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

Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$.

Answer: $\sqrt{2}$


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

In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C_{1}$ are given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). Point $M$ moves on curve $C_{1}$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of point $P$ forms the curve $C_{2}$.

(I) Find the equation of $C_{2}$;

(II) In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the ray $\theta = \dfrac{\pi}{3}$ intersects $C_{1}$ at point $A$ and $C_{2}$ at point $B$. Find the length of the segment $|AB|$.

Answer: $2 \sqrt{3}$


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

Arrange 8 people, including A and B, to work for 4 days. If 2 people are arranged each day, the probability that A and B are arranged on the same day is ______. (Express the result as a fraction)

Answer: $\frac{1}{7}$


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

Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them.

Answer: $19.17$


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

In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively. If $a\cos \left(B-C\right)+a\cos A=2\sqrt{3}c\sin B\cos A$ and $b^{2}+c^{2}-a^{2}=2$, then the area of $\triangle ABC$ is ____.

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


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

Teams A and B are playing a volleyball match. Currently, Team A needs to win one more game to become the champion, while Team B needs to win two more games to become the champion. If the probability of each team winning each game is 0.5, then the probability of Team A becoming the champion is $\boxed{\text{answer}}$.

Answer: $0.75$


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

Given that the function f(x) (x ∈ R) satisfies f(x + π) = f(x) + sin(x), and f(x) = 0 when 0 ≤ x ≤ π. Find f(23π/6).

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>.

Compute the number of geometric sequences of length  $3$  where each number is a positive integer no larger than  $10$ .

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 a new arcade game, the "monster" is the shaded region of a semicircle with radius 2 cm as shown in the diagram. The mouth, which is an unshaded piece within the semicircle, subtends a central angle of 90°. Compute the perimeter of the shaded region.
A) $\pi + 3$ cm
B) $2\pi + 2$ cm
C) $\pi + 4$ cm
D) $2\pi + 4$ cm
E) $\frac{5}{2}\pi + 2$ cm

Answer: $\pi + 4$


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>.

Given $a \gt 0$, $b \gt 0$, and $a+2b=1$, find the minimum value of $\frac{{b}^{2}+a+1}{ab}$.

Answer: $2\sqrt{10} + 6$


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

In each part of this problem, cups are arranged in a circle and numbered \(1, 2, 3, \ldots\). A ball is placed in cup 1. Then, moving clockwise around the circle, a ball is placed in every \(n\)th cup. The process ends when cup 1 contains two balls.

(a) There are 12 cups in the circle and a ball is placed in every 5th cup, beginning and ending with cup 1. List, in order, the cups in which the balls are placed.

(b) There are 9 cups in the circle and a ball is placed in every 6th cup, beginning and ending with cup 1. List the numbers of the cups that do not receive a ball.

(c) There are 120 cups in the circle and a ball is placed in every 3rd cup, beginning and ending with cup 1. How many cups do not contain at least one ball when the process is complete? Explain how you obtained your answer.

(d) There are 1000 cups in the circle and a ball is placed in every 7th cup, beginning and ending with cup 1. Determine the number of the cup into which the 338th ball is placed.

Answer: $360$


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

Given that the coefficient of the second term in the expansion of $((x+2y)^{n})$ is $8$, find the sum of the coefficients of all terms in the expansion of $((1+x)+(1+x)^{2}+…+(1+x)^{n})$.

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 integer sequence \(a_1, a_2, a_3, \dots\) is defined as follows: \(a_1 = 1\). For \(n \geq 1\), \(a_{n+1}\) is the smallest integer greater than \(a_n\) such that for all \(i, j, k \in \{1, 2, \dots, n+1\}\), the condition \(a_i + a_j \neq 3a_k\) is satisfied. Find the value of \(a_{22006}\).

Answer: $66016$


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

Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle.

Answer: $300$


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

Two arithmetic sequences $C$ and $D$ both begin with 50. Sequence $C$ has a common difference of 12 and is increasing, while sequence $D$ has a common difference of 8 and is decreasing. What is the absolute value of the difference between the 31st term of sequence $C$ and the 31st term of sequence $D$?

