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

Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$.

Answer: $\frac{9\pi}{2} - 9$


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

Given that $\sin(α + \frac{π}{6}) - \cos α = \frac{1}{3}$, find the value of $2 \sin α \cos(α + \frac{π}{6})$.

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


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

There are 5 different books to be distributed among three people, with each person receiving at least 1 book and at most 2 books. Calculate the total number of different distribution methods.

Answer: $90$


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

Given the function $f(x)=\sin (\omega x+\varphi)$ with $\omega > 0$ and $|\varphi| < \frac {\pi}{2}$, the function has a minimum period of $4\pi$ and, after being shifted to the right by $\frac {2\pi}{3}$ units, becomes symmetric about the $y$-axis. Determine the value of $\varphi$.

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>.

Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.
(1) Determine the range of the real number $a$ and the expression for $S_n$;
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain why.

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


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

In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$.

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>.

Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \]

Answer: $3691$


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

If the function $f(x)$ is monotonic in its domain $(-\infty, +\infty)$, and for any real number $x$, it satisfies $f(f(x)+e^{x})=1-e$, where $e$ is the base of the natural logarithm, determine the value of $f(\ln 2)$.

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>.

In an isosceles triangle $ABC$ with $AB = AC = 6$ units and $BC = 5$ units, a point $P$ is randomly selected inside the triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than to either vertex $A$ or vertex $B$?

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


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

Given that the function $f(x)$ is defined on $\mathbb{R}$ and has a period of $4$ as an even function. When $x \in [2,4]$, $f(x) =|\log_4 \left(x-\frac{3}{2}\right)|$. Determine the value of $f\left(\frac{1}{2}\right)$.

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


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

Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Assume two of the roots of $f(x)$ are $s + 2$ and $s + 5,$ and two of the roots of $g(x)$ are $s + 4$ and $s + 8.$ Given that:
\[ f(x) - g(x) = 2s \] for all real numbers $x.$ Find $s.$

Answer: $3.6$


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

The letters \( A, J, H, S, M, E \) and the numbers \( 1, 9, 8, 9 \) are "rotated" as follows:
\begin{tabular}{rrr}
AJHSME & 1989 & \\
1. JHSMEA & 9891 & (1st rotation) \\
2. HSMEAJ & 8919 & (2nd rotation) \\
3. SMEAJH & 9198 & (3rd rotation) \\
..... & &
\end{tabular}

To make AJHSME1989 reappear, the minimum number of rotations needed is:

Answer: $12$


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

In the circle \(x^{2} + y^{2} \leq R^{2}\), the two-dimensional probability density is \(f(x, y) = C\left(R - \sqrt{x^{2} + y^{2}}\right)\); outside the circle \(f(x, y) = 0\). Find: a) the constant \(C\); b) the probability that a random point \((X, Y)\) falls within a circle of radius \(r = 1\) centered at the origin, if \(R = 2\).

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


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

Given that $\cos \alpha$ is a root of the equation $3x^2 - x - 2 = 0$ and $\alpha$ is an angle in the third quadrant, find the value of $\frac{\sin (-\alpha + \frac{3\pi}{2}) \cos (\frac{3\pi}{2} + \alpha) \tan^2 (\pi - \alpha)}{\cos (\frac{\pi}{2} + \alpha) \sin (\frac{\pi}{2} - \alpha)}$.

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


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

Let \(C\) be the circle with the equation \(x^2 - 4y - 18 = -y^2 + 6x + 26\). Find the center \((a, b)\) and radius \(r\) of the circle, and compute \(a + b + r\).

Answer: $5 + \sqrt{57}$


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

Given two circles $A:(x+4)^2+y^2=25$ and $B:(x-4)^2+y^2=1$, a moving circle $M$ is externally tangent to both fixed circles. Let the locus of the center of moving circle $M$ be curve $C$.

(I) Find the equation of curve $C$;

(II) If line $l$ intersects curve $C$ at points $P$ and $Q$, and $OP \perp OQ$. Is $\frac{1}{|OP|^2}+\frac{1}{|OQ|^2}$ a constant value? If it is, find the value; if not, explain the reason.

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


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

(1) Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.

(2) The equation of one of the asymptotes of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{9} = 1$ $(a > 0)$ is $y = \dfrac{3}{5}x$, then $a=$ ______.

