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

A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.

Answer: $2/3$


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

Calculate the sum of the distances from one vertex of a rectangle with length 3 and width 4 to the midpoints of each of its sides.
A) $6 + \sqrt{5}$
B) $7 + \sqrt{12}$
C) $7.77 + \sqrt{13}$
D) $9 + \sqrt{15}$

Answer: $7.77 + \sqrt{13}$


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>.

Find $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$

Answer: $\sqrt[4]{27}$


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>.

Please choose one of the following two sub-questions to answer. If multiple choices are made, the score will be based on the first chosen question.

$(①)$ The sum of the internal angles of a regular hexagon is $ $ degrees.

$(②)$ Xiaohua saw a building with a height of $(137)$ meters at its signboard. From the same horizontal plane at point $B$, he measured the angle of elevation to the top of the building $A$ to be $30^{\circ}$. The distance from point $B$ to the building is $ $ meters (rounded to the nearest whole number, and ignore the measuring instrument error, $\sqrt{3} \approx 1.732$).

Answer: $237$


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

A pyramid has a base which is an equilateral triangle with side length  $300$  centimeters. The vertex of the pyramid is  $100$  centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of  $134$  centimeters, how many centimeters above the base of the pyramid is the mouse?

Answer: $67$


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

Let \( y = (17 - x)(19 - x)(19 + x)(17 + x) \), where \( x \) is a real number. Find the smallest possible value of \( y \).

Answer: $-1296$


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

In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\).

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>.

Two circles with radius  $2$  and radius  $4$  have a common center at P.  Points  $A, B,$  and  $C$  on the larger circle are the vertices of an equilateral triangle.  Point  $D$  is the intersection of the smaller circle and the line segment  $PB$ . Find the square of the area of triangle  $ADC$ .

Answer: $192$


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

Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?

Answer: $999000000000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds.

Answer: $19$


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

Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6.

Answer: $\frac{18}{7}$


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>.

If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters.

Answer: $24$


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

Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?

Answer: $15$


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

Given that $\sin \alpha$ is a root of the equation $5x^2 - 7x - 6 = 0$, find the value of $\frac{\sin \left(\alpha - \frac{\pi}{2}\right)\sin^2\left(-\alpha + \pi\right)\sin \left( \frac{3\pi}{2}-\alpha\right)\tan \left(\alpha - \pi\right)}{\sin \left( \frac{\pi}{2}-\alpha\right)\cos \left(\alpha + \frac{\pi}{2}\right)\cos \left(\pi + \alpha\right)\tan \left(-\alpha + 5\pi\right)}$.

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


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>.

Inside a unit cube, eight equal spheres are placed. Each sphere is inscribed in one of the trihedral angles of the cube and touches the three spheres corresponding to the adjacent vertices of the cube. Find the radii of the spheres.

Answer: $1/4$


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

Given the event "Randomly select a point P on the side CD of rectangle ABCD, such that the longest side of ΔAPB is AB", with a probability of 1/3, determine the ratio of AD to AB.

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


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

The length of the hypotenuse of a right-angled triangle is 2cm longer than one of its legs, and the other leg is 6cm long. Find the length of its hypotenuse.

Answer: $10$


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

A couch cost 62,500 rubles. Its price was adjusted by 20% each month, either increasing or decreasing. Over six months, the price increased three times and decreased three times (the order of these changes is unknown). Can you uniquely determine the price of the couch after six months? If so, what was its price?

Answer: $55296$


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

The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere.

Answer: $171:113$


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

Calculate: \(\frac{2 \times 4.6 \times 9 + 4 \times 9.2 \times 18}{1 \times 2.3 \times 4.5 + 3 \times 6.9 \times 13.5} =\)

Answer: $\frac{18}{7}$


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>.

Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book?

Answer: $40$


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

The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$.

Answer: $84$


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

Let $S=1+ \frac{1}{ \sqrt {2}}+ \frac{1}{ \sqrt {3}}+...+ \frac{1}{ \sqrt {1000000}}$. Determine the integer part of $S$.

