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

Round to the nearest thousandth and then subtract 0.005: 18.48571.

Answer: $18.481$


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

A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?
A) 12
B) 15
C) 18
D) 20
E) 24

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>.

Find the product of the divisors of \(72\).

Answer: $72^6$


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>.

Suppose that  $PQ$  and  $RS$  are two chords of a circle intersecting at a point  $O$ . It is given that  $PO=3 \text{cm}$  and  $SO=4 \text{cm}$ . Moreover, the area of the triangle  $POR$  is  $7 \text{cm}^2$ . Find the area of the triangle  $QOS$ .

Answer: $112/9$


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>.

Points with integer coordinates (including zero) are called lattice points (or grid points). Find the total number of lattice points (including those on the boundary) in the region bounded by the x-axis, the line \(x=4\), and the parabola \(y=x^2\).

Answer: $35$


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

In $\triangle ABC$, $\sin ^{2}A-\sin ^{2}C=(\sin A-\sin B)\sin B$, then angle $C$ equals to $\dfrac {\pi}{6}$.

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


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

There are two rows of seats, with 11 seats in the front row and 12 seats in the back row. Now, we need to arrange for two people, A and B, to sit down. It is stipulated that the middle 3 seats of the front row cannot be occupied, and A and B cannot sit next to each other. How many different arrangements are there?

Answer: $346$


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

Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$.

Answer: $413.5$


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 geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\frac{1}{2}$. Let $\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\Pi_{n}$ maximized?

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>.

Consider the decimal function denoted by $\{ x \} = x - \lfloor x \rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\{x\} = \frac{1}{\lfloor x \rfloor}$. Express your answer as a mixed number.

Answer: $21\frac{9}{20}$


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>.

Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.

Answer: $\frac{670}{2011}$


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>.

In $\triangle ABC$, $\angle A= \frac {\pi}{3}$, $BC=3$, $AB= \sqrt {6}$, find $\angle C=$ \_\_\_\_\_\_ and $AC=$ \_\_\_\_\_\_.

Answer: $\frac{\sqrt{6} + 3\sqrt{2}}{2}$


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

Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$.

Answer: $4\pi$


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

Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars.

Answer: $\frac{300}{13}$


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>.

Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men.

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 $\alpha \in \left(0,\pi \right)$, $tan2\alpha=\frac{sin\alpha}{2+cos\alpha}$, find the value of $\ tan \alpha$.

Answer: $-\sqrt{15}$


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>.

Evaluate: \( \frac {\tan 150^{\circ} \cos (-210^{\circ}) \sin (-420^{\circ})}{\sin 1050^{\circ} \cos (-600^{\circ})} \).

Answer: $-\sqrt{3}$


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

If $a^{2}-4a+3=0$, find the value of $\frac{9-3a}{2a-4} \div (a+2-\frac{5}{a-2})$ .

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


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

(1) Given that $x < 3$, find the maximum value of $f(x) = \frac{4}{x - 3} + x$;
(2) Given that $x, y \in \mathbb{R}^+$ and $x + y = 4$, find the minimum value of $\frac{1}{x} + \frac{3}{y}$.

Answer: $1 + \frac{\sqrt{3}}{2}$


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>.

The constant term in the expansion of the binomial \\((x \sqrt {x}- \dfrac {1}{x})^{5}\\) is \_\_\_\_\_\_ . (Answer with a number)

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>.

Let $p$, $q$, and $r$ be the roots of $x^3 - 2x^2 - x + 3 = 0$. Find $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$.

Answer: $-3$


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

In the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; (a > b > 0)$, a line with slope $k(k > 0)$ intersects the ellipse at the left vertex $A$ and another point $B$. The projection of point $B$ on the $x$-axis is exactly the right focus $F$. If the eccentricity of the ellipse $e= \frac{1}{3}$, then the value of $k$ is _____.

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


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

Three people, A, B, and C, stand on a staircase with 7 steps. If each step can accommodate at most 2 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is.