(3) A tangent line to the circle $x^{2} + y^{2} = \dfrac{1}{4}a^{2}$ passing through the left focus $F$ of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ intersects the right branch of the hyperbola at point $P$. If $\overrightarrow{OE} = \dfrac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, then the eccentricity of the hyperbola is ______.

(4) A line passing through the focus $F$ of the parabola $y^{2} = 2px$ $(p > 0)$ with an inclination angle of $\dfrac{\pi}{4}$ intersects the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through point $(0,2)$, then $p=$ ______.

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


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>.

Solve for $x$: $\sqrt[4]{40x + \sqrt[4]{40x + 24}} = 24.$

Answer: $8293.8$


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>.

How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'?

Answer: $1260$


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

Determine with proof the number of positive integers  $n$  such that a convex regular polygon with  $n$  sides has interior angles whose measures, in degrees, are integers.

Answer: $22$


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

A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 88 and the floor forms a perfect square with an even side length, find the total number of tiles that cover the floor.

Answer: $1936$


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

Given non-zero vectors \\(a\\) and \\(b\\) satisfying \\(|b|=2|a|\\) and \\(a \perp (\sqrt{3}a+b)\\), find the angle between \\(a\\) and \\(b\\).

Answer: $\dfrac{5\pi}{6}$


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>.

Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}=(1+\cos ^{2} \frac {nπ}{2})a_{n}+\sin ^{2} \frac {nπ}{2}$, find the sum of the first 12 terms of the sequence.

Answer: $147$


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

Given quadrilateral $ABCD$ where $AC \perp BD$ and $AC=2$, $BD=3$, find the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{CD}$.

Answer: $- \dfrac{13}{4}$


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

A function, defined on the set of positive integers, is such that \( f(x y) = f(x) + f(y) \) for all \( x \) and \( y \). It is known that \( f(10) = 14 \) and \( f(40) = 20 \).

What is the value of \( f(500) \)?

Answer: $39$


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

Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder.

Answer: $2\pi$


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

Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$.

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>.

Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole number if necessary.

Answer: $350$


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

The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence of numbers $\{a_n\}$ is known as the "Fibonacci sequence". Determine which term in the Fibonacci sequence is represented by $$\frac { a_{ 1 }^{ 2 }+ a_{ 2 }^{ 2 }+ a_{ 3 }^{ 2 }+…+ a_{ 2015 }^{ 2 }}{a_{2015}}$$.

Answer: $2016$


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

Given the plane rectangular coordinate system $(xOy)$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C_{1}$ is $\rho = 4\cos \theta$, and the line $l$: $\begin{cases} x = 1 - \frac{2\sqrt{5}}{5}t \\ y = 1 + \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is a parameter)

(I) Find the rectangular coordinate equation of the curve $C_{1}$ and the general equation of the line $l$;

(II) If the parametric equation of the curve $C_{2}$ is $\begin{cases} x = 2\cos \alpha \\ y = \sin \alpha \end{cases}$ ($\alpha$ is a parameter), $P$ is a point on the curve $C_{1}$ with a polar angle of $\frac{\pi}{4}$, and $Q$ is a moving point on the curve $C_{2}$, find the maximum value of the distance from the midpoint $M$ of $PQ$ to the line $l$.

Answer: $\frac{\sqrt{10}}{5}$


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

Let $h(x) = x - 3$ and $k(x) = x / 4$. Also denote the inverses to these functions as $h^{-1}$ and $k^{-1}$. Compute \[h(k^{-1}(h^{-1}(h^{-1}(k(h(27)))))).\]

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>.

Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves.

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


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

Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)?

Answer: $16$


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

Calculate the value of $\frac{2468_{10}}{111_{3}} - 3471_{9} + 1234_{7}$. Express your answer in base 10.

Answer: $-1919$


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

Given that the two distinct square roots of a positive number $a$ are $2x-2$ and $6-3x$, find the cube root of $a$.

Answer: $\sqrt[3]{36}$


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>.

A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \( r \) and \( s \), with \( r < s \). Find the ratio \( r : s \).

Answer: $1: 3$


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

Let $O$ be the origin, the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ and the hyperbola $C_{2}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ have a common focus $F$. The line passing through $F$ and perpendicular to the $x$-axis intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$ in the first quadrant at point $M$. If $\overrightarrow{OM}=m\overrightarrow{OA}+n\overrightarrow{OB}\left(m,n\in R\right)$ and $mn=\frac{1}{8}$, find the eccentricity of the hyperbola $C_{2}$.