Answer: $600$


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

Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pattern of increasing distance continues, what is the number of unit squares in the $50^{th}$ ring, if each ring’s distance from the center increases linearly (i.e., $n^{th}$ ring is at $n$ units from the center), and every second ring starting from the first is colored red?

Answer: $400$


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

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

[asy]
draw(circle((0,0),13));
draw(circle((5,-6.2),5));
draw(circle((-5,-6.2),5));
label(" $B$ ", (9.5,-9.5), S);
label(" $A$ ", (-9.5,-9.5), S);
[/asy]

Answer: $69$


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

There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, calculate the number of different arrangements.

Answer: $72$


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

Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ satisfying $\overrightarrow{a} + \overrightarrow{b} + 2\overrightarrow{c} = \overrightarrow{0}$, and the magnitudes $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=3$, $|\overrightarrow{c}|=2$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b} + 2\overrightarrow{a} \cdot \overrightarrow{c} + 2\overrightarrow{b} \cdot \overrightarrow{c}$.

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 the expansion of $(1+x)^3 + (1+x)^4 + \ldots + (1+x)^{12}$, the coefficient of the term containing $x^2$ is _____. (Provide your answer as a number)

Answer: $285$


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

Let $x, y, z$ be positive real numbers such that $x + 2y + 3z = 1$. Find the maximum value of $x^2 y^2 z$.

Answer: $\frac{4}{16807}$


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

Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circle $P$ and circle $R$.

Answer: $2\pi$


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

Which of the following numbers is equal to 33 million?

Answer: $33000000$


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

Ria writes down the numbers  $1,2,\cdots, 101$  in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen?

Answer: $68$


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

Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, where $S_n=n^2+n$, and the general term of the sequence $\{b_n\}$ is given by $b_n=x^{n-1}$.

(1) Find the general term formula for the sequence $\{a_n\}$;

(2) Let $c_n=a_nb_n$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$.

   (i) Find $T_n$;

   (ii) If $x=2$, find the minimum value of the sequence $\left\{\dfrac{nT_{n+1}-2n}{T_{n+2}-2}\right\}$.

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


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

Start with a three-digit positive integer  $A$ . Obtain  $B$  by interchanging the two leftmost digits of  $A$ . Obtain  $C$  by doubling  $B$ . Obtain  $D$  by subtracting  $500$  from  $C$ . Given that  $A + B + C + D = 2014$ , find  $A$ .

Answer: $344$


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

Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$. Let $P$ be a point on the ellipse, and $\triangle F\_1 P F\_2$ have centroid $G$ and incenter $I$. If $\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentricity $e$ of the ellipse.

A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{1}{4}$
D) $\frac{\sqrt{5}-1}{2}$

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


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

In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$.

Answer: $1675$


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

Determine the number of different arrangements for assigning 6 repatriated international students to work in Jinan and Qingdao, given that at least 2 students must be assigned to Jinan and at least 3 students must be assigned to Qingdao.

Answer: $35$


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

In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?

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>.

Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$?

Answer: $16$


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

In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge between the losing teams on Saturday to decide the starting server for Sunday's consolation match. One possible ranking of the team from first to fourth at the end of the tournament is EGHF. Determine the total number of possible four-team ranking sequences at the end of the tournament.

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>.

Using the digits 1, 2, 3, 4, how many distinct four-digit even numbers can be formed?

Answer: $12$


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

Given that $\sin \alpha = 3 \sin \left(\alpha + \frac{\pi}{6}\right)$, find the value of $\tan \left(\alpha + \frac{\pi}{12}\right)$.

Answer: $2 \sqrt{3} - 4$


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

In a household, when someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability of the phone call being answered within the first four rings.

Answer: $0.9$


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

Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time.

Answer: $1.25$


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>.

The cost price of a certain product is 1360 yuan. When it is sold at 80% of the marked price, the profit margin is 15%. What should be the marked price of the product in yuan?

