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

The noon temperatures for ten consecutive days were $78^{\circ}$, $80^{\circ}$, $82^{\circ}$, $85^{\circ}$, $88^{\circ}$, $90^{\circ}$, $92^{\circ}$, $95^{\circ}$, $97^{\circ}$, and $95^{\circ}$ Fahrenheit. The increase in temperature over the weekend days (days 6 to 10) is attributed to a local summer festival. What is the mean noon temperature, in degrees Fahrenheit, for these ten days?

Answer: $88.2$


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

Simplify and find the value: $4(a^{2}b+ab^{2})-3(a^{2}b-1)+2ab^{2}-6$, where $a=1$, $b=-4$.

Answer: $89$


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

Example: The ancients used Heavenly Stems and Earthly Branches to keep track of order. There are 10 Heavenly Stems: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui; and 12 Earthly Branches: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai. The 10 characters of the Heavenly Stems and the 12 characters of the Earthly Branches are arranged in two rows in a cyclic manner as follows:
Jia Yi Bing Ding Wu Ji Geng Xin Ren Gui Jia Yi Bing Ding Wu Ji Geng Xin Ren Gui…
Zi Chou Yin Mao Chen Si Wu Wei Shen You Xu Hai Zi Chou Yin Mao Chen Si Wu Wei Shen You Xu Hai…
Counting from left to right, the first column is Jia Zi, the third column is Bing Yin…, the question is, when Jia and Zi are in the same column for the second time,
what is the column number?

Answer: $61$


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

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho^{2}-2\rho\cos \theta-4\rho\sin \theta+4=0$, and the equation of line $l$ is $x-y-1=0$.
$(1)$ Write the parametric equation of curve $C$;
$(2)$ Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value.

Answer: $1+ \sqrt{2}$


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

Among the four-digit numbers composed of the digits $0$, $1$, $2$, $3$, $4$, $5$ without repetition, there are a total of \_\_\_\_\_ numbers that are not divisible by $5$.

Answer: $192$


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

Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20?

Answer: $\frac{11}{221}$


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

Given \( f(x) = a \sin x + b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^{2} + 1}\right) + 1003 \) (where \( a \), \( b \), and \( c \) are real numbers), and \( f\left(\lg^{2} 10\right) = 1 \), what is \( f(\lg \lg 3) \)?

Answer: $2005$


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

A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it was found that for every $1$ yuan reduction in price per shirt, the mall can sell an additional $2$ shirts on average.

$(1)$ If the price reduction per shirt is set at $x$ yuan, and the average daily profit is $y$ yuan, find the functional relationship between $y$ and $x$.
$(2)$ At what price reduction per shirt will the mall have the maximum average daily profit?
$(3)$ If the mall needs an average daily profit of $1200$ yuan, how much should the price per shirt be reduced?

Answer: $20$


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

A game wheel is divided into six regions labeled $A$, $B$, $C$, $D$, $E$, and $F$. The probability of the wheel stopping on region $A$ is $\frac{1}{3}$, the probability it stops on region $B$ is $\frac{1}{6}$, and the probability of it stopping on regions $C$ and $D$ are equal, as are the probabilities for regions $E$ and $F$. What is the probability of the wheel stopping in region $C$?

Answer: $\frac{1}{8}$


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

Given the function $f(x)=|x-1|+|x+1|$.
(I) Solve the inequality $f(x) < 3$;
(II) If the minimum value of $f(x)$ is $m$, let $a > 0$, $b > 0$, and $a+b=m$, find the minimum value of $\frac{1}{a}+ \frac{2}{b}$.

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


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

Given that the golden ratio $m = \frac{{\sqrt{5}-1}}{2}$, calculate the value of $\frac{{\sin{42}°+m}}{{\cos{42}°}}$.

Answer: $\sqrt{3}$


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

In the diagram, \( B, C \) and \( D \) lie on a straight line, with \(\angle ACD=100^{\circ}\), \(\angle ADB=x^{\circ}\), \(\angle ABD=2x^{\circ}\), and \(\angle DAC=\angle BAC=y^{\circ}\). The value of \( x \) is:

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>.

