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

Given acute angles $\alpha$ and $\beta$ such that $\sin \alpha= \frac { \sqrt {5}}{5}$ and $\sin(\alpha-\beta)=- \frac { \sqrt {10}}{10}$, determine the value of $\beta$.

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


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 certain item is always sold with a 30% discount, and the profit margin is 47%. During the shopping festival, the item is sold at the original price, and there is a "buy one get one free" offer. Calculate the profit margin at this time. (Note: Profit margin = (selling price - cost) ÷ cost)

Answer: $5\%$


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>.

Find $5273_{8} - 3614_{8}$. Express your answer in base $8$.

Answer: $1457_8$


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>.

At a national competition, 31 participants are accommodated in a hotel where each participant gets his/her own room. The rooms are numbered from 1 to 31. When all participants have arrived, except those assigned to rooms 15, 16, and 17, what is the median room number of the remaining 28 participants?

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>.

A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and radius equal to the upper base of the frustum attached to the top of the frustum. Calculate the total surface area excluding the bases.

Answer: $39\pi\sqrt{5} + 20\pi$


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 a circle $C$: $x^{2}+y^{2}-2x-2ay+a^{2}-24=0$ ($a\in\mathbb{R}$) whose center lies on the line $2x-y=0$.

$(1)$ Find the value of the real number $a$;
$(2)$ Find the minimum length of the chord formed by the intersection of circle $C$ and line $l$: $(2m+1)x+(m+1)y-7m-4=0$ ($m\in\mathbb{R}$).

Answer: $4\sqrt{5}$


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

In right triangle $DEF$, $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?

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>.

Let the number of elements in the set \( S \) be denoted by \( |S| \), and the number of subsets of the set \( S \) be denoted by \( n(S) \). Given three non-empty finite sets \( A, B, C \) that satisfy the following conditions:
$$
\begin{array}{l}
|A| = |B| = 2019, \\
n(A) + n(B) + n(C) = n(A \cup B \cup C).
\end{array}
$$

Determine the maximum value of \( |A \cap B \cap C| \) and briefly describe the reasoning process.

Answer: $2018$


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

Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\sin \alpha$ and $\cos \alpha$ (where $0 < \alpha < \frac{\pi}{2}$). What is the length of the hypotenuse?

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


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

Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?

Answer: $\frac{5}{18}$


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>.

Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$.

Answer: $110.8333$


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

The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?

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>.

There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.

Answer: $1260$


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

In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, find the distance from $F$ to the midpoint of segment $DE$.

Answer: $7.5$


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

Given that $P$ is a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\frac{{\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}}}{{|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|}}=\frac{1}{2}$, then the area of $\triangle F_{1}PF_{2}$ is ______.

Answer: $3\sqrt{3}$


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

The graph of $y = ax^2 + bx + c$ has a maximum value of 72, and passes through the points $(0, -1)$ and $(6, -1)$. Find $a + b + c$.

Answer: $\frac{356}{9}$


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 the lines $l_{1}$: $x+\left(m-3\right)y+m=0$ and $l_{2}$: $mx-2y+4=0$.
$(1)$ If line $l_{1}$ is perpendicular to line $l_{2}$, find the value of $m$.
$(2)$ If line $l_{1}$ is parallel to line $l_{2}$, find the distance between $l_{1}$ and $l_{2}$.

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


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

If $9:y^3 = y:81$, what is the value of $y$?

Answer: $3\sqrt{3}$


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

The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$?

Answer: $\frac{11175}{7499}$


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

The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ has the property that
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_6 = 60,$ then find $b_7.$

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>.

Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes.

Answer: $80$


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

Points \(A = (2,8)\), \(B = (2,2)\), and \(C = (6,2)\) lie in the first quadrant and are vertices of triangle \(ABC\). Point \(D=(a,b)\) is also in the first quadrant, and together with \(A\), \(B\), and \(C\), forms quadrilateral \(ABCD\). The quadrilateral formed by joining the midpoints of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\) is a square. Additionally, the diagonal of this square has the same length as the side \(\overline{AB}\) of triangle \(ABC\). Find the sum of the coordinates of point \(D\).
A) 12
B) 14
C) 15
D) 16
E) 18

Answer: $14$


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

Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum possible distance between $P$ and $Q$.