Answer: $1955$


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

Rectangle  $ABCD$  has  $AB = 8$  and  $BC = 13$ . Points  $P_1$  and  $P_2$  lie on  $AB$  and  $CD$  with  $P_1P_2 \parallel BC$ . Points  $Q_1$  and  $Q_2$  lie on  $BC$  and  $DA$  with  $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral  $P_1Q_1P_2Q_2$ .

Answer: $52$


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

To meet market demand, a supermarket purchased a brand of zongzi before the arrival of the Dragon Boat Festival on May 5th. The cost of each box is $40. The supermarket stipulates that the selling price of each box must not be less than $45. Based on past sales experience, it was found that when the selling price is set at $45 per box, 700 boxes can be sold per day. For every $1 increase in the selling price per box, 20 fewer boxes are sold per day.

$(1)$ Find the functional relationship between the daily sales volume $y$ (boxes) and the selling price per box $x$ (in dollars).

$(2)$ At what price per box should the selling price be set to maximize the daily profit $P$ (in dollars)? What is the maximum profit?

$(3)$ To stabilize prices, the relevant management department has set a maximum selling price of $58 per box for this type of zongzi. If the supermarket wants to make a profit of at least $6000 per day, how many boxes of zongzi must be sold per day at least?

Answer: $440$


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

Given that the slant height of a certain cone is $4$ and the height is $2\sqrt{3}$, calculate the total surface area of the cone.

Answer: $12\pi$


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

Given the function $y=\sin (2x+\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$.

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


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

The graph of the function $y=\sin (2x+\varphi)$ is translated to the left by $\dfrac {\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\varphi$.

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


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

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle?

Answer: $\frac{119.84}{\pi^2}$


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

Right $\triangle PQR$ has sides $PQ = 5$, $QR = 12$, and $PR = 13$. Rectangle $ABCD$ is inscribed in $\triangle PQR$ such that $A$ and $B$ are on $\overline{PR}$, $D$ on $\overline{PQ}$, and $C$ on $\overline{QR}$. If the height of the rectangle (parallel to side $\overline{PQ}$) is half its length (parallel to side $\overline{PR}$), find the length of the rectangle.

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>.

A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$?

Answer: $\frac{27}{220}$


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

Calculate: ${2}^{0}-|-3|+(-\frac{1}{2})=\_\_\_\_\_\_$.

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


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

The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased?

Answer: $1598$


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

Given a point $Q$ on a rectangular piece of paper $DEF$, where $D, E, F$ are folded onto $Q$. Let $Q$ be a fold point of $\triangle DEF$ if the creases, which number three unless $Q$ is one of the vertices, do not intersect within the triangle. Suppose $DE=24, DF=48,$ and $\angle E=90^\circ$. Determine the area of the set of all possible fold points $Q$ of $\triangle DEF$.

Answer: $147$


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

In a WeChat group, five members simultaneously grab for four red envelopes, each person can grab at most one, and all red envelopes are claimed. Among the four red envelopes, there are two containing 2 yuan and two containing 3 yuan. Determine the number of scenarios in which both members A and B have grabbed a red envelope.

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>.

Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear.

Answer: $-2$


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

Twelve chairs are evenly spaced around a round table and numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse or next to someone of the same profession. Determine the number of seating arrangements possible.

Answer: $2880$


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

Points are drawn on the sides of a square, dividing each side into \( n \) equal parts. The points are joined to form several small squares and some triangles. How many small squares are formed when \( n=7 \)?

Answer: $84$


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

For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ a point $P$ on the ellipse makes lines connecting it to the two foci $F1$ and $F2$ perpendicular to each other. Then, the area of $\triangle PF1F2$ is ________.

Answer: $12$


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

In quadrilateral $EFGH$, $\angle F$ is a right angle, diagonal $\overline{EG}$ is perpendicular to $\overline{GH}$, $EF=20$, $FG=24$, and $GH=16$. Find the perimeter of $EFGH$.