Answer: $336$


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

In a box, there are two red balls, two yellow balls, and two blue balls. If a ball is randomly drawn from the box, at least how many balls need to be drawn to ensure getting balls of the same color? If one ball is drawn at a time without replacement until balls of the same color are obtained, let $X$ be the number of different colors of balls drawn during this process. Find $E(X)=$____.

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


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

In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$?

Answer: $59$


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

Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center of mass) of $\triangle ABC$?

Answer: $18\pi$


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

Draw a perpendicular line from the left focus $F_1$ of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ to the $x$-axis meeting the ellipse at point $P$, and let $F_2$ be the right focus. If $\angle F_{1}PF_{2}=60^{\circ}$, calculate the eccentricity of the ellipse.

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>.

Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$.

Answer: $54.6428571$


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

The mathematical giant Euler in history was the first to represent polynomials in terms of $x$ using the notation $f(x)$. For example, $f(x) = x^2 + 3x - 5$, and the value of the polynomial when $x$ equals a certain number is denoted by $f(\text{certain number})$. For example, when $x = -1$, the value of the polynomial $x^2 + 3x - 5$ is denoted as $f(-1) = (-1)^2 + 3 \times (-1) - 5 = -7$. Given $g(x) = -2x^2 - 3x + 1$, find the values of $g(-1)$ and $g(-2)$ respectively.

Answer: $-1$


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

When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young.

Answer: $18$


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

Simplify and find the value of:
$(1)$ If $a=2$ and $b=-1$, find the value of $(3{a^2}b+\frac{1}{4}a{b^2})-(\frac{3}{4}a{b^2}-{a^2}b)$.
$(2)$ If the value of the algebraic expression $(2x^{2}+ax-y+6)-(2bx^{2}-3x+5y-1)$ is independent of the variable $x$, find the value of the algebraic expression $5ab^{2}-[a^{2}b+2(a^{2}b-3ab^{2})]$.

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>.

An urn initially contains two red balls and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation five times: he draws a ball from the urn at random and then takes a ball of the same color from the box and adds those two matching balls to the urn. After the five iterations, the urn contains eight balls. What is the probability that the urn contains three red balls and five blue balls?
A) $\frac{1}{10}$
B) $\frac{1}{21}$
C) $\frac{4}{21}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$

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


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

Given $0 \leq x_0 < 1$, for all integers $n > 0$, let
$$
x_n = \begin{cases}
2x_{n-1}, & \text{if } 2x_{n-1} < 1,\\
2x_{n-1} - 1, & \text{if } 2x_{n-1} \geq 1.
\end{cases}
$$
Find the number of initial values of $x_0$ such that $x_0 = x_6$.

Answer: $64$


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 that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$.

Answer: $-170$


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

Given $\cos \left(\alpha- \frac {\beta}{2}\right)=- \frac {1}{9}$ and $\sin \left( \frac {\alpha}{2}-\beta\right)= \frac {2}{3}$, with $0 < \beta < \frac {\pi}{2} < \alpha < \pi$, find $\sin \frac {\alpha+\beta}{2}=$ ______.

Answer: $\frac {22}{27}$


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>.

For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000$ with $S(n)$ even. Find $|a-b|.$

Answer: $54$


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

For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4?

Answer: $34$


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

Given an ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{1}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse is tangent to the line $\sqrt{7}x - \sqrt{5}y + 12 = 0$.
(1) Find the equation of ellipse C;
(2) Let A(-4, 0), and a line l, which does not coincide with the x-axis, passes through point R(3, 0) and intersects ellipse C at points P and Q. Lines AP and AQ intersect the line $x = \frac{16}{3}$ at points M and N, respectively. If the slopes of lines MR and NR are $k_1$ and $k_2$ respectively, is $k_1k_2$ a constant? If so, find the constant. If not, explain why.

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


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

First, find the derivative of the following functions and calculate the derivative at \\(x=\pi\\).

\\((1) f(x)=(1+\sin x)(1-4x)\\)    \\((2) f(x)=\ln (x+1)-\dfrac{x}{x+1}\\).

Answer: $\dfrac{\pi}{(\pi+1)^{2}}$


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 the fractional equation about $x$: $\frac{x+m}{x+2}-\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\left\{\begin{array}{l}{m-6y>2}\\{y-4\leq 3y+4}\end{array}\right.$ has exactly four integer solutions, then the sum of the integers $m$ that satisfy the conditions is ____.