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


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

Given that $\frac{5+7+9}{3} = \frac{4020+4021+4022}{M}$, find $M$.

Answer: $1723$


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

In the sequence $\{a_{n}\}$, $a_{1}=1$, $\sqrt{{a}_{n+1}}-\sqrt{{a}_{n}}=1$ ($n\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\{b_{n}\}$ is $S_{n}=2^{n}-m$. For $n\in N^{*}$, the smallest value of the real number $\lambda$ that satisfies $\lambda b_{n}\geqslant a_{n}$ for all $n$ is ______.

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


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 vectors $\overrightarrow{a}=(\cos x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, 2\cos x-\sqrt{3}\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the maximum value of the function $g(x)=f(x-\frac{\pi}{6})+af(\frac{x}{2}-\frac{\pi}{6})-af(\frac{x}{2}+\frac{\pi}{12})$ on the interval $[0,\pi]$ is $6$, determine the value of the real number $a$.

Answer: $2\sqrt{2}$


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

Let positive integers $a$, $b$, $c$ satisfy $ab + bc = 518$ and $ab - ac = 360$. The maximum value of $abc$ is ____.

Answer: $1008$


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

A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. Thus, there are a total of     different orders for this performance.

Answer: $210$


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

The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle.

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 a class of 50 students, it is decided to use systematic sampling to select 10 students out of these 50. The students are randomly assigned numbers from 1 to 50 and grouped, with the first group being numbers 1 to 5, the second group 6 to 10, and so on, up to the tenth group which is 46 to 50. If a student with the number 12 is selected from the third group, then the student with the number $\_\_\_$ will be selected from the eighth group.

Answer: $37$


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

How many even numbers are greater than 300 and less than 600?

Answer: $149$


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

The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o'clock at the beginning of the five-minute interval of correct movement?

Answer: $28$


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

The real numbers $x$ and $y$ satisfy the equation $2\cos ^{2}(x+y-1)= \frac {(x+1)^{2}+(y-1)^{2}-2xy}{x-y+1}$. Find the minimum value of $xy$.

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>.

Find the largest positive integer  $N$  so that the number of integers in the set  $\{1,2,\dots,N\}$  which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

Answer: $65$


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

Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \geq 1$. Find
\[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\]

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


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

The average of the seven numbers in a list is 62. The average of the first four numbers is 58. What is the average of the last three numbers?

Answer: $67.\overline{3}$


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

If \( g(x) = \frac{x^5 - 1}{4} \), find \( g^{-1}(-7/64) \).

Answer: $\left(\frac{9}{16}\right)^{\frac{1}{5}}$


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>.

The mean of the numbers 3, 7, 10, and 15 is twice the mean of $x$, 20, and 6. What is the value of $x$?

Answer: $-12.875$


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

Simplify the expression $\frac{{2x+4}}{{{x^2}-1}}÷\frac{{x+2}}{{{x^2}-2x+1}}-\frac{{2x}}{{x+1}}$, then substitute an appropriate number from $-2$, $-1$, $0$, $1$ to evaluate.

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>.

Let $a$ and $b$ be integers such that $ab = 72.$  Find the minimum value of $a + b.$

Answer: $-73$


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

A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\triangle PF_2Q$ is.

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>.

A square with a side length of 10 centimeters is rotated about its horizontal line of symmetry. Calculate the volume of the resulting cylinder in cubic centimeters and express your answer in terms of $\pi$.

Answer: $250\pi$


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

Athletes A and B have probabilities of successfully jumping over a 2-meter high bar of 0.7 and 0.6, respectively. The outcomes of their jumps are independent of each other. Find:

(Ⅰ) The probability that A succeeds on the third attempt.

(Ⅱ) The probability that at least one of A or B succeeds on the first attempt.

(Ⅲ) The probability that A succeeds exactly one more time than B in two attempts for each.

Answer: $0.3024$


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

Given that three numbers are randomly selected from the set {1, 2, 3, 4, 5}, find the probability that the sum of the remaining two numbers is odd.

Answer: $0.6$


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>.