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


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>.

What is the sum of the odd integers from 21 through 65, inclusive?

Answer: $989$


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

Suppose we have 12 dogs and need to divide them into three groups: one with 4 dogs, one with 5 dogs, and one with 3 dogs. Determine how many ways the groups can be formed if Rocky, a notably aggressive dog, must be in the 4-dog group, and Bella must be in the 5-dog group.

Answer: $4200$


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

If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: $48$


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

Given $\sin \left(\frac{3\pi }{2}+\theta \right)=\frac{1}{4}$,

find the value of $\frac{\cos (\pi +\theta )}{\cos \theta [\cos (\pi +\theta )-1]}+\frac{\cos (\theta -2\pi )}{\cos (\theta +2\pi )\cos (\theta +\pi )+\cos (-\theta )}$.

Answer: $\frac{32}{15}$


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>.

A piece of wood of uniform density in the shape of a right triangle with base length $3$ inches and hypotenuse $5$ inches weighs $12$ ounces. Another piece of the same type of wood, with the same thickness, also in the shape of a right triangle, has a base length of $5$ inches and a hypotenuse of $7$ inches. Calculate the approximate weight of the second piece.

Answer: $24.5$


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

In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho^2 - 2m\rho\cos\theta + 4\rho\sin\theta = 1 - 2m$.
(1) Find the rectangular coordinate equation of $C$ and its radius.
(2) When the radius of $C$ is the smallest, the curve $y = \sqrt{3}|x - 1| - 2$ intersects $C$ at points $A$ and $B$, and point $M(1, -4)$. Find the area of $\triangle MAB$.

Answer: $2 + \sqrt{3}$


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

Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book.

Answer: $315$


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

A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Express this two-digit number algebraically.

Answer: $28$


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

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

Answer: $-73$


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

Let
\[f(x) = \left\{
\begin{array}{cl}
x + 5 & \text{if $x < 15$}, \\
3x - 1 & \text{if $x \ge 15$}.
\end{array}
\right.\]Find $f^{-1}(10) + f^{-1}(50).$

Answer: $22$


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

Given that the equation of line $l_{1}$ is $y=x$, and the equation of line $l_{2}$ is $y=kx-k+1$, find the value of $k$ for which the area of triangle $OAB$ is $2$.

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


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

The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd?

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


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

Given vectors $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(-2,2)$.
(1) If $\overrightarrow{a}\cdot \overrightarrow{b}= \frac {14}{5}$, find the value of $(\sin α+\cos α)^{2}$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\sin (π-α)\cdot\sin ( \frac {π}{2}+α)$.

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


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

Given that $\cos \theta = \frac{12}{13}, \theta \in \left( \pi, 2\pi \right)$, find the values of $\sin \left( \theta - \frac{\pi}{6} \right)$ and $\tan \left( \theta + \frac{\pi}{4} \right)$.

Answer: $\frac{7}{17}$


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

Find the smallest natural number \( n \) that satisfies the following conditions:
1. The units digit of \( n \) is 6.
2. If the units digit 6 is moved to the front of the number, the new number is 4 times \( n \).

Answer: $153846$


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

Given that $\tan\left(\alpha + \frac{\pi}{3}\right)=2$, find the value of $\frac{\sin\left(\alpha + \frac{4\pi}{3}\right) + \cos\left(\frac{2\pi}{3} - \alpha\right)}{\cos\left(\frac{\pi}{6} - \alpha\right) - \sin\left(\alpha + \frac{5\pi}{6}\right)}$.

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>.

Given that $y < 1$ and \[(\log_{10} y)^2 - \log_{10}(y^3) = 75,\] compute the value of \[(\log_{10}y)^3 - \log_{10}(y^4).\]

Answer: $\frac{2808 - 336\sqrt{309}}{8} - 6 + 2\sqrt{309}$


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

Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \).

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


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 $\alpha$ is an angle in the third quadrant, the function $f(\alpha)$ is defined as:
$$f(\alpha) = \frac {\sin(\alpha - \frac {\pi}{2}) \cdot \cos( \frac {3\pi}{2} + \alpha) \cdot \tan(\pi - \alpha)}{\tan(-\alpha - \pi) \cdot \sin(-\alpha - \pi)}.$$
1. Simplify $f(\alpha)$.
2. If $\cos(\alpha - \frac {3\pi}{2}) = \frac {1}{5}$, find $f(\alpha + \frac {\pi}{6})$.

