sin x+1$ are $\_\_\_\_\_\_$ and $\_\_\_\_\_\_$, respectively.

Answer: $\frac{17}{8}$


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>.

Ellen wants to color some of the cells of a $4 \times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color?

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>.

Let  $ABC$  be a triangle where $\angle$ **B=55** and  $\angle$  **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**.  $\angle$ **FDO** = ?

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>.

The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances.

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


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 high school with 2000 students held a "May Fourth" running and mountain climbing competition in response to the call for "Sunshine Sports". Each student participated in only one of the competitions. The number of students from the first, second, and third grades participating in the running competition were \(a\), \(b\), and \(c\) respectively, with \(a:b:c=2:3:5\). The number of students participating in mountain climbing accounted for \(\frac{2}{5}\) of the total number of students. To understand the students' satisfaction with this event, a sample of 200 students was surveyed. The number of second-grade students participating in the running competition that should be sampled is \_\_\_\_\_.

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>.

For the set $M$, define the function $f_M(x) = \begin{cases} -1, & x \in M \\ 1, & x \notin M \end{cases}$. For two sets $M$ and $N$, define the set $M \triangle N = \{x | f_M(x) \cdot f_N(x) = -1\}$. Given $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 4, 8, 16\}$.

(1) List the elements of the set $A \triangle B = \_\_\_\_\_$;
(2) Let $\text{Card}(M)$ represent the number of elements in a finite set $M$. When $\text{Card}(X \triangle A) + \text{Card}(X \triangle B)$ takes the minimum value, the number of possible sets $X$ is $\_\_\_\_\_$.

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>.

Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F.

Answer: $36\pi$


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>.

Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$.

Answer: $171$


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

In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.

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>.

A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$.

Answer: $44$


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

Given the function \[f(x) = \left\{ \begin{aligned} x+3 & \quad \text{ if } x < 2 \\ x^2 & \quad \text{ if } x \ge 2 \end{aligned} \right.\] determine the value of \(f^{-1}(-5) + f^{-1}(-4) + \dots + f^{-1}(2) + f^{-1}(3). \)

Answer: $-35 + \sqrt{2} + \sqrt{3}$


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

What is the smallest positive integer with exactly 12 positive integer divisors?

Answer: $96$


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

Given a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimeter of one of the small triangles to the perimeter of the large rectangle.

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


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

A bus ticket is numbered with six digits: from 000000 to 999999. You buy one ticket. What is the probability that you will get a ticket whose digits are in ascending (or descending) order?

Answer: $0.00042$


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

There is a unique two-digit positive integer $u$ for which the last two digits of $15\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$.

Answer: $43$


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

Find the greatest real  $k$  such that, for every tetrahedron  $ABCD$  of volume  $V$ , the product of areas of faces  $ABC,ABD$  and  $ACD$  is at least  $kV^2$ .

Answer: $9/2$


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

How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two properties?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

Answer: $143$


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

Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \).

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 school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books?

Answer: $2142$


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

Given points \(A=(8,15)\) and \(B=(16,9)\) are on a circle \(\omega\), and the tangent lines to \(\omega\) at \(A\) and \(B\) meet at a point \(P\) on the x-axis, calculate the area of the circle \(\omega\).

Answer: $250\pi$


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>.

How many ways can the integers from  $-7$  to  $7$  inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?

Answer: $128$


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

In triangle $ABC$, let vector $\vec{a} = (1, \cos B)$ and vector $\vec{b} = (\sin B, 1)$, and suppose $\vec{a}$ is perpendicular to $\vec{b}$. Find the magnitude of angle $B$.

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


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

The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?

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>.

A man chooses two positive integers \( m \) and \( n \). He defines a positive integer \( k \) to be good if a triangle with side lengths \( \log m \), \( \log n \), and \( \log k \) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \( mn \).

Answer: $134$


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

In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$.

Answer: $7\sqrt{3}$


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 $(100110_2 + 1001_2) \times 110_2 \div 11_2$. Express your answer in base 2.

Answer: $1011110_2$


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 dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(x)=\dfrac{1260}{x+1}$; If $x$ is greater than or equal to $180$, then the sales volume is zero; When $20\leqslant x\leqslant 180$, $q(x)=a-b\sqrt{x}$ ($a,b$ are real constants).

$(1)$ Find the expression for the function $q(x)$;

$(2)$ At what value of $x$ does the total profit (unit: Yuan) reach its maximum value, and what is this maximum value?

Answer: $240000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 five-digit numbers formed from the digits 1, 2, 3, 4, 5 without repeating any digits, and where the digits 1 and 2 are not adjacent to the digit 5.

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>.

In the "five rows by six columns" examination room, if two students A and B from the same school are to be seated with no two adjacent seats in any direction (front, back, left, right), calculate the number of different seating arrangements for students A and B.

Answer: $772$


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

Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$.

Answer: $\frac{2865 - 120\sqrt{165}}{49}$