$(1)$ Calculate: $tan60°×{({-2})^{-1}}-({\sqrt{\frac{3}{4}}-\sqrt[3]{8}})+|{-\frac{1}{2}\sqrt{12}}|$;<br/>$(2)$ Simplify and find the value: $({\frac{{x+2}}{{{x^2}-2x}}-\frac{{x-1}}{{{x^2}-4x+4}}})÷\frac{{x-4}}{x}$, where $x=\sqrt{2}+2$.

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>.

Calculate \(3 \cdot 15 + 20 \div 4 + 1\).

Then add parentheses to the expression so that the result is:

1. The largest possible integer,
2. The smallest possible integer.

Answer: $13$


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

Given $f\left( \alpha \right)=\frac{\cos \left( \frac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\frac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \frac{3}{2}\pi +\alpha \right)}$.

$(1)$ Simplify $f\left( \alpha \right)$; $(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos \left( \alpha -\frac{3}{2}\pi \right)=\frac{1}{5}$, find the value of $f\left( \alpha \right)$.

Answer: $\frac{2 \sqrt{6}}{5}$


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>.

A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$.

Answer: $32.5$


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 how many different ways can 900 be expressed as the product of two (possibly equal) positive integers?  Regard  $m \cdot n$  and  $n \cdot m$  as the same product.

Answer: $14$


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

Given the equation of the Monge circle of the ellipse $\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\Gamma$.

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


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

Simplify $\frac{{1+\cos{20}°}}{{2\sin{20}°}}-\sin{10°}\left(\frac{1}{{\tan{5°}}}-\tan{5°}\right)=\_\_\_\_\_\_$.

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>.

Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]

Answer: $14$


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

Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\frac{S_{n}}{T_{n}} = \frac{2n+1}{4n-2}$ for all positive integers $n$, find the value of $\frac{a_{10}}{b_{3}+b_{18}} + \frac{a_{11}}{b_{6}+b_{15}}$.

Answer: $\frac{41}{78}$


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

What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form.

Answer: $\frac{1}{\sqrt[3]{12}}$


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 circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______.

Answer: $-8$


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

Given a four-digit number $\overline{ABCD}$ such that $\overline{ABCD} + \overline{AB} \times \overline{CD}$ is a multiple of 1111, what is the minimum value of $\overline{ABCD}$?

Answer: $1729$


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

Given $3\sin \left(-3\pi +\theta \right)+\cos \left(\pi -\theta \right)=0$, then the value of $\frac{sinθcosθ}{cos2θ}$ is ____.

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


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>.

What is the greatest common factor of the numbers 2750 and 9450?

Answer: $50$


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

The perpendicular bisectors of the sides  $AB$  and  $CD$  of the rhombus  $ABCD$  are drawn. It turned out that they divided the diagonal  $AC$  into three equal parts. Find the altitude of the rhombus if  $AB = 1$ .

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>.

Liam needs to add 125 to 96 and then round the result to the nearest hundred. He plans to round the numbers before adding them. He rounds 125 to 100 and 96 to 100, resulting in an answer of $100 + 100 = 200$. Is this answer correct? What is the correct answer?

Answer: $200$


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

Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$.

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


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

A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\frac{m}{n}$, and find the sum of the numerator and denominator.

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>.

For a positive integer  $n$ , let  $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest  $n$  such that  $\left|f_n^{\prime \prime}(0)\right|>2023$ .

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>.

A curious archaeologist is holding a competition where participants must guess the age of a unique fossil. The age of the fossil is formed from the six digits 2, 2, 5, 5, 7, and 9, and the fossil's age must begin with a prime number.

Answer: $90$


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

Simplify first, then evaluate: $3a^{2}+\left[a^{2}+\left(5a^{2}-2a\right)-3\left(a^{2}-3a\right)\right]$, where $a=-2$.

Answer: $10$


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

The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$.

Answer: $12$


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

Given the new operation $n\heartsuit m=n^{3+m}m^{2+n}$, evaluate $\frac{2\heartsuit 4}{4\heartsuit 2}$.

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 the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same unit length. It is known that the polar equation of line $l$ is $\rho(2\cos \theta - \sin \theta) = 6$.
$(1)$ After extending the x-coordinate of all points on curve $C_{1}$ by $\sqrt {3}$ times and the y-coordinate by $2$ times to obtain curve $C_{2}$, write down the Cartesian equation of line $l$ and the parametric equation of curve $C_{2}$;
$(2)$ Let $P$ be any point on curve $C_{2}$. Find the maximum distance from point $P$ to line $l$.