Answer: $60 + 8\sqrt{19}$


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

Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube.

Answer: $1059$


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

The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$.

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>.

How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?

Note that we consider the case of rotation and turn over as distinct way.

Answer: $72$


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

I'm going to dinner at a large restaurant which my friend recommended, unaware that I am vegan and have both gluten and dairy allergies. Initially, there are 6 dishes that are vegan, which constitutes one-sixth of the entire menu. Unfortunately, 4 of those vegan dishes contain either gluten or dairy. How many dishes on the menu can I actually eat?

Answer: $\frac{1}{18}$


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

A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square?

Answer: $110$


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

In the country Betia, there are 125 cities, some of which are connected by express trains that do not stop at intermediate stations. It is known that any four cities can be visited in a circular order. What is the minimum number of city pairs connected by express trains?

Answer: $7688$


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

Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order.

Answer: $74$


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

In the triangle \( \triangle ABC \), it is given that the angles are in the ratio \(\angle A : \angle B : \angle C = 3 : 5 : 10\). Also, it is known that \(\triangle A'B'C \cong \triangle ABC\). What is the ratio \(\angle BCA' : \angle BCB'\)?

Answer: $1:4$


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

Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:

|          | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150         | 50           | 200   |
| Machine B | 120         | 80           | 200   |
| Total    | 270         | 130          | 400   |

$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?

$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?

Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.

| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$                   | 3.841 | 6.635 | 10.828 |

Answer: $99\%$


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

Each square of an  $n \times n$  grid is coloured either blue or red, where  $n$  is a positive integer. There are  $k$  blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form  $S_B$ . He then performs the same calculation on the red cells to compute  $S_R$ .
If  $S_B-  S_R = 50$ , determine (with proof) all possible values of  $k$ .

Answer: $313$


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

Technology changes the world. Express sorting robots have become popular from Weibo to Moments. According to the introduction, these robots can not only automatically plan the optimal route, accurately place packages in the corresponding compartments, sense and avoid obstacles, automatically return to the team to pick up packages, but also find charging piles to charge themselves when they run out of battery. A certain sorting warehouse plans to sort an average of 200,000 packages per day. However, the actual daily sorting volume may deviate from the plan. The table below shows the situation of sorting packages in the third week of October in this warehouse (the part exceeding the planned amount is recorded as positive, and the part that does not reach the planned amount is recorded as negative):

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Sorting Situation (in 10,000s) | +6 | +4 | -6 | +8 | -1 | +7 | -4 |

$(1)$ The day with the most sorted packages in the warehouse this week is ______; the day with the least sorted packages is ______; the day with the most sorted packages has ______ more packages than the day with the least sorted packages;<br/>$(2)$ How many packages, on average, did the warehouse actually sort per day this week?

Answer: $22$


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

Given that the Riemann function defined on the interval $\left[0,1\right]$ is: $R\left(x\right)=\left\{\begin{array}{l}{\frac{1}{q}, \text{when } x=\frac{p}{q} \text{(p, q are positive integers, } \frac{p}{q} \text{ is a reduced proper fraction)}}\\{0, \text{when } x=0,1, \text{or irrational numbers in the interval } (0,1)}\end{array}\right.$, and the function $f\left(x\right)$ is an odd function defined on $R$ with the property that for any $x$ we have $f\left(2-x\right)+f\left(x\right)=0$, and $f\left(x\right)=R\left(x\right)$ when $x\in \left[0,1\right]$, find the value of $f\left(-\frac{7}{5}\right)-f\left(\frac{\sqrt{2}}{3}\right)$.

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


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

Given the data in the table, where the number of No. 5 batteries is $x$ and the number of No. 7 batteries is $y$, the masses of one No. 5 battery and one No. 7 battery are $x$ grams and $y$ grams, respectively. By setting up and solving a system of equations, express the value of $x$ obtained by elimination.

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>.

When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a "dragon" of length $k." Find the shortest dragon length.