Answer: $1998$


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

Consider a regular tetrahedron $ABCD$. Find $\sin \angle BAC$.

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


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

Given the ellipse $\Gamma$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $F(3,0)$, and its upper and lower vertices at $A$ and $B$ respectively. The line $AF$ intersects $\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at the point $N(12,0)$, find the eccentricity of $\Gamma$.

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 35 groups of students, each group containing 3 students. Among these groups:
- 10 groups have only 1 boy.
- 19 groups have at least 2 boys.
- The number of groups with 3 boys is twice the number of groups with 3 girls.

How many boys are there?

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>.

Find the integer that is closest to $2000\sum_{n=2}^{5000}\frac{1}{n^2-1}$.

Answer: $1500$


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

Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart.

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


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

Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$.

Answer: $119$


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

Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits.

Answer: $142857$


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

Given an increasing sequence $\{a_n\}$ that satisfies $a_{n+1}a_{n-1} = a_n^2$ (for $n \geq 2$, $n \in \mathbb{N}$), the sum of the first 10 terms equals 50, and the sum of the first 15 terms is 210, calculate the sum of the first 5 terms ($S_5$).

Answer: $10$


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

Given the sequence \\(\{a_n\}\) satisfies \\(a_{n+1}= \dfrac {2016a_n}{2014a_n+2016}(n\in N_+)\), and \\(a_1=1\), find \\(a_{2017}= \) ______.

Answer: $\dfrac {1008}{1007\times 2017+1}$


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>.

Evaluate the following expression:   $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$   The terms with minus signs are exactly the powers of two.

Answer: $1996906$


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

Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\).

Answer: $24/5$


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 reconnaissance team has 12 soldiers, including 3 radio operators. The 12 soldiers are randomly divided into three groups, with group sizes of 3, 4, and 5 soldiers, respectively. What is the probability that each group has exactly 1 radio operator?

Answer: $3/11$


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

Write $-\left(-3\right)-4+\left(-5\right)$ in the form of algebraic sum without parentheses.

Answer: $3-4-5$


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

Given $l_{1}$: $ρ \sin (θ- \frac{π}{3})= \sqrt {3}$, $l_{2}$: $ \begin{cases} x=-t \\ y= \sqrt {3}t \end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse $\frac{x^{2}}{4}+y^{2}=1$. $O$ is the coordinate origin, and $∠AOB=∠BOC=∠COA=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}+ \frac{1}{|OC|^{2}}$.

Answer: $\frac{15}{8}$


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>.

Calculate the lengths of arcs of curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2(t-\sin t) \\
y=2(1-\cos t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$

Answer: $8 - 4\sqrt{2}$


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 average density of pathogenic microbes in one cubic meter of air is 100. A sample of 2 cubic decimeters of air is taken. Find the probability that at least one microbe will be found in the sample.

Answer: $0.181$


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

What is the area of a square inscribed in a semicircle of radius 1, with one of its sides flush with the diameter of the semicircle?

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


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

A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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>.

Add 78.621 to 34.0568 and round to the nearest thousandth.

Answer: $112.678$


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

Given $f(x) = kx + \frac {2}{x^{3}} - 3$ $(k \in \mathbb{R})$, and it is known that $f(\ln 6) = 1$. Find $f\left(\ln \frac {1}{6}\right)$.

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>.

The decimal number corresponding to the binary number $111011001001_2$ is to be found.

Answer: $3785$


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

How many paths are there from point $A$ to point $B$, if every step must be up or to the right in a grid where $A$ is at the bottom left corner and $B$ is at the top right corner of a 7x7 grid?

Answer: $3432$


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

Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________.

Answer: $111111_{(2)}$


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

In triangle \(ABC\), \(AB = 13\) and \(BC = 15\). On side \(AC\), point \(D\) is chosen such that \(AD = 5\) and \(CD = 9\). The angle bisector of the angle supplementary to \(\angle A\) intersects line \(BD\) at point \(E\). Find \(DE\).