Answer: $-2$


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

Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $5x$ inches. Square C has side lengths each measuring $2x$ inches. What is the ratio of the area of Square A to the area of Square B, and what is the ratio of the area of Square C to the area of Square B? Express each answer as a common fraction.

Answer: $\frac{4}{25}$


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>.

Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.

Answer: $1 + 3\sqrt{2}$


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

Given: The lengths of the three sides of a triangle, $a$, $b$, and $c$, are integers, and $a \leq b < c$, where $b = 5$. Calculate the number of such triangles.

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>.

How many groups of integer solutions are there for the equation $xyz = 2009$?

Answer: $72$


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

If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls.

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 vectors $\overrightarrow{\alpha}$, $\overrightarrow{\beta}$, $\overrightarrow{\gamma}$ satisfy $|\overrightarrow{\alpha}|=1$, $|\overrightarrow{\alpha}-\overrightarrow{\beta}|=|\overrightarrow{\beta}|$, $(\overrightarrow{\alpha}-\overrightarrow{\gamma}) \cdot (\overrightarrow{\beta}-\overrightarrow{\gamma})=0$. If for every determined $\overrightarrow{\beta}$, the maximum and minimum values of $|\overrightarrow{\gamma}|$ are $m$ and $n$ respectively, then for any $\overrightarrow{\beta}$, the minimum value of $m-n$ is \_\_\_\_\_\_\_\_.

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


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

In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\).

Answer: $\frac{11}{15}$


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>.

The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her school. Betsy had 15 pieces of candy left over. Find the least number of pieces of candy the Tasty Candy Company could have placed in each one-pound bag.

Answer: $302$


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

The constant term in the expansion of the binomial $\left(\frac{1}{\sqrt{x}} - x^2\right)^{10}$ is ______.

Answer: $45$


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

Consider four functions, labelled from (6) to (9). The domain of function (7) is $\{-6, -5, -4, -3, -2, -1, 0, 1\}$. Determine the product of the labels of the functions that are invertible. The graphs are as follows:
- Function (6): $y = x^3 - 3x$
- Function (7): A set of discrete points: $\{(-6, 3), (-5, -5), (-4, 1), (-3, 0), (-2, -2), (-1, 4), (0, -4), (1, 2)\}$
- Function (8): $y = \tan(x)$, defined over $(-\frac{\pi}{2}, \frac{\pi}{2})$
- Function (9): $y = \frac{5}{x}$, defined over all real numbers except $x = 0$

Answer: $504$


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

Given the function $y=\cos (2x-\frac{\pi }{4})$, determine the horizontal translation of the graph of the function $y=\sin 2x$.

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


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>.

The Great Eighteen Hockey League is divided into two divisions, with nine teams in each division. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled?

Answer: $378$


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

Guangcai Kindergarten has a total of 180 books, of which 40% are given to the senior class. The remaining books are divided between the junior and middle classes in a ratio of 4:5. How many books does each of the junior and middle classes get?

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>.

There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school  can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.

Answer: $\binom{2003}{11}$


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>.

Point P is any point on the surface of the circumscribed sphere of a cube ABCD-A1B1C1D1 with edge length 2. What is the maximum volume of the tetrahedron P-ABCD?

Answer: $\frac{4(1+\sqrt{3})}{3}$


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>.

Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed?

Answer: $14$


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

An Englishman owns a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250,000 rubles per hectare. It is known that 1 acre = 4840 square yards, 1 yard = 0.9144 meters, and 1 hectare = 10,000 square meters. Calculate how much the Englishman will earn from the sale.

Answer: $303514$


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

We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)?

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>.

Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage?

Answer: $33.33$


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

Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. It is known that $a=5$ and $\sin A= \frac{\sqrt{5}}{5}$.
(1) If the area of triangle $ABC$ is $\sqrt{5}$, find the minimum value of the perimeter $l$.
(2) If $\cos B= \frac{3}{5}$, find the value of side $c$.