Answer: $39$


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

Calculate the volumes of the bodies bounded by the surfaces.

$$
z = 2x^2 + 18y^2, \quad z = 6
$$

Answer: $6\pi$


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 points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\frac{3}{4}$.
$(1)$ Find the equation of the locus $C$ for point $P$.
$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at another point $Q$. Let point $R$ be the midpoint of segment $PF_2$, and let $O$ be the origin. Let $S$ represent the sum of the areas of triangles $QF_1O$ and $PF_1R$. Find the maximum value of $S$.

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


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

Given that $0 < \alpha < \frac{\pi}{2}$ and $0 < \beta < \frac{\pi}{2}$, if $\sin\left(\frac{\pi}{3}-\alpha\right) = \frac{3}{5}$ and $\cos\left(\frac{\beta}{2} - \frac{\pi}{3}\right) = \frac{2\sqrt{5}}{5}$,
(I) find the value of $\sin \alpha$;
(II) find the value of $\cos\left(\frac{\beta}{2} - \alpha\right)$.

Answer: $\frac{11\sqrt{5}}{25}$


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

In right triangle $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 15$, $BC = 20$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\omega$. Find the ratio of the perimeter of $\triangle ABI$ to the length $AB$ and express it in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers.

Answer: $97$


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

Matt now needs to place five identical, dotless dominoes (shaded 1 by 2 rectangles) on a 6 by 5 grid. He wishes to create a path from the upper left-hand corner $C$ to the lower right-hand corner $D$. In the path, consecutive dominoes must touch at their sides and not just at their corners. No domino may be placed diagonally; each domino must cover exactly two of the grid squares shown. Determine how many such distinct domino arrangements are possible.

Answer: $126$


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

Let \( D \) be a point inside the acute triangle \( \triangle ABC \). Given that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).

Answer: $\sqrt{2}$


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

Express the quotient $2033_4 \div 22_4$ in base 4.

Answer: $11_4$


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>.

On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?

Answer: $34$


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

Given that $\cos (\frac{\pi}{4}-a) = \frac{12}{13}$ and $(\frac{\pi}{4}-a)$ is a first-quadrant angle, find the value of $\frac{\sin (\frac{\pi}{2}-2a)}{\sin (\frac{\pi}{4}+a)}$: \_\_\_\_\_\_.

Answer: $\frac{119}{144}$


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

Compute without using a calculator: $12!/11!$

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>.

Find the number of integers  $ c$  such that  $ \minus{}2007 \leq c \leq 2007$  and there exists an integer  $ x$  such that  $ x^2 \plus{} c$  is a multiple of  $ 2^{2007}$ .

Answer: $670$


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

Find the sum of the roots of the equation $\tan^2x - 8\tan x + 2 = 0$ that are between $x = 0$ and $x = 2\pi$ radians.

Answer: $3\pi$


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

The triangle  $ABC$  is isosceles with  $AB=BC$ . The point F on the side  $[BC]$  and the point  $D$  on the side  $AC$  are the feets of the the internals bisectors drawn from  $A$  and altitude drawn from  $B$  respectively so that  $AF=2BD$ . Fine the measure of the angle  $ABC$ .

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>.

Let the roots of the cubic equation \(27x^3 - 81x^2 + 63x - 14 = 0\) be in geometric progression. Find the difference between the square of the largest root and the square of the smallest root.

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


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

In triangle $XYZ,$ points $G,$ $H,$ and $I$ are on sides $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:3.$ Line segments $\overline{XG},$ $\overline{YH},$ and $\overline{ZI}$ intersect at points $S,$ $T,$ and $U,$ respectively. Compute $\frac{[STU]}{[XYZ]}.$

Answer: $\frac{9}{55}$


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

One angle of a parallelogram is 150 degrees, and two consecutive sides have lengths of 10 inches and 20 inches. What is the area of the parallelogram? Express your answer in simplest radical form.

Answer: $100\sqrt{3}$


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>.