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>.

If $x > 0$, $y > 0$, and $\frac{1}{2x+y} + \frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$.

Answer: $7 + 2\sqrt{6}$


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

In the following diagram, \(ABCD\) is a square, \(BD \parallel CE\) and \(BE = BD\). Let \(\angle E = x^{\circ}\). Find \(x\).

Answer: $30$


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

Given that the sequence $\{a\_n\}$ is an arithmetic sequence with the sum of its first $n$ terms denoted as $S\_n$. It is known that $a\_1 + a\_5 + a\_9 = 27$. Find $a\_5$ and $S\_9$.

Answer: $81$


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

Given the function \( f(x)=\{\begin{array}{ll}x+\frac{1}{2} & 0 \leqslant x \leqslant \frac{1}{2}, \\ 2(1-x) & \frac{1}{2}<x \leqslant 1,\end{array} \), define \( f_{n}(x)=\underbrace{f(f(\cdots f}_{n \uparrow 1}(x) \cdots)), n \in \mathbf{N}^{*} \). Find the value of \( f_{2006}\left(\frac{2}{15}\right) \).

Answer: $\frac{19}{30}$


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

Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $\pi < \alpha < \frac{3}{2}\pi$.

$(1)$ Find the value of $\cos \alpha + \sin \alpha$.

$(2)$ Find the value of $\frac{\sin (\alpha-3\pi)+\cos (\pi-\alpha)+\sin ( \frac{3\pi}{2}-\alpha)-2\cos ( \frac{\pi}{2}+\alpha)}{\sin (-\alpha)+\cos (\pi+\alpha)}$.

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


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

Triangle \(ABC\) is isosceles with \(AB = AC\) and \(BC = 65 \, \text{cm}\). \(P\) is a point on \(BC\) such that the perpendicular distances from \(P\) to \(AB\) and \(AC\) are \(24 \, \text{cm}\) and \(36 \, \text{cm}\), respectively. Find the area of \(\triangle ABC\).

Answer: $2535$


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

Given that $\sin \alpha$ is a root of the equation $5x^{2}-7x-6=0$, calculate the value of $\frac {\sin (-\alpha- \frac {3}{2}\pi)\cdot \sin ( \frac {3}{2}\pi-\alpha)\cdot \tan ^{2}(2\pi-\alpha)}{\cos ( \frac {\pi}{2}-\alpha)\cdot \cos ( \frac {\pi}{2}+\alpha)\cdot \cos ^{2}(\pi-\alpha)}$.

Answer: $\frac{25}{16}$


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>.

If \( AC = 1.5 \, \text{cm} \) and \( AD = 4 \, \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle DBC \)?

Answer: $3/5$


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>.

We say that an ordered pair  $(a,b)$  of positive integers with  $a>b$  is square-ish if both  $a+b$  and  $a-b$  are perfect squares. For example,  $(17,8)$  is square-ish because  $17+8=25$  and  $17-8=9$  are both perfect squares. How many square-ish pairs  $(a,b)$  with  $a+b<100$  are there?

*Proposed by Nathan Xiong*

Answer: $16$


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

Given the function $f(x)=\left\{\begin{array}{l}{x^2}-2ax+8, x\leq 1\\ x+\frac{4}{x}+2a, x>1\end{array}\right.$, if the minimum value of $f(x)$ is $f(1)$, find the value of the real number $a$.

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


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

Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, their sums of the first $n$ terms are denoted as $S_n$ and $T_n$, respectively. If $\frac {S_{n}}{T_{n}}= \frac {n}{n+7}$, then calculate $\frac {a_{7}}{b_{7}}$.

Answer: $\frac{13}{20}$


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

Given that \(a, b, c, d\) are within the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin a + \sin b + \sin c + \sin d = 1\), and \(\cos 2a + \cos 2b + \cos 2c + \cos 2d \geq \frac{10}{3}\), what is the maximum value of \(a\)? \(\quad\).