Answer: $\frac{6\sqrt{2} - 1}{10}$


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>.

In a regular triangle $ABC$ with side length $3$, $D$ is a point on side $BC$ such that $\overrightarrow{CD}=2\overrightarrow{DB}$. Calculate the dot product $\overrightarrow{AB} \cdot \overrightarrow{AD}$.

Answer: $\frac{15}{2}$


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>.

Fill the numbers 1, 2, 3 into a 3×3 grid such that each row and each column contains no repeated numbers. How many different ways can this be done?

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 polynomial with integer coefficients given by:
\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]
Find the number of different possible rational roots of this polynomial.

Answer: $28$


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

Let \( N = 34 \times 34 \times 63 \times 270 \). The ratio of the sum of all odd factors of \( N \) to the sum of all even factors of \( N \) is ( ).

Answer: $1: 14$


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>.

A person rides a bicycle from place A to place B. If they increase their speed by 6 km/h, they can arrive 5 minutes earlier; if they decrease their speed by 5 km/h, they will be 6 minutes late. What is the distance between place A and place B in kilometers?

Answer: $15$


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

In triangle $PQR$, $PQ = 8$, $PR = 17$, and the length of median $PM$ is 12. Additionally, the angle $\angle QPR = 60^\circ$. Find the area of triangle $PQR$.

Answer: $34\sqrt{3}$


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

What is the 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>.

Find the distance between the vertices of the hyperbola given by the equation $4x^2 + 16x - 9y^2 + 18y - 23 = 0.$

Answer: $\sqrt{30}$


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>.

Let there be three individuals, labeled A, B, and C, to be allocated across seven laboratories in preparation for experiments. Each laboratory can accommodate no more than two people. Calculate the total number of distinct allocation schemes.

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>.

Given that $x > 0$, $y > 0$, and $x+y=1$, find the minimum value of $\frac{x^{2}}{x+2}+\frac{y^{2}}{y+1}$.

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


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

If $a$, $b$, and $c$ are natural numbers, and $a < b$, $a + b = 719$, $c - a = 915$, then the largest possible value of $a + b + c$ is.

Answer: $1993$


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

A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colourings are there for the circle?

Answer: $20$


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

Given that $\alpha$ is an angle in the third quadrant, $f(\alpha) = \frac {\sin(\pi-\alpha)\cdot \cos(2\pi-\alpha)\cdot \tan(-\alpha-\pi)}{\tan(-\alpha )\cdot \sin(-\pi -\alpha)}$.
1. Simplify $f(\alpha)$;
2. If $\cos\left(\alpha- \frac {3}{2}\pi\right) = \frac {1}{5}$, find the value of $f(\alpha)$;
3. If $\alpha=-1860^\circ$, find the value of $f(\alpha)$.

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


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

A population consists of $20$ individuals numbered $01$, $02$, $\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select two numbers from left to right each time. If the two selected numbers are not within the population, remove them and continue selecting two numbers to the right. Then, the number of the $4$th individual selected is ______.

$7816$	$6572$	$0802$	$6314$	$0702$	$4369$	$9728$	$0198$

$3204$	$9234$	$4935$	$8200$	$3623$	$4869$	$6938$	$7481$

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>.

The eccentricity of the ellipse given by the parametric equations $\begin{cases} x=3\cos\theta \\ y=4\sin\theta\end{cases}$ is $\frac{\sqrt{7}}{\sqrt{3^2+4^2}}$, calculate this value.

Answer: $\frac { \sqrt {7}}{4}$


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 value of $({-\frac{4}{5}})^{2022} \times ({\frac{5}{4}})^{2021}$.

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


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

Selected Exercise $4-4$: Coordinate Systems and Parametric Equations

In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\begin{cases} & x=\cos \theta \\ & y=\sin \theta \end{cases}$, where $\theta$ is the parameter. In the polar coordinate system with the same unit length as the rectangular coordinate system $xOy$, taking the origin $O$ as the pole and the non-negative half of the $x$-axis as the polar axis, the equation of the line $l$ is given by $\sqrt{2}p \sin (\theta - \frac{\pi}{4}) = 3$.