Let \( x, y, z \) be positive real numbers such that \( xyz = 3 \). Compute the minimum value of
\[ x^2 + 4xy + 12y^2 + 8yz + 3z^2. \]

Answer: $162$


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

Given that the area of a cross-section of sphere O is $\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________.

Answer: $\frac{8\sqrt{2}}{3}\pi$


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

If $x^{2y}=16$ and $x = 16$, what is the value of $y$? Express your answer as a common fraction.

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


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

Consider two lines $p$ and $q$ in a coordinate plane with equations $y = -3x + 9$ and $y = -6x + 9$, respectively. Determine the probability that a point randomly selected in the first quadrant and below line $p$ will fall between $p$ and $q$.

Answer: $0.5$


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

The average weight of 8 boys is 160 pounds, and the average weight of 6 girls is 130 pounds. Calculate the average weight of these 14 children.

Answer: $147$


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

What integer $n$ satisfies $0 \leq n < 151$ and $$100n \equiv 93 \pmod {151}~?$$

Answer: $29$


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

Compute the sum of the geometric series $-3 + 6 - 12 + 24 - \cdots - 768$.

Answer: $514$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 value of $k$ for which it is possible to mark $k$ cells on a $9 \times 9$ board such that any placement of a three-cell corner touches at least two marked cells?

Answer: $56$


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

Find the product of the roots and the sum of the roots of the equation $24x^2 + 60x - 600 = 0$.

Answer: $-2.5$


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

In recent years, live streaming e-commerce has gradually become an emerging marketing model, bringing new growth points to the e-commerce industry. At the beginning of the first year, a certain live streaming platform had an initial capital of 5 million yuan. Due to the participation of some well-known hosts, the platform's annual average capital growth rate can reach 40%. At the end of each year, operating costs of $a$ million yuan are deducted, and the remaining funds are reinvested in the live streaming platform.
$(1)$ If $a=100$, how many million yuan will the live streaming platform have at the end of the third year after deducting operating costs?
$(2)$ How much should the maximum annual operating cost be controlled at most, in order for the live streaming platform to reach 3000 million yuan at the end of the sixth year after deducting operating costs? (Give the result accurate to 0.1 million yuan)

Answer: $46.8$


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 Jo and Blair take turns counting from 1, with Jo adding 2 to the last number said and Blair subtracting 1 from the last number said, determine the 53rd number said.

Answer: $79$


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

Given a rectangular yard containing two congruent isosceles right triangles in the form of flower beds and a trapezoidal remainder, with the parallel sides of the trapezoid having lengths $15$ and $25$ meters.

Answer: $\frac{1}{5}$


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

Suppose $a$, $b$, $c$, and $d$ are positive integers satisfying $a + b + c + d = 3000$. Calculate $a!b!c!d! = m \cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by 10. What is the smallest possible value of $n$?
A) 745
B) 748
C) 751
D) 754
E) 757

Answer: $748$


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

 $21$  Savage has a  $12$  car garage, with a row of spaces numbered  $1,2,3,\ldots,12$ . How many ways can he choose  $6$  of them to park his  $6$  identical cars in, if no  $3$  spaces with consecutive numbers may be all occupied?

*2018 CCA Math Bonanza Team Round #9*

Answer: $357$


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

\(5, 6, 7\)

Answer: $21$


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

Let \( a_0 = -3 \), \( b_0 = 2 \), and for \( n \geq 0 \), let:
\[
\begin{align*}
a_{n+1} &= 2a_n + 2b_n + 2\sqrt{a_n^2 + b_n^2}, \\
b_{n+1} &= 2a_n + 2b_n - 2\sqrt{a_n^2 + b_n^2}.
\end{align*}
\]
Find \( \frac{1}{a_{2023}} + \frac{1}{b_{2023}} \).

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>.

From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?

(2004 National Girls' Olympiad problem)

Answer: $1006009$


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

A four-digit number \((xyzt)_B\) is called a stable number in base \(B\) if \((xyzt)_B = (dcba)_B - (abcd)_B\), where \(a \leq b \leq c \leq d\) are the digits \(x, y, z, t\) arranged in ascending order. Determine all the stable numbers in base \(B\).