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


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

In trapezoid \( KLMN \), the lengths of the bases are \( KN = 25 \), \( LM = 15 \), and the lengths of the legs are \( KL = 6 \), \( MN = 8 \). Find the length of the segment connecting the midpoints of the bases.

Answer: $20$


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

Calculate the definite integral:

$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$

Answer: $\frac{5\pi}{64}$


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

Given a circle $O: x^{2}+y^{2}=4$.
$(1)$ A tangent line is drawn from point $P(2,1)$ to circle $O$, find the equation of the tangent line $l$;
$(2)$ Let $A$ and $B$ be the points where circle $O$ intersects the positive $x$-axis and positive $y$-axis, respectively. A moving point $Q$ satisfies $QA=\sqrt{2}QB$. Is the locus of the moving point $Q$ intersecting circle $O$ at two points? If yes, find the length of the common chord; if not, explain the reason.

Answer: $\frac{8\sqrt{5}}{5}$


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>.

Simplify first, then evaluate: $(1-\frac{m}{{m-3}})\div \frac{{{m^2}-3m}}{{{m^2}-6m+9}}$, where $m=4\sqrt{3}$.

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


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

Given the task of selecting 10 individuals to participate in a quality education seminar from 7 different schools, with the condition that at least one person must be chosen from each school, determine the total number of possible allocation schemes.

Answer: $84$


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

At the mountain hut, the coach said, "If we continue at this comfortable pace of $4 \mathrm{~km}$ per hour, we will arrive at the station 45 minutes after the departure of our train."

Then he pointed to a group that had just passed us: "They are using poles, and therefore achieve an average speed of $6 \mathrm{~km}$ per hour. They will arrive at the station half an hour before the departure of our train."

How far was the station from the mountain hut?

Answer: $15$


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

Given that $x$, $y$, $z$ are positive real numbers, find the maximum value of $\dfrac{xy+yz}{x^{2}+y^{2}+z^{2}}$.

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


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>.

Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count while descending with a policeman on a stationary escalator?

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>.

Given the sequence $1, 4, 9, -16, 25, -36,\ldots$, whose $n$th term is $(-1)^{n+1}\cdot n^2$, find the average of the first $100$ terms of the sequence.

Answer: $-50.5$


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 $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and line $l$ passes through points $(a, 0)$ and $(0, b)$. The sum of the distances from $F_1$ and $F_2$ to line $l$ is $\frac{4c}{5}$. Determine the eccentricity of the hyperbola.

Answer: $\frac{5\sqrt{21}}{21}$


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>.

The angle can be represented by the two uppercase letters on its sides and the vertex letter. The angle in the diagram $\angle A O B$ symbol ("$\angle$" represents angle) can also be represented by $\angle O$ (when there is only one angle). In the triangle $\mathrm{ABC}$ below, given $\angle B A O = \angle C A O$, $\angle C B O = \angle A B O$, $\angle A C O = \angle B C O$, and $\angle A O C = 110^{\circ}$, find $\angle C B O =$.

Answer: $20$


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

What is the largest power of 2 by which the number \(10^{10} - 2^{10}\) is divisible?

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>.

Assume that $p$, $q$, $r$, and $s$ are positive integers such that $p^3 = q^2$, $r^5 = s^4$, and $r - p = 31$. Determine $s - q$.

Answer: $-2351$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \overrightarrow{OP}+ \overrightarrow{OQ}|$.

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


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>.

Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?

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>.

Six students sign up for three different intellectual competition events. How many different registration methods are there under the following conditions? (Not all six students must participate)

(1) Each person participates in exactly one event, with no limit on the number of people per event;

(2) Each event is limited to one person, and each person can participate in at most one event;

(3) Each event is limited to one person, but there is no limit on the number of events a person can participate in.

Answer: $216$


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

Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods.