Answer: $2 \sqrt {5}$


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

$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. Given that $f(6) = 6$, determine $f(2012)$.

Answer: $-2000$


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

The math teacher wants to arrange 6 multiple-choice questions into a large practice exam paper, requiring that the two easy questions, $A$ and $B$, must be adjacent, and the two difficult questions, $E$ and $F$, cannot be adjacent. There are a total of $\_\_\_\_\_\_\_\_\_$ different arrangements. (Answer with a number)

Answer: $144$


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

Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean score of all the students in both sections.

Answer: $\frac{1006}{12}$


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>.

Detached calculation.

327 + 46 - 135

1000 - 582 - 128

(124 - 62) × 6

500 - 400 ÷ 5

Answer: $420$


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

Evaluate the infinite geometric series:
$$\frac{5}{3} - \frac{5}{4} + \frac{25}{48} - \frac{125}{384} + \dots$$

Answer: $\frac{20}{21}$


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 \(ABCD\) be a convex quadrilateral, and let \(M_A,\) \(M_B,\) \(M_C,\) \(M_D\) denote the midpoints of sides \(BC,\) \(CA,\) \(AD,\) and \(DB,\) respectively. Find the ratio \(\frac{[M_A M_B M_C M_D]}{[ABCD]}.\)

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>.

Twelve million added to twelve thousand equals what number?

Answer: $12012000$


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

If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____.

Answer: $\ln 2$


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

Given the polynomial $f(x)=x^{6}-5x^{5}+6x^{4}+x^{2}+0.3x+2$, use Horner's method to calculate $f(-2)$ and find the value of $v_{1}$.

Answer: $-7$


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

Given a quadratic polynomial \( P(x) \). It is known that the equations \( P(x) = x - 2 \) and \( P(x) = 1 - x / 2 \) each have exactly one root. What is the discriminant of \( P(x) \)?

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


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

If real numbers \(a\), \(b\), and \(c\) satisfy \(a^{2} + b^{2} + c^{2} = 9\), then what is the maximum value of the algebraic expression \((a - b)^{2} + (b - c)^{2} + (c - a)^{2}\)?

Answer: $27$


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

Given sets A = {x | $x^2 - ax + a^2 - 19 = 0$}, B = {x | $x^2 - 5x + 6 = 0$}, C = {x | $x^2 + 2x - 8 = 0$} such that $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$. Find the value of the real number $a$.

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>.

The sum of three numbers $x$, $y$, and $z$ is 150. If we increase $x$ by 7, decrease $y$ by 12 and multiply $z$ by 4, the three resulting numbers are equal. What is the value of $y$?

Answer: $\frac{688}{9}$


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>.

In Perfectville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Jane runs around the block on the $500$-foot side of the street, while John runs on the opposite side of the street. How many more feet than Jane does John run for every lap around the block?

Answer: $240$


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

Given that the solution set for the inequality $ax^2+ax+2>0$ is $\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.
(1) Find the set $M$.
(2) If $t>0$, for all $a \in M$, it holds that $(a^2-2a)t \leq t^2 + 3t - 46$. Find the minimum value of $t$.

Answer: $46$


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

Given $\cos\alpha + \cos\beta = \frac{1}{2}$, $\sin\alpha + \sin\beta = \frac{\sqrt{3}}{2}$, then $\cos(\alpha - \beta) =$ \_\_\_\_\_\_.

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>.

Given $\sqrt{15129}=123$ and $\sqrt{x}=0.123$, calculate the value of $x$.

Answer: $0.015129$


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

In a jar, there are 5 salted duck eggs of the same size and shape, among which 3 have green shells, and 2 have white shells. If two eggs are taken out one after another without replacement, calculate:
(1) The probability of the first egg taken out being green-shelled;
(2) The probability that both the first and the second eggs taken out are green-shelled;
(3) Given that the first egg taken out is green-shelled, the probability that the second egg is also green-shelled.

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


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

If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.

The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.

a) Find the probability that under this condition, event \( B \) did not occur at all.

b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).

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


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>.