Compute $20\left(\frac{256}{4} + \frac{64}{16} + \frac{16}{64} + 2\right)$.

Answer: $1405$


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

A store increased the price of a certain Super VCD by 40% and then advertised a "10% discount and a free 50 yuan taxi fare" promotion. As a result, each Super VCD still made a profit of 340 yuan. What was the cost price of each Super VCD?

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>.

The equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ is to be simplified to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\] for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$

Answer: $14$


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

Given a circle \\(O: x^2 + y^2 = 2\\) and a line \\(l: y = kx - 2\\).

\\((1)\\) If line \\(l\\) intersects circle \\(O\\) at two distinct points \\(A\\) and \\(B\\), and \\(\angle AOB = \frac{\pi}{2}\\), find the value of \\(k\\).

\\((2)\\) If \\(EF\\) and \\(GH\\) are two perpendicular chords of the circle \\(O: x^2 + y^2 = 2\\), with the foot of the perpendicular being \\(M(1, \frac{\sqrt{2}}{2})\\), find the maximum area of the quadrilateral \\(EGFH\\).

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


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

For what real number \\(m\\) is the complex number \\(z=m^{2}+m-2+(m^{2}-1)i\\)
\\((1)\\) a real number; \\((2)\\) an imaginary number; \\((3)\\) a pure imaginary number?

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>.

A gardener plans to place potted plants along both sides of a 150-meter-long path (including at both ends), with one pot every 2 meters. In total, \_\_\_\_\_\_ pots are needed.

Answer: $152$


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

If \( a = \log 25 \) and \( b = \log 49 \), compute
\[
5^{a/b} + 7^{b/a}.
\]

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>.

Calculate $[x]$, where $x = -3.7 + 1.5$.

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 $cos({\frac{π}{4}-α})=\frac{3}{5}$, $sin({\frac{{5π}}{4}+β})=-\frac{{12}}{{13}}$, $α∈({\frac{π}{4},\frac{{3π}}{4}})$, $β∈({0,\frac{π}{4}})$, calculate the value of $\sin \left(\alpha +\beta \right)$.

Answer: $\frac{56}{65}$


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

Given \(\frac{x+ \sqrt{2}i}{i}=y+i\), where \(x\), \(y\in\mathbb{R}\), and \(i\) is the imaginary unit, find \(|x-yi|\).

Answer: $\sqrt{3}$


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

1. Given $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$.
2. Given $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\cos\alpha = -\frac{5}{13}$, $\tan \frac{\beta}{2} = \frac{1}{3}$, find the value of $\cos\left(\frac{\alpha}{2} + \beta\right)$.

Answer: $-\frac{17\sqrt{13}}{65}$


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

Given the angle $\frac {19\pi}{5}$, express it in the form of $2k\pi+\alpha$ ($k\in\mathbb{Z}$), then determine the angle $\alpha$ that makes $|\alpha|$ the smallest.

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


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 \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(a=2\\), \\(c=3\\), and \\(\cos B= \dfrac {1}{4}\\),
\\((1)\\) find the value of \\(b\\);
\\((2)\\) find the value of \\(\sin C\\).

Answer: $\dfrac {3 \sqrt {6}}{8}$


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

Given that $(a + \frac{1}{a})^3 = 3$, find the value of $a^4 + \frac{1}{a^4}$.
A) $9^{1/3} - 4 \cdot 3^{1/3} + 2$
B) $9^{1/3} - 2 \cdot 3^{1/3} + 2$
C) $9^{1/3} + 4 \cdot 3^{1/3} + 2$
D) $4 \cdot 3^{1/3} - 9^{1/3}$

Answer: $9^{1/3} - 4 \cdot 3^{1/3} + 2$


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>.

(1) Simplify: $\dfrac {\tan (3\pi-\alpha)\cos (2\pi-\alpha)\sin (-\alpha+ \dfrac {3\pi}{2})}{\cos (-\alpha-\pi)\sin (-\pi+\alpha)\cos (\alpha+ \dfrac {5\pi}{2})}$;
(2) Given $\tan \alpha= \dfrac {1}{4}$, find the value of $\dfrac {1}{2\cos ^{2}\alpha -3\sin \alpha \cos \alpha }$.