Answer: $36$


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

Given $(2x-1)^{5}=a\_{0}+a\_{1}x+a\_{2}x^{2}+…+a\_{5}x^{5}$, find the values of:

(1) $a\_{0}+a\_{1}+a\_{2}+a\_{3}+a\_{4}$,

(2) $|a\_{0}|+|a\_{1}|+|a\_{2}|+|a\_{3}|+|a\_{4}|+|a\_{5}|$,

(3) $a\_{1}+a\_{3}+a\_{5}$,

(4) $(a\_{0}+a\_{2}+a\_{4})^{2}-(a\_{1}+a\_{3}+a\_{5})^{2}$.

Answer: $-243$


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

In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively.

Answer: $10$


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

Given the function \( f(x) = 5(x+1)^{2} + \frac{a}{(x+1)^{5}} \) for \( a > 0 \), find the minimum value of \( a \) such that \( f(x) \geqslant 24 \) when \( x \geqslant 0 \).

Answer: $2 \sqrt{\left(\frac{24}{7}\right)^7}$


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 number  $\overline{abcd}$ , where  $a$ ,  $b$ ,  $c$ , and  $d$ , represent the digits of  $\overline{abcd}$ , find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where  $a$ ,  $b$ ,  $c$ , and  $d$  are distinct
Answer$\overline{abcd}=1089$ , minimum value of  $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$

Answer: $60.5$


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 $\cos (\frac{\pi}{4}-\frac{\theta}{2})=\frac{2}{3}$, find the value of $\sin \theta$.

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>.

When $1 + 3 + 3^2 + \cdots + 3^{1004}$ is divided by $500$, what is the remainder?

Answer: $121$


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

A six-digit number can be tripled by reducing the first digit by three and appending a three at the end. What is this number?

Answer: $428571$


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

Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)?

Answer: $1440$


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

How many integers between 1000 and 3000 have all three of the numbers 18, 24, and 36 as factors?

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>.

A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes?

Answer: $40$


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

Given $(1+x) + (1+x)^2 + \ldots + (1+x)^5 = a_0 + a_1(1-x) + a_2(1-x)^2 + \ldots + a_5(1-x)^5$, calculate the value of $a_1 + a_2 + a_3 + a_4 + a_5$.

Answer: $-57$


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

Let the positive divisors of \( 2014^2 \) be \( d_{1}, d_{2}, \cdots, d_{k} \). Then
$$
\frac{1}{d_{1}+2014}+\frac{1}{d_{2}+2014}+\cdots+\frac{1}{d_{k}+2014} =
$$

Answer: $\frac{27}{4028}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 sum of the distances from point $P$ to two points $(0,-\sqrt{3})$ and $(0,\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be $C$.
(I) Write the equation of $C$;
(II) Given that the line $y=kx+1$ intersects $C$ at points $A$ and $B$, for what value of $k$ is $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?

Answer: $\frac{4\sqrt{65}}{17}$


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>.

Four people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Calculate the probability that no two adjacent people stand up.

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


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

On November 15, a dodgeball tournament took place. In each game, two teams competed. A win was awarded 15 points, a tie 11 points, and a loss 0 points. Each team played against every other team once. At the end of the tournament, the total number of points accumulated was 1151. How many teams participated?

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>.

How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement?

Answer: $55440$


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

For the "Skillful Hands" club, Pavel needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Pavel took a wire piece of 10 meters and was able to cut only 15 pieces of the required length. Then, Pavel took a piece that was 40 centimeters longer, but it was also enough for only 15 pieces. What length should the pieces be? Express the answer in centimeters.

Answer: $66$


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

Factorize \( n^{5} - 5n^{3} + 4n \). What can be concluded in terms of divisibility?

Answer: $120$


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

Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be?