Simplify first, then find the value of $\frac{{{a^2}-{b^2}}}{{{a^2}b-a{b^2}}}÷(1+\frac{{{a^2}+{b^2}}}{2ab})$, where $a=\sqrt{3}-\sqrt{11}$ and $b=\sqrt{3}+\sqrt{11}$.

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


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

There is infinite sequence of composite numbers  $a_1,a_2,...,$  where  $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$  ;  $p_n$  is smallest prime divisor of  $a_n$ . It is known, that  $37|a_n$  for every  $n$ .
Find possible values of  $a_1$

Answer: $37^2$


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

A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?

A) $\frac{1}{9}$
B) $\frac{1}{4}$
C) $\frac{1}{36}$
D) 97

Answer: $\frac{1}{9}$


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>.

In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.

A Nine-Headed Bird discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 4 times the number of heads. A Nine-Tailed Fox discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 3 times the number of heads. How many Nine-Tailed Foxes are there on the mountain?

Answer: $14$


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

For what value of $n$ does $|6 + ni| = 6\sqrt{5}$?

Answer: $12$


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

Let the complex numbers \( z_1 \) and \( z_2 \) correspond to the points \( A \) and \( B \) on the complex plane respectively, and suppose \( \left|z_1\right| = 4 \) and \( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \). Let \( O \) be the origin. Calculate the area of triangle \( \triangle OAB \).

Answer: $8\sqrt{3}$


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

Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$.

Answer: $\sqrt{3}$


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

The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $5$. Another die is marked with $2$, $3$, $4$, $6$, $7$, and $9$. What is the probability that the sum of the top two numbers will be $6$, $8$, or $10$?
A) $\frac{8}{36}$
B) $\frac{11}{36}$
C) $\frac{15}{36}$
D) $\frac{18}{36}$

Answer: $\frac{11}{36}$


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

The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Determine the $150^{\mbox{th}}$ term of this sequence.

Answer: $2280$


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

Given that the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F\left(4,0\right)$, and $O$ is the origin.
$(1)$ Find the equation of the parabola.
$(2)$ A line with a slope of $1$ passes through point $F$ and intersects the parabola at points $A$ and $B$. Find the area of $\triangle AOB$.

Answer: $32\sqrt{2}$


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>.

There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?

Answer: $12$


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

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a common difference of $d$. If $a_{1}=190$, $S_{20} \gt 0$, and $S_{24} \lt 0$, then one possible value for the integer $d$ is ______.

Answer: $-17$


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

Let the triangle  $ABC$  have area  $1$ . The interior bisectors of the angles  $\angle BAC,\angle ABC, \angle BCA$  intersect the sides  $(BC), (AC), (AB) $  and the circumscribed circle of the respective triangle  $ABC$  at the points  $L$  and  $G, N$  and  $F, Q$  and  $E$ . The lines  $EF, FG,GE$  intersect the bisectors  $(AL), (CQ) ,(BN)$  respectively at points  $P, M, R$ . Determine the area of the hexagon  $LMNPR$ .

Answer: $1/2$


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

Given that $2ax^3 - 3bx + 8 = 18$ when $x = -1$, determine the value of $9b - 6a + 2$.

Answer: $32$


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

The line RS does not intersect with △ABC. Perpendiculars are drawn from the vertices A, B, and C of △ABC to line RS, and the corresponding feet of the perpendiculars are D, E, and F respectively. It is given that AD = 10, BE = 6, and CF = 24. Let H be the foot of the perpendicular drawn from the centroid G of △ABC to line RS. If x denotes the length of the segment GH, find x.

Answer: $\frac{40}{3}$


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 positive integers \( a, b, c, d \) satisfy \( a > b > c > d \) and \( a+b+c+d=2004 \), \( a^2 - b^2 + c^2 - d^2 = 2004 \). Find the minimum value of \( a \).

Answer: $503$


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

What is the largest divisor of 540 that is less than 80 and also a factor of 180?

Answer: $60$


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

The lengths of the three sides of a triangle are 11, 15, and \( k \) \((k \in Z^{+})\). The number of values of \( k \) that make the triangle an obtuse triangle is:

Answer: $13$


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

Given that the random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, and $P(\xi \leq 4) = 0.84$, find the probability $P(\xi \leq -2)$.

Answer: $0.16$


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>.