Answer: $\dfrac {17}{20}$


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>.

The volume of a given sphere is \(72\pi\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \(\pi\).

Answer: $36\pi 2^{2/3}$


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>.

Add $36_7 + 274_7.$ Express your answer in base 7.

Answer: $343_7$


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

The area of the region in the  $xy$ -plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is  $k\pi$ , for some integer  $k$ . Find  $k$ .

*Proposed by Michael Tang*

Answer: $210$


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

In the sequence \(1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \cdots, 200, 200, \cdots, 200\), each number \(n\) appears \(n\) times consecutively, where \(n \in \mathbf{N}\) and \(1 \leq n \leq 200\). What is the median of this sequence?

Answer: $142$


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

Calculate $f(x) = 3x^5 + 5x^4 + 6x^3 - 8x^2 + 35x + 12$ using the Horner's Rule when $x = -2$. Find the value of $v_4$.

Answer: $83$


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

Two concentric circles have radii  $1$  and  $4$ .  Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown.  The three lightly shaded circles are internally tangent to the circle with radius  $4$  while the three darkly shaded circles are externally tangent to the circle with radius  $1$ .  The radius of the six congruent circles can be written  $\textstyle\frac{k+\sqrt m}n$ , where  $k,m,$  and  $n$  are integers with  $k$  and  $n$  relatively prime.  Find  $k+m+n$ .

[asy]
size(150);
defaultpen(linewidth(0.8));
real r = (sqrt(133)-9)/2;
draw(circle(origin,1)^^circle(origin,4));
for(int i=0;i<=2;i=i+1)
{
filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
}
for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy]

Answer: $126$


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

How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$?

Answer: $600$


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

Let  $x, y$  be two positive integers, with  $x> y$ , such that  $2n = x + y$ , where n is a number two-digit integer. If  $\sqrt{xy}$  is an integer with the digits of  $n$  but in reverse order, determine the value of  $x - y$

Answer: $66$


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

Let \( A \) be a point on the parabola \( y = x^2 - 4x \), and let \( B \) be a point on the line \( y = 2x - 3 \). Find the shortest possible distance \( AB \).

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


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

Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $\frac{\sqrt{3}c}{\cos C} = \frac{a}{\cos(\frac{3\pi}{2} + A)}$.
(I) Find the value of $C$;
(II) If $\frac{c}{a} = 2$, $b = 4\sqrt{3}$, find the area of $\triangle ABC$.

Answer: $2\sqrt{15} - 2\sqrt{3}$


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>.

The equations of the asymptotes of the hyperbola \\( \frac {x^{2}}{3}- \frac {y^{2}}{6}=1 \\) are \_\_\_\_\_\_, and its eccentricity is \_\_\_\_\_\_.

Answer: $\sqrt {3}$


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

Given that $a = \sin (2015\pi - \frac {\pi}{6})$ and the function $f(x) = \begin{cases} a^{x}, & x > 0 \\ f(-x), & x < 0 \end{cases}$, calculate the value of $f(\log_{2} \frac {1}{6})$.

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>.

In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(15,20)$ that does not go inside the circle $(x-7)^2 + (y-9)^2 = 36$?
A) $2\sqrt{94} + 3\pi$
B) $2\sqrt{130} + 3\pi$
C) $2\sqrt{94} + 6\pi$
D) $2\sqrt{130} + 6\pi$
E) $94 + 3\pi$

Answer: $2\sqrt{94} + 3\pi$


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

How many five-character license plates consist of two consonants, followed by two vowels, and ending with a digit? (For this problem, consider Y is not a vowel.)

Answer: $110,250$


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

Calculate:
$(1)-3+5-\left(-2\right)$;
$(2)-6÷\frac{1}{4}×(-4)$;
$(3)(\frac{5}{6}-\frac{3}{4}+\frac{1}{3})×(-24)$;
$(4)-1^{2023}-[4-(-3)^2]÷(\frac{2}{7}-1)$.