(Problem from the 26th International Mathematical Olympiad, 1985)

Answer: $(1001)_2, (3021)_4, (3032)_5, (3B/5, B/5-1, 4B/5-1, 2B/5)_B, 5 | B$


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

Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 5.\) Find the minimum value of
\[
\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.
\]

Answer: $30$


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

Given that 600 athletes are numbered from 001 to 600 and divided into three color groups (red: 001 to 311, white: 312 to 496, and yellow: 497 to 600), calculate the probability of randomly drawing an athlete wearing white clothing.

Answer: $\frac{8}{25}$


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

Henry walks $\tfrac{3}{4}$ of the way from his home to his gym, which is $2$ kilometers away from Henry's home, and then walks $\tfrac{3}{4}$ of the way from where he is back toward home. Determine the difference in distance between the points toward which Henry oscillates from home and the gym.

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


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>.

What is the greatest integer less than 150 for which the greatest common factor of that integer and 24 is 3?

Answer: $147$


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

Calculate the sum $2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} + 2^{-7} \pmod{17}$.

Express your answer as an integer from $0$ to $16$, inclusive.

Answer: $10$


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

Given a 12-hour digital clock with a glitch where every '2' is displayed as a '7', determine the fraction of the day that the clock shows the correct time.

Answer: $\frac{55}{72}$


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

How many ten digit positive integers with distinct digits are multiples of  $11111$ ?

Answer: $3456$


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

Let $PQRST$ be a convex pentagon with $PQ \parallel RT, QR \parallel PS, QS \parallel PT, \angle PQR=100^\circ, PQ=4, QR=7,$ and $PT = 21.$ Given that the ratio between the area of triangle $PQR$ and the area of triangle $RST$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Answer: $232$


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

Find the area of the region enclosed by the graph of \( |x-75| + |y| = \left|\frac{x}{3}\right| \).

Answer: $703.125$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 with side lengths 5 by 8 is folded along the dashed lines so that the folded flaps just touch at the corners as indicated by the dotted lines. Find the area of the resulting trapezoid.

Answer: $55/2$


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>.

Solve for $x$:
\[\arcsin 3x - \arccos (2x) = \frac{\pi}{6}.\]

Answer: $-\frac{1}{\sqrt{7}}$


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

For any $x \in (0, +\infty)$, the inequality $(x-a+\ln \frac{x}{a})(-2x^2+ax+10) \leq 0$ always holds. Then, the range of the real number $a$ is ______.

Answer: $\sqrt{10}$


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>.

Consider a square flag with a red cross of uniform width and a blue triangular central region on a white background. The cross is symmetric with respect to each of the diagonals of the square. Let's say the entire cross, including the blue triangle, occupies 45% of the area of the flag. Calculate the percentage of the flag's area that is blue if the triangle is an equilateral triangle centered in the flag and the side length of the triangle is half the width of the red cross arms.

Answer: $1.08\%$


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

A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe.

Answer: $1/4$


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

A rectangular garden measuring 60 feet by 20 feet is enclosed by a fence. In a redesign to maximize the area using the same amount of fencing, its shape is changed to a circle. How many square feet larger or smaller is the new garden compared to the old one?

Answer: $837.62$


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

If two numbers are randomly chosen without replacement from the set $\{6, 8, 9, 12\}$, what is the probability that their product will be a multiple of 36? Express your answer as a common fraction.

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>.

The first operation divides the bottom-left square of diagram $\mathrm{a}$ into four smaller squares, as shown in diagram b. The second operation further divides the bottom-left smaller square of diagram b into four even smaller squares, as shown in diagram c; continuing this process, after the sixth operation, the resulting diagram will contain how many squares in total?

Answer: $29$


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

(from The Good Soldier Svejk) Senior military doctor Bautze exposed  $abccc$  malingerers among  $aabbb$  draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?

Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.

*(1 point)*

Answer: $10999$


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

Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \).

Answer: $240$