Answer: $\frac{30!}{(10!)^3}$


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

Given triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $\overrightarrow{BC} \cdot \overrightarrow{BA} + 2\overrightarrow{AC} \cdot \overrightarrow{AB} = \overrightarrow{CA} \cdot \overrightarrow{CB}$.
$(1)$ Find the value of $\frac{{\sin A}}{{\sin C}}$;
$(2)$ If $2a \cdot \cos C = 2b - c$, find the value of $\cos B$.

Answer: $\frac{3\sqrt{2} - \sqrt{10}}{8}$


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>.

Consider the sequence \(\left\{a_{n}\right\}\) defined by \(a_{0}=\frac{1}{2}\) and \(a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}\) for \(n = 0, 1, \ldots\). Find the integer \(k\) such that \(a_{k} < 1 < a_{k+1}\).

Answer: $2023$


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

Let \( A \), \( B \), and \( C \) be pairwise independent events with equal probabilities, and \( A \cap B \cap C = \varnothing \). Find the maximum possible value for the probability \( \mathrm{P}(A) \).

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 $\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\frac {\sin (2A+B)}{\sin A}=2+2\cos (A+B)$.
(I) Find the value of $\frac {b}{a}$;
(II) If $a=1$ and $c= \sqrt {7}$, find the area of $\triangle ABC$.

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>.

Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new trailer homes were added five years ago?

Answer: $20$


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

A full container holds 150 watermelons and melons with a total value of 24,000 rubles. The total value of all watermelons is equal to the total value of all melons. How much does one watermelon cost in rubles, given that the container can hold 120 melons (without watermelons) or 160 watermelons (without melons)?

Answer: $100$


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

Find the degree measure of the angle

$$
\delta=\arccos \left(\left(\sin 2541^{\circ}+\sin 2542^{\circ}+\cdots+\sin 6141^{\circ}\right)^{\cos 2520^{\circ}}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$

Answer: $69$


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

$AB$ and $AC$ are tangents to a circle, $\angle BAC = 60^{\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$.

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>.

Let \((x_1, y_1), (x_2, y_2),\) and \((x_3, y_3)\) be three different real solutions to the system of equations:
\[
\begin{cases}
x^3 - 5xy^2 = 21 \\
y^3 - 5x^2y = 28
\end{cases}
\]
Find the value of \(\left(11 - \frac{x_1}{y_1}\right)\left(11 - \frac{x_2}{y_2}\right)\left(11 - \frac{x_3}{y_3}\right)\).

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>.

Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?

Answer: $6311/624$


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

Given the function $f(x)=3^{x}$, where $f(a+2)=27$, and the function $g(x)=λ \cdot 2^{ax}-4^{x}$ with a domain of $[0,2]$, find:

(1) The value of $a$;
(2) The value of the real number $λ$ if the maximum value of function $g(x)$ is $\frac{1}{3}$.

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


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

If $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2011} + a_{2012} > 0$, and $a_{2011} \cdot a_{2012} < 0$, determine the natural number $n$ that maximizes the sum of the first $n$ terms $S_n$.

Answer: $2011$


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

Given the function $f(x)=x^{3}+3mx^{2}+nx+m^{2}$ has an extreme value of $0$ at $x=-1$, find the value of $m+n$.

Answer: $11$


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

Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method.

Answer: $36$


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

If $M = 2098 \div 2$, $N = M \times 2$, and $X = M + N$, what is the value of $X$?

Answer: $3147$


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

1. If $A_{10}^{m} =10×9×…×5$, then $m=$ ______.
2. The number of ways for A, B, C, and D to take turns reading the same book, with A reading first, is ______.
3. If five boys and two girls are to be arranged in a row for a photo, with boy A required to stand in the middle and the two girls required to stand next to each other, the total number of arrangements that meet these conditions is ______.

Answer: $192$


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

Given a three-digit natural number \(abc\) with digits \(a\), \(b\), and \(c\) in the hundreds, tens, and units places respectively, and where \(a > b\) and \(c > b\), with \(a, b, c \in \{0, 2, 3, 4, 5\}\) and \(a, b, c\) distinct, calculate the number of concave numbers.