Answer: $-8$


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

For what value of the parameter \( p \) will the sum of the squares of the roots of the equation

\[
x^{2}+(3 p-2) x-7 p-1=0
\]

be minimized? What is this minimum value?

Answer: $\frac{53}{9}$


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>.

Determine the number of perfect cubic divisors in the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 6!$.

Answer: $10$


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

A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one book is in the library and at least one is with students?

Answer: $254$


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

What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square?

Answer: $337$


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

Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $\frac{a_{12}}{a_{11}} < -1$, find the maximum value of $n$ for which the sum of its first $n$ terms, $s_n$, is greater than $0$.

Answer: $21$


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

For two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are given by $S_n$ and $T_n$ respectively, and $\frac{S_{n}}{T_{n}} = \frac{3n - 1}{2n + 3}$. Determine the ratio $\frac{a_{7}}{b_{7}}$.

Answer: $\frac{38}{29}$


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>.

A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element?

Answer: $65$


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

For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?

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 point $P$ moves on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\sqrt{5}=0$.

Answer: $\frac{\sqrt{10}}{2}$


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>.

Ket  $f(x) = x^{2} +ax + b$ . If for all nonzero real  $x$   $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$  and the roots of  $f(x) = 0$  are integers, what is the value of  $a^{2}+b^{2}$ ?

Answer: $13$


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

On a line \( l \) in space, points \( A \), \( B \), and \( C \) are sequentially located such that \( AB = 18 \) and \( BC = 14 \). Find the distance between lines \( l \) and \( m \) if the distances from points \( A \), \( B \), and \( C \) to line \( m \) are 12, 15, and 20, respectively.

Answer: $12$


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

Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $\frac{{a^2 + c^2 - b^2}}{{\cos B}} = 4$. Find:

$(1)$ $ac$;

$(2)$ If $\frac{{2b\cos C - 2c\cos B}}{{b\cos C + c\cos B}} - \frac{c}{a} = 2$, find the area of $\triangle ABC$.

Answer: $\frac{\sqrt{15}}{4}$


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

Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, whose sums of the first $n$ terms are $A_n$ and $B_n$ respectively, and $\frac {A_{n}}{B_{n}} = \frac {7n+1}{4n+27}$, determine $\frac {a_{6}}{b_{6}}$.

Answer: $\frac{78}{71}$


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

A set of numbers $\{-3, 1, 5, 8, 10, 14\}$ needs to be rearranged with new rules:

1. The largest isn't in the last position, but it is in one of the last four places.
2. The smallest isn’t in the first position, but it is in one of the first four places.
3. The median isn't in the middle positions.

What is the product of the second and fifth numbers after rearrangement?
A) $-21$
B) $24$
C) $-24$
D) $30$
E) None of the above

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 a sequence ${a_n}$ whose first $n$ terms have a sum of $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y = \frac{1}{2}x + \frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \in \mathbb{N}^*$), and $b_3 = 11$, with the sum of the first 9 terms being 153.
(I) Find the general term formulas for the sequences ${a_n}$ and ${b_n}$;
(II) Let $c_n = \frac{3}{(2a_n - 11)(2b_n - 1)}$. The sum of the first $n$ terms of the sequence ${c_n}$ is $T_n$. Find the maximum positive integer value $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n \in \mathbb{N}^*$.

Answer: $18$


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

Given a geometric sequence $\{a_n\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\{a_n\}$.

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>.

Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\_1|=3|PF\_2|$ and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}=a^{2}$, calculate the eccentricity of the hyperbola $C$.

Answer: $\sqrt{2}$


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

The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction.

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>.

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1800$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$.

Answer: $500$


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

Given that the odd function $f(x)$ is an increasing function defined on $\mathbb{R}$, and the sequence $x_n$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, then the value of $x_{2011}$ is equal to.

Answer: $4003$


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

Given $\cos \left(a- \frac{\pi}{6}\right) + \sin a = \frac{4 \sqrt{3}}{5}$, find the value of $\sin \left(a+ \frac{7\pi}{6}\right)$.