Answer: $11$


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

In the expansion of $(x^{4}+y^{2}+\frac{1}{2xy})^{7}$, the constant term is ______.

Answer: $\frac{105}{16}$


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>.

Consider the set $\{0.34,0.304,0.034,0.43\}$. The sum of the smallest and largest numbers in the set is

Answer: $0.464$


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

$2016$  bugs are sitting in different places of  $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts, which can be reached by bugs?

Answer: $1008^2$


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

A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \_\_\_\_\_\_.

Answer: $3.2$


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>.

What is the maximum possible value of  $k$  for which  $2013$  can be written as a sum of  $k$  consecutive positive integers?

Answer: $61$


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

Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value.

Answer: $3456^2$


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>.

For every positive integer \( n \), define \( a_{n} \) as the last digit of the sum of the digits of the number formed by writing "2005" \( n \) times consecutively. For example, \(\mathrm{a}_{1}=7\) and \(\mathrm{a}_{2}=4\).

a) What are the positive integers \( n \) such that \( a_{n}=0 \)?

b) Calculate \( a_{1}+a_{2}+\cdots+a_{2005} \).

Answer: $9025$


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

Among the four students A, B, C, and D participating in competitions in mathematics, writing, and English, each subject must have at least one participant (and each participant can only choose one subject). If students A and B cannot participate in the same competition, the total number of different participation schemes is _____. (Answer with a number)

Answer: $30$


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

The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere?

Answer: $100\pi$


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

Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$.

Answer: $3\sqrt{2}$


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>.

Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.

Answer: $36$


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

In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is the same. After several trials of drawing balls, Xiaogang found that the probability of drawing a red or black ball is stable at 15% and 45%, respectively. What could be the possible number of white balls in the bag?

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>.

Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 times. Afterward, he pours all that is left in the bottle into the jug. What is the proportion of orange juice in the resulting drink in the jug? If necessary, round the answer to the nearest 0.01.

Answer: $0.05$


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

In the polar coordinate system, find the length of the segment cut by the curve $\rho=1$ from the line $\rho\sin\theta-\rho\cos\theta=1$.

Answer: $\sqrt{2}$


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

For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal?

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 5 balls with 2 identical black balls and one each of red, white, and blue, calculate the number of different arrangements of 4 balls in a row.

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>.

Let \\(α\\) and \\(β\\) be in \\((0,π)\\), and \\(\sin(α+β) = \frac{5}{13}\\), \\(\tan \frac{α}{2} = \frac{1}{2}\\). Find the value of \\(\cos β\\).

Answer: $-\frac{16}{65}$


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

The maximum distance from a point on the ellipse $$\frac {x^{2}}{16}+ \frac {y^{2}}{4}=1$$ to the line $$x+2y- \sqrt {2}=0$$ is \_\_\_\_\_\_.

Answer: $\sqrt{10}$


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

Determine how many regions of space are divided by:
a) The six planes of a cube's faces.
b) The four planes of a tetrahedron's faces.

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>.

How many subsets of the set $\{1, 2, 3, 4, 5, 6\}$ must contain the number 6 and at least one of the numbers 1 or 2?

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 sum of all two-digit positive integers whose squares end with the digits 25?

Answer: $495$


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

If $P = 3012 \div 4$, $Q = P \div 2$, and $Y = P - Q$, then what is the value of $Y$?

Answer: $376.5$


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 the numbers $1, 2, 3, 4, 5, 6$ are randomly arranged in a row, represented as $a, b, c, d, e, f$, what is the probability that the number $a b c + d e f$ is odd?

Answer: $1/10$


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>.

Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\circ$.

Answer: $\sqrt{289 + 120\sqrt{3}}$


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

Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers?

Answer: $23$


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

Nine positive integers  $a_1,a_2,...,a_9$  have their last  $2$ -digit part equal to  $11,12,13,14,15,16,17,18$  and  $19$  respectively. Find the last  $2$ -digit part of the sum of their squares.

Answer: $85$


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

Given the function $f\left( x \right)=\frac{{{x}^{2}}}{1+{{x}^{2}}}$.