What is the minimum value of $w$ if $w=3x^2 + 4y^2 - 12x + 8y + 15$?

Answer: $-1$


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

On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction.

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>.

If $\overrightarrow{a} = (2, 3)$, $\overrightarrow{b} = (-4, 7)$, and $\overrightarrow{a} + \overrightarrow{c} = 0$, then the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{b}$ is \_\_\_\_\_\_.

Answer: $-\frac{\sqrt{65}}{5}$


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

Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$.

Answer: $88$


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

Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$.

Answer: $25$


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

A student, Alex, is required to do a specified number of homework assignments to earn homework points using a different system: for the first four points, each point requires one homework assignment; for the next four points (points 5-8), each requires two homework assignments; then, for points 9-12, each requires three assignments, etc. Calculate the smallest number of homework assignments necessary for Alex to earn a total of 20 homework points.

Answer: $60$


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

Given the lines $x-y-1=0$ and $x-y-5=0$ both intersect circle $C$ creating chords of length 10, find the area of circle $C$.

Answer: $27\pi$


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>.

In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dana, refuse to sit in four consecutive seats?

Answer: $3507840$


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

Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$.

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>.

Given $m+n=2$ and $mn=-2$. Find the value of:

1. $2^{m}\cdot 2^{n}-(2^{m})^{n}$
2. $(m-4)(n-4)$
3. $(m-n)^{2}$.

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>.

What is the sum of the digits of the square of the number 22222?

Answer: $46$


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

Mrs. Johnson recorded the following scores for a test taken by her 120 students. Calculate the average percent score for these students.

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&10\\\hline
85&20\\\hline
75&40\\\hline
65&30\\\hline
55&15\\\hline
45&3\\\hline
0&2\\\hline
\end{tabular}

Answer: $71.33$


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

What is $3.57 - 1.14 - 0.23$?

Answer: $2.20$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $c=2$, $b=\sqrt{2}a$. The maximum area of $\triangle ABC$ is ______.

Answer: $2\sqrt{2}$


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

Given an arithmetic sequence $\{a_{n}\}$, $\{b_{n}\}$, with their sums of the first $n$ terms being $S_{n}$ and $T_{n}$ respectively, and satisfying $\frac{S_n}{T_n}=\frac{n+3}{2n-1}$, find $\frac{a_5}{T_9}$.

Answer: $\frac{4}{51}$


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

If the function $f(x)=\frac{1}{2}(2-m)x^{2}+(n-8)x+1$ $(m > 2)$ is monotonically decreasing in the interval $[-2,-1]$, find the maximum value of $mn$.

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>.

Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$.

Answer: $\frac {175}{264}$


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

Given a sequence of distinct positive integers $(i\_1, i\_2, \ldots, i\_n)$ where $n$ is a positive integer greater than or equal to 2, if $i\_p > i\_q$ when $p > q$, then $i\_p$ and $i\_q$ are called a "good order" of the sequence. The number of "good orders" in a sequence is called the "good order number" of that sequence. For example, the sequence (1, 3, 4, 2) has "good orders" "1, 3", "1, 4", "1, 2", "3, 4", so its "good order number" is 4. If the "good order number" of the sequence of distinct positive integers $(a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7)$ is 3, then the "good order number" of $(a\_7, a\_6, a\_5, a\_4, a\_3, a\_2, a\_1)$ is $\_\_\_\_\_\_\_.$

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>.

Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(1000,0),(1000,1005),$ and $(0,1005)$. What is the probability that $x > 5y$? Express your answer as a common fraction.

Answer: $\frac{20}{201}$


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

For a given point $P$ on the curve $x^2 - y - \ln x = 0$, what is the minimum distance from point $P$ to the line $y = x - 2$?

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>.

The number of right-angled triangles with integer side lengths $a$ and $b$ (where $b < 2011$), and hypotenuse length $b + 1$ is     .

Answer: $31$


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

What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$?

Answer: $104060401$


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

A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram.

Answer: $35 + 2\sqrt{34}$


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>.

In how many ways can a President, a Vice-President, and a Secretary be chosen from a group of 6 people, given the following constraints:
- The President must be one of the first three members of the group.
- The Vice-President must be one of the last four members of the group.
- No person can hold more than one office.

Answer: $36$