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>.

Given that the sum of the first 10 terms of a geometric sequence $\{a_n\}$ is 32 and the sum of the first 20 terms is 56, find the sum of the first 30 terms.

Answer: $74$


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

A clueless ant makes the following route: starting at point  $ A $  goes  $ 1$  cm north, then  $ 2$  cm east, then  $ 3$  cm south, then  $ 4$  cm west, immediately  $ 5$  cm north, continues  $ 6$  cm east, and so on, finally  $ 41$  cm north and ends in point  $ B $ . Calculate the distance between  $ A $  and  $ B $  (in a straight line).

Answer: $29$


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

We have  $ 23^2 = 529 $  ordered pairs  $ (x, y) $  with  $ x $  and  $ y $  positive integers from 1 to 23, inclusive. How many of them have the property that  $ x^2 + y^2 + x + y $  is a multiple of 6?

Answer: $225$


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

There are 4 male students and 2 female students taking the photo. The male student named "甲" cannot stand at either end, and the female students must stand next to each other. Find the number of ways to arrange the students.

Answer: $144$


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

Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n + \frac{1}{3}a_n = 1 (n \in \mathbb{N}^*)$.
(I) Find the general term formula of the sequence ${a_n}$;
(II) Let $b_n = \log_4(1 - S_{n+1}) (n \in \mathbb{N}^*)$, $T_n = \frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \dots + \frac{1}{b_nb_{n+1}}$, find the smallest positive integer $n$ such that $T_n \geq \frac{1007}{2016}$.

Answer: $2014$


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

Given the parametric equation of line C1 as $$\begin{cases} x=2+t \\ y=t \end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis.

1. Find the general equation of line C1 and the standard equation of ellipse C2 in rectangular coordinates.
2. If line C1 intersects with ellipse C2 at points A and B, and intersects with the x-axis at point E, find the value of |EA + EB|.

Answer: $\frac{6\sqrt{3}}{5}$


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>.

Investigate the function $f(x)=e^{x}-e^{-x}+\sin x+1$, and let $a=\frac{2023+2024}{2}=2023.5$. Determine the value of $f(-2023)+f(-2022)+\ldots +f(2022)+f(2023)$.

Answer: $4047$


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

Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle.

Answer: $24\pi$


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

In the geometric sequence ${a_n}$, there exists a positive integer $m$ such that $a_m = 3$ and $a_{m+6} = 24$. Find the value of $a_{m+18}$.

Answer: $1536$


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

In \\(\triangle ABC\\), \\(AB=BC\\), \\(\cos B=-\dfrac{7}{18}\\). If an ellipse with foci at points \\(A\\) and \\(B\\) passes through point \\(C\\), find the eccentricity of the ellipse.

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


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>.

Given that $y$ is a multiple of $3456$, what is the greatest common divisor of $g(y) = (5y+4)(9y+1)(12y+6)(3y+9)$ and $y$?

Answer: $216$


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

Among the subsets of the set $\{1,2, \cdots, 100\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other.

Answer: $67$


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

The equation ${{a}^{2}}{{x}^{2}}+(a+2){{y}^{2}}+2ax+a=0$ represents a circle. Find the possible values of $a$.

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>.

Determine the number of ways to place 7 identical balls into 4 distinct boxes such that each box contains at least one ball.

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>.

Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$?

Answer: $\frac{7}{33}$


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

The parametric equation of line $l$ is
$$
\begin{cases}
x= \frac { \sqrt {2}}{2}t \\
y= \frac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of circle $c$ is $\rho=2\cos(\theta+ \frac{\pi}{4})$. Tangent lines are drawn from the points on the line to the circle; find the minimum length of these tangent lines.

Answer: $2\sqrt{6}$


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

On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; the "Two Gardens" refer to the Theme Carnival Park and the Agricultural Experience Park; the "One Belt" refers to the Strawberry Leisure Experience Belt; and the "One Valley" refers to the Yanshou Ecological Sightseeing Valley. Due to limited time, a group of students plans to visit "One Pavilion, One Garden, One Belt, and One Valley." How many different routes can they take for their visit? (Answer with a number)

Answer: $144$


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

A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume the side of the larger hexagon is twice the radius of the circle.