(1) Find the values of $f\left( 2 \right)+f\left( \frac{1}{2} \right),f\left( 3 \right)+f\left( \frac{1}{3} \right),f\left( 4 \right)+f\left( \frac{1}{4} \right)$ and conjecture a general conclusion (proof is not required).

(2) Evaluate: $2f\left(2\right)+2f\left(3\right)+⋯+2f\left(2017\right)+f\left( \frac{1}{2}\right)+f\left( \frac{1}{3}\right)+⋯+f\left( \frac{1}{2017}\right)+ \frac{1}{{2}^{2}}f\left(2\right)+ \frac{1}{{3}^{2}}f\left(3\right)+⋯ \frac{1}{{2017}^{2}}f\left(2017\right)$

Answer: $4032$


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

A certain company has two research and development teams, Team A and Team B. The probability of success for developing a new product by Team A is $\frac{4}{5}$, and for Team B is $\frac{3}{4}$. Team A is assigned to develop a new product $A$, and Team B is assigned to develop a new product $B$. It is assumed that the research and development of Teams A and B are independent of each other.
$(1)$ Find the probability that exactly one new product is successfully developed.
$(2)$ If the development of new product $A$ is successful, the company will make a profit of $150$ thousand dollars, otherwise it will incur a loss of $60$ thousand dollars. If the development of new product $B$ is successful, the company will make a profit of $120$ thousand dollars, otherwise it will incur a loss of $40$ thousand dollars. Find the probability distribution and the mathematical expectation $E(\xi)$ of the company's profit (in thousand dollars).

Answer: $188$


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

Given \( \alpha, \beta \in (0, \pi) \), and \( \tan \alpha, \tan \beta \) are the roots of the equation \( x^{2} + 3x + 1 = 0 \), find the value of \( \cos(\alpha - \beta) \).

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


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

Given $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = 6,$ find
\[\begin{vmatrix} p & 9p + 4q \\ r & 9r + 4s \end{vmatrix}.\]

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>.

Compute $63 \times 57$ in your head.

Answer: $3591$


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

A function $f(x) = a \cos ωx + b \sin ωx (ω > 0)$ has a minimum positive period of $\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \frac{π}{6}$.

1. Find the values of $a$, $b$, and $ω$.
2. If $\frac{π}{4} < x < \frac{3π}{4}$ and $f(x + \frac{π}{6}) = \frac{4}{3}$, find the value of $f(\frac{x}{2} + \frac{π}{6})$.

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


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>.

Draw five lines \( l_1, l_2, \cdots, l_5 \) on a plane such that no two lines are parallel and no three lines pass through the same point.
(1) How many intersection points are there in total among these five lines? How many intersection points are there on each line? How many line segments are there among these five lines?
(2) Considering these line segments as sides, what is the maximum number of isosceles triangles that can be formed? Please briefly explain the reasoning and draw the corresponding diagram.

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 that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$.

Answer: $91$


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

Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$

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>.

The real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_{k} - y_{k+1} \right| \).

Answer: $2000$


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

The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area.

Answer: $4\pi$


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

The inclination angle of the line $x- \sqrt {3}y+3=0$ is \_\_\_\_\_\_.

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>.

A and B are running on a circular track at their respective constant speeds. If both start running from point A in opposite directions, and after their first meeting, B takes another 8 minutes to return to the starting point. Given that A takes 6 minutes to complete a lap, how many minutes does it take for B to complete a lap?

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>.

If $\sqrt[3]{0.3}\approx 0.6694$ and $\sqrt[3]{3}\approx 1.442$, then $\sqrt[3]{300}\approx$____.

Answer: $6.694$


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

Determine how many divisors of \(9!\) are multiples of 3.

Answer: $128$


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

Given triangle $ABC$ where $AB=6$, $\angle A=30^\circ$, and $\angle B=120^\circ$, find the area of $\triangle ABC$.

Answer: $9\sqrt{3}$


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

How many minutes are needed at least to finish these tasks: washing rice for 2 minutes, cooking porridge for 10 minutes, washing vegetables for 3 minutes, and chopping vegetables for 5 minutes.

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>.