Answer: $20$


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

Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, let vectors $\overrightarrow{m}=(\cos B,\sin C)$ and $\overrightarrow{n}=(\cos C,-\sin B)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=\frac{1}{2}$.

$(1)$ Find the magnitude of the interior angle $A$;

$(2)$ If the radius of the circumcircle of $\triangle ABC$ is $R=2$, find the maximum value of the perimeter $L$ of $\triangle ABC$.

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


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

Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction.

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


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

Find the distance from vertex \( A_4 \) to the face \( A_1 A_2 A_3 \).

\( A_1(2, 1, 4) \)

\( A_2(-1, 5, -2) \)

\( A_3(-7, -3, 2) \)

\( A_4(-6, -3, 6) \)

Answer: $5 \sqrt{\frac{2}{11}}$


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

Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,100\pi).$

Answer: $100$


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

Given that the two distinct square roots of a positive number $x$ are $a+3$ and $2a-15$, and $\sqrt [3] {x+y-2}=4$. Find the value of $x-2y+2$.

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>.

Given the equations \(3x - 4y - 2z = 0\) and \(x - 2y + 5z = 0\) with \(z \neq 0\), find the numerical value of \(\frac{2x^2 - xy}{y^2 + 4z^2}\).

Answer: $\frac{744}{305}$


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

Represent the number 36 as the product of three whole number factors, the sum of which is equal to 4. What is the smallest of these factors?

Answer: $-4$


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

The area of a quadrilateral is 3 cm², and the lengths of its diagonals are 6 cm and 2 cm. Find the angle between the diagonals.

Answer: $30$


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

Given that the base of the tetrahedron \( S-ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). Assume that the points \( S, A, B, \) and \( C \) lie on a spherical surface with center \( O \). Find the distance from point \( O \) to the plane \( ABC \).

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>.

If $a\in[0,\pi]$, $\beta\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$, $\lambda\in\mathbb{R}$, and $\left(\alpha -\frac{\pi}{2}\right)^{3}-\cos \alpha -2\lambda =0$, $4\beta^{3}+\sin \beta \cos \beta +\lambda =0$, then the value of $\cos \left(\frac{\alpha}{2}+\beta \right)$ is ______.

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>.

If $x\%$ of five-digit numbers have at least one repeated digit, then what is $x$? Express your answer as a decimal to the nearest tenth.

Answer: $69.8$


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>.

Suppose $w$ is a complex number such that $w^3 = 64 - 48i$. Find $|w|$.

Answer: $2\sqrt[3]{10}$


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

A store sells chalk in three types of packages: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 2:3:6. After some packages of regular and unusual chalk—totaling no more than 100 packages—were delivered to the store, and $40\%$ of the excellent chalk packages were sold, the ratio changed to 5:7:4. How many total chalk packages were sold in the store?

Answer: $24$


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

What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$?

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>.

Consider a board on  $2013 \times 2013$  squares, what is the maximum number of chess knights that can be placed so that no  $2$  attack each other?

Answer: $2026085$


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

In the cells of an $8 \times 8$ chessboard, there are 8 white and 8 black pieces arranged such that no two pieces are in the same cell. Additionally, no pieces of the same color are in the same row or column. For each white piece, the distance to the black piece in the same column is calculated. What is the maximum value that the sum of these distances can take? The distance between the pieces is considered to be the distance between the centers of the cells they occupy.

Answer: $32$


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

A shopkeeper set up incorrect scales in his shop, where one side of the balance beam is longer than the other. During one weighing, 3 cans balanced with 8 packets, and during another, 1 packet balanced with 6 cans. Given that the true weight of one can is 1 kg, how much do 8 packets weigh?

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 $-765^\circ$, convert this angle into the form $2k\pi + \alpha$ ($0 \leq \alpha < 2\pi$), where $k \in \mathbb{Z}$.

Answer: $-6\pi + \frac{7\pi}{4}$