(I) Find the Cartesian equation of the curve $C$ and the equation of the line $l$ in rectangular coordinates.

(II) Let $P$ be any point on the curve $C$. Find the maximum distance from the point $P$ to the line $l$.

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


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>.

Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).

Answer: $81$


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

There is a question: "If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\left(a+b\right)+4\left(2a+b\right)$?" Tang, who loves to use his brain, solved the problem as follows:
Original expression $=2a+2b+8a+4b=10a+6b=2\left(5a+3b\right)=2\times \left(-4\right)=-8$
Tang treated $5a+3b$ as a whole to solve it. Treating expressions as a whole is an important method in solving problems in high school mathematics. Please follow the method above to complete the following questions:
$(1)$ Given $a^{2}+a=3$, then $2a^{2}+2a+2023=$______;
$(2)$ Given $a-2b=-3$, find the value of $3\left(a-b\right)-7a+11b+2$;
$(3)$ Given $a^{2}+2ab=-5$, $ab-2b^{2}=-3$, find the value of the algebraic expression $a^{2}+ab+2b^{2}$.

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>.

I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation?

Answer: $31.25$


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>.

Define an operation between sets A and B: $A*B = \{x | x = x_1 + x_2, \text{ where } x_1 \in A, x_2 \in B\}$. If $A = \{1, 2, 3\}$ and $B = \{1, 2\}$, then the sum of all elements in $A*B$ is ____.

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>.

In triangle  $ABC$  with  $\overline{AB}=\overline{AC}=3.6$ , a point  $D$  is taken on  $AB$  at a distance  $1.2$  from  $A$ .  Point  $D$  is joined to  $E$  in the prolongation of  $AC$  so that triangle  $AED$  is equal in area to  $ABC$ .  Then  $\overline{AE}$  is:

Answer: $10.8$


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

For the set $E=\{a_1, a_2, \ldots, a_{100}\}$, define a subset $X=\{a_1, a_2, \ldots, a_n\}$, and its "characteristic sequence" as $x_1, x_2, \ldots, x_{100}$, where $x_1=x_{10}=\ldots=x_n=1$. The rest of the items are 0. For example, the "characteristic sequence" of the subset $\{a_2, a_3\}$ is $0, 1, 0, 0, \ldots, 0$
(1) The sum of the first three items of the "characteristic sequence" of the subset $\{a_1, a_3, a_5\}$ is     ;
(2) If the "characteristic sequence" $P_1, P_2, \ldots, P_{100}$ of a subset $P$ of $E$ satisfies $p_1=1$, $p_i+p_{i+1}=1$, $1\leq i\leq 99$; and the "characteristic sequence" $q_1, q_2, \ldots, q_{100}$ of a subset $Q$ of $E$ satisfies $q_1=1$, $q_j+q_{j+1}+q_{j+2}=1$, $1\leq j\leq 98$, then the number of elements in $P\cap Q$ is     .

Answer: $17$


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

In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 3$, then what is $BC$?

Answer: $10\sqrt{10}$


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>.

Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer  $N$  for which there exists a rectangular prism with integer side lengths and *bigness* $N$  and another one with integer side lengths and *bigness* $N + 1$ .

Answer: $55$


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

In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3\cos \alpha, \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho\sin \left( \theta- \frac{\pi}{4} \right)= \sqrt{2}$.

$(1)$ Find the general equation of $C$ and the inclination angle of $l$;

$(2)$ Let point $P(0,2)$, $l$ and $C$ intersect at points $A$ and $B$, find the value of $|PA|+|PB|$.

Answer: $\frac{18 \sqrt{2}}{5}$


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

For all positive numbers $a,b \in \mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\frac{1}{2a}-\frac{2}{b}$.

Answer: $-\frac{9}{2}$


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

Given that \\(\alpha\\) is an angle in the second quadrant, and \\(\sin (π+α)=- \frac {3}{5}\\), find the value of \\(\tan 2α\\).

Answer: $- \frac {24}{7}$


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

Given the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, find the value of $a^{2015} + b^{2016}$.

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>.

In a rectangular configuration $ABCD$, there are three squares with non-overlapping interiors. One of them, which is shaded, has an area of 4 square inches. The side length of the larger square is twice the side length of the shaded square. What is the area of rectangle $ABCD$, given that all squares fit exactly within $ABCD$ with no other spaces?