Consider a two-digit number in base 12 represented by $AB_{12}$, where $A$ and $B$ are duodecimal digits (0 to 11), and $A \neq B$. When this number is reversed to $BA_{12}$, under what condition is a particular prime number a necessary factor of the difference $AB_{12} - BA_{12}$?

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>.

Let $(2x+1)^6 = a_0x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6$, which is an identity in $x$ (i.e., it holds for any $x$). Try to find the values of the following three expressions:
(1) $a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6$;
(2) $a_1 + a_3 + a_5$;
(3) $a_2 + a_4$.

Answer: $300$


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

Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$.

Answer: $\frac{47\sqrt{3}}{4}$


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 \\(\sin \theta + \cos \theta = \frac{3}{4}\\), where \\(\theta\\) is an angle of a triangle, the value of \\(\sin \theta - \cos \theta\\) is \_\_\_\_\_.

Answer: $\frac{\sqrt{23}}{4}$


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>.

$ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\sin \angle GAC$.

Answer: $\frac{3}{\sqrt{14}}$


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

Evaluate the expression given by $$2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8}}}.$$

Answer: $\frac{137}{52}$


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

A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers  $n < 100$  going through and "flipping" every  $n$ th locker. How many lockers are open after this process?

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>.

Eighty bricks, each measuring $3''\times9''\times18''$, are to be stacked one on top of another to form a tower 80 bricks tall. Each brick can be oriented so it contributes $3''$, $9''$, or $18''$ to the total height of the tower. How many different tower heights can be achieved using all eighty of the bricks?

Answer: $401$


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

$(1)$ Calculate: $\sqrt{9}+2\sin30{}°-(π-3){}°$;<br/>$(2)$ Solve the equation: $\left(2x-3\right)^{2}=2\left(2x-3\right)$.

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


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

What is the base five product of the numbers $203_{5}$ and $14_{5}$?

Answer: $3402_5$


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 $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda \cdot \overrightarrow{BC}$, determine the value of the real number $\lambda$.

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


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

The minimum value of the function $y=|x-1|$ is 0, the minimum value of the function $y=|x-1|+|x-2|$ is 1, and the minimum value of the function $y=|x-1|+|x-2|+|x-3|$ is 2. What is the minimum value of the function $y=|x-1|+|x-2|+\ldots+|x-10|$?

Answer: $25$


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

A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis.

Answer: $6.4$


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

Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).

\( A_{1}(1, 0, 2) \)

\( A_{2}(1, 2, -1) \)

\( A_{3}(2, -2, 1) \)

\( A_{4}(2, 1, 0) \)

Answer: $\sqrt{\frac{7}{11}}$


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>.

Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$.

Answer: $\frac{5}{9}$


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{2\sqrt{3}\sin A}{3\sin C}$.

(1) Find the value of $b$;

(2) If $B = \frac{\pi}{3}$, find the maximum area of triangle $ABC$.

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


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

Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used?

Answer: $430$


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

In the sequence $\{a_n\}$, $a_n$ is the closest positive integer to $\sqrt{n}$ ($n \in \mathbb{N}^*$). Compute the sum $\sum_{i=1}^{100}\frac{1}{a_i} = \_\_\_\_\_\_\_\_$.

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>.

Given $\tan (\alpha +\beta )=7$ and $\tan (\alpha -\beta )=1$, find the value of $\tan 2\alpha$.

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


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>.

Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane that contains this given edge.

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>.

How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?

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>.

Consider a $3 \times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring?

Answer: $88$


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

In a pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second card is drawn randomly. Let the sum of the numbers on the cards drawn the first and second times be $\xi$.

$(1)$ When is the probability of $\xi$ the greatest? Please explain your reasoning.

$(2)$ Calculate the expected value $E(\xi)$ of the random variable $\xi$.

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


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>.

Calculate $7 \cdot 9\frac{2}{5}$.

Answer: $65\frac{4}{5}$


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>.

Calculate using a simple method:
$(1)100.2\times 99.8$;
$(2)103^{2}$.

Answer: $10609$


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

Calculate $3(72+76+80+84+88+92+96+100+104+108)$.

Answer: $2700$