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>.

Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate the total number of different distribution schemes.

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>.

Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser.

Answer: $22$


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

Given the planar vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ that satisfy $|\overrightarrow {e_{1}}| = |3\overrightarrow {e_{1}} + \overrightarrow {e_{2}}| = 2$, determine the maximum value of the projection of $\overrightarrow {e_{1}}$ onto $\overrightarrow {e_{2}}$.

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


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

Express $326_{13} + 4C9_{14}$ as a base 10 integer, where $C = 12$ in base 14.

Answer: $1500$


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

Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and $g(x) \neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{ \frac{f(n)}{g(n)}\right\} (n=1,2,…,10)$ is greater than $\frac{15}{16}$.

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


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

A square is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1,\]
such that its sides are parallel to the coordinate axes. Find the area of this square.

Answer: $\frac{32}{3}$


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>.

Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $JKL$, the altitude splits the triangle into two triangles with areas in the ratio of 4:5. What is the ratio of the area of triangle $GHI$ to the area of triangle $JKL$? Express your answer as a common fraction.

Answer: $\dfrac{7}{15}$


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

In the three-dimensional Cartesian coordinate system, given points $A(2,a,-1)$, $B(-2,3,b)$, $C(1,2,-2)$.
$(1)$ If points $A$, $B$, and $C$ are collinear, find the values of $a$ and $b$;
$(2)$ Given $b=-3$, $D(-1,3,-3)$, and points $A$, $B$, $C$, and $D$ are coplanar, find the value of $a$.

Answer: $a=1$


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>.

If the complex number $z$ satisfies $z(1-i)=|1-i|+i$, then the imaginary part of $\overline{z}$ is ______.

Answer: $-\dfrac{\sqrt{2}+1}{2}$


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>.

A basketball team consists of 18 players, including a set of 3 triplets: Bob, Bill, and Ben; and a set of twins: Tim and Tom. In how many ways can we choose 7 starters if exactly two of the triplets and one of the twins must be in the starting lineup?

Answer: $4290$


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

Let \\(f(x)=ax^{2}-b\sin x\\) and \\(f′(0)=1\\), \\(f′\left( \dfrac {π}{3}\right)= \dfrac {1}{2}\\). Find the values of \\(a\\) and \\(b\\).

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>.

Given the function $$f(x)=\sin(x+ \frac {\pi}{6})+2\sin^{2} \frac {x}{2}$$.
(1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$.
(2) Determine the intervals of monotonicity for the function $f(x)$.
(3) In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and $a= \sqrt {3}$, $f(A)= \frac {3}{2}$, the area of triangle $ABC$ is $\frac { \sqrt {3}}{2}$. Find the value of $\sin B + \sin C$.

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


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

Given three rays $AB$, $BC$, $BB_{1}$ are not coplanar, and the diagonals of quadrilaterals $BB_{1}A_{1}A$ and $BB_{1}C_{1}C$ bisect each other, and $\overrightarrow{AC_{1}}=x\overrightarrow{AB}+2y\overrightarrow{BC}+3z\overrightarrow{CC_{1}}$, find the value of $x+y+z$.

Answer: $\frac{11}{6}$


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>.

Given the hyperbola $x^{2}- \frac{y^{2}}{24}=1$, let the focal points be F1 and F2, respectively. If P is a point on the left branch of the hyperbola such that $|PF_{1}|=\frac{3}{5}|F_{1}F_{2}|$, find the area of triangle $\triangle PF_{1}F_{2}$.

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>.

Find the minimum value of the function $f(x)=27x-x^{3}$ in the interval $[-4,2]$.

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>.

Let $\mathbf{v} = \begin{pmatrix} 4 \\ -5 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 13 \\ 2 \end{pmatrix}$. Find the area of the parallelogram formed by vectors $\mathbf{v}$ and $2\mathbf{w}$.

Answer: $146$


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

The increasing sequence \(1, 3, 4, 9, 10, 12, 13, \cdots\) consists of some positive integers that are either powers of 3 or sums of distinct powers of 3. Find the value of the 2014th term.

Answer: $88329$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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. Given $a(4-2 \sqrt {7}\cos B)=b(2 \sqrt {7}\cos A-5)$, find the minimum value of $\cos C$.

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


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

Find the maximum value of the function \( f(x) \), which is defined as the minimum of the three functions \( 4x + 1 \), \( x + 2 \), and \( -2x + 4 \) for each real number \( x \).

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


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

A line passes through the vectors $\mathbf{a}$ and $\mathbf{b}$. For a certain value of $k$, the vector
\[ k \mathbf{a} + \frac{5}{8} \mathbf{b} \]
must also lie on the line. Find $k$.

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>.

A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?

Answer: $115$


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

An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle?

Answer: $1.5$


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

The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$.

Answer: $-49$


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

Ben is throwing darts at a circular target with diameter 10.  Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target.  Ben gets  $\lceil 5-x \rceil$  points for having the dart land  $x$  units away from the center of the target.  What is the expected number of points that Ben can earn from throwing a single dart?  (Note that  $\lceil y \rceil$  denotes the smallest integer greater than or equal to  $y$ .)

Answer: $11/5$


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>.

Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$ with eccentricity $e$, and the parabola $x=2py^{2}$ with focus at $(e,0)$, find the value of the real number $p$.

Answer: $\frac{1}{16}$


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

Given the function $f\left(x\right)=x^{3}+ax^{2}+x+1$ achieves an extremum at $x=-1$. Find:
$(1)$ The equation of the tangent line to $f\left(x\right)$ at $\left(0,f\left(0\right)\right)$;
$(2)$ The maximum and minimum values of $f\left(x\right)$ on the interval $\left[-2,0\right]$.

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>.

Given that $| \overrightarrow{a}|=6$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$.

Answer: $-4$


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

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\)

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


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

Given the function $f(x) = 2\sin\omega x\cos\omega x - 2\sqrt{3}\sin^2\omega x + \sqrt{3}(\_\omega (\_ > 0)), the lines $x = \_x\_{1}$ and $x = \_x\_{2}$ are any two symmetry axes of the graph of the function $y = f(x)$, and the minimum value of $|x\_1 - x\_2|$ is $\frac{\pi}{2}$.

1. Find the value of $\omega$;
2. Find the intervals where the function $f(x)$ is increasing;
3. If $f(\alpha) = \frac{2}{3}$, find the value of $\sin(\frac{5}{6}\pi - 4\alpha)$.

Answer: $-\frac{7}{9}$


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>.

The function defined on the set of real numbers, \(f(x)\), satisfies \(f(x-1) = \frac{1 + f(x+1)}{1 - f(x+1)}\). Find the value of \(f(1) \cdot f(2) \cdot f(3) \cdots f(2008) + 2008\).

Answer: $2009$


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

How many distinct trees with exactly 7 vertices exist?

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>.

Simplify:
$(1)(-\frac{1}{2}+\frac{2}{3}-\frac{1}{4})÷(-\frac{1}{24})$;
$(2)3\frac{1}{2}×(-\frac{5}{7})-(-\frac{5}{7})×2\frac{1}{2}-\frac{5}{7}×(-\frac{1}{2})$.

Answer: $-\frac{5}{14}$


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>.

Simplify first, then evaluate: $(\frac{{x-1}}{{x-3}}-\frac{{x+1}}{x})÷\frac{{{x^2}+3x}}{{{x^2}-6x+9}}$, where $x$ satisfies $x^{2}+2x-6=0$.

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


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

Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$.
(1) Find the smallest positive period and the interval where the function is decreasing;
(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\frac{A}{2}-\frac{\pi}{6})=\sqrt{3}$, and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of triangle $ABC$.

Answer: $10\sqrt{3}$


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

Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$

Answer: $25000$


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

Given the curve $x^{2}-y-2\ln \sqrt{x}=0$ and the line $4x+4y+1=0$, find the shortest distance from any point $P$ on the curve to the line.

Answer: $\dfrac{\sqrt{2}(1+\ln2)}{2}$


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

In $\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\overline{QR}$. What is the sum of the radii of the circles inscribed in $\triangle PQS$ and $\triangle PRS$?
A) $\frac{14\sqrt{5}}{13}$
B) $\frac{14\sqrt{5}}{6.5 + \sqrt{29}}$
C) $\frac{12\sqrt{4}}{8.5}$
D) $\frac{10\sqrt{3}}{7 + \sqrt{24}}$

Answer: $\frac{14\sqrt{5}}{6.5 + \sqrt{29}}$


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

Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______.

Answer: $8-\sqrt{51}$