Answer: $25\%$


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

Given $f(x)=x^{3}+ax^{2}+bx+a^{2}$, the extreme value at $x=1$ is $10$. Find the value of $a+b$.

Answer: $-7$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $\sin (A-B)+ \sin C= \sqrt {2}\sin A$.
(I) Find the value of angle $B$;
(II) If $b=2$, find the maximum value of $a^{2}+c^{2}$, and find the values of angles $A$ and $C$ when the maximum value is obtained.

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


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

Consider the graph on  $1000$  vertices  $v_1, v_2, ...v_{1000}$  such that for all  $1 \le i < j  \le 1000$ ,  $v_i$  is connected to  $v_j$  if and only if  $i$  divides  $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.

Answer: $10$


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

Given that the area of $\triangle ABC$ is $2 \sqrt {3}$, $BC=2$, $C=120^{\circ}$, find the length of side $AB$.

Answer: $2 \sqrt {7}$


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

What is the sum of all the odd integers between $200$ and $600$?

Answer: $80000$


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

Given $\overrightarrow{a}=(1+\cos \omega x,-1)$, $\overrightarrow{b}=( \sqrt {3},\sin \omega x)$ ($\omega > 0$), and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\pi$.
(1) Find the expression of the function $f(x)$.
(2) Let $\theta\in(0, \frac {\pi}{2})$, and $f(\theta)= \sqrt {3}+ \frac {6}{5}$, find the value of $\cos \theta$.

Answer: $\frac {3 \sqrt {3}+4}{10}$


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>.

Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$.

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>.

James used a calculator to find the product $0.005 \times 3.24$. He forgot to enter the decimal points, and the calculator showed $1620$. If James had entered the decimal points correctly, what would the answer have been?
A) $0.00162$
B) $0.0162$
C) $0.162$
D) $0.01620$
E) $0.1620$

Answer: $0.0162$


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

Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap.

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


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

Given the function $f(x) = 4\cos(\omega x - \frac{\pi}{6})\sin \omega x - \cos(2\omega x + \pi)$, where $\omega > 0$.
(I) Find the range of the function $y = f(x)$.
(II) If $f(x)$ is an increasing function on the interval $[-\frac{3\pi}{2}, \frac{\pi}{2}]$, find the maximum value of $\omega$.

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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. It is known that $c^{2}=a^{2}+b^{2}-4bc\cos C$, and $A-C= \frac {\pi}{2}$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) Find the value of $\cos \left(B+ \frac {\pi}{3}\right)$.

Answer: $\frac {4-3 \sqrt {3}}{10}$


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

Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\).

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>.

Let  $ ABC$  be an isosceles triangle with  $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$  and  $ \left|BC\right| \equal{} 12$ .  $ P$  and  $ R$  are points on  $ \left[BC\right]$  such that  $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$ .  $ S$  and  $ T$  are midpoints of  $ \left[AB\right]$  and  $ \left[AC\right]$ , respectively. If  $ M$  and  $ N$  are the foot of perpendiculars from  $ S$  and  $ R$  to  $ PT$ , then find  $ \left|MN\right|$ .

Answer: $ \frac {10\sqrt {13} }{13} $


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

If the letters in the English word "good" are written in the wrong order, then the total number of possible mistakes is _____

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>.

A 12-hour digital clock has a glitch such that whenever it is supposed to display a 5, it mistakenly displays a 7. For example, when it is 5:15 PM the clock incorrectly shows 7:77 PM. What fraction of the day will the clock show the correct time?

Answer: $\frac{33}{40}$


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

A nine-digit integer is formed by repeating a three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?

Answer: $1001001$


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

Given $\tan \alpha = 3$, evaluate the expression $2\sin ^{2}\alpha + 4\sin \alpha \cos \alpha - 9\cos ^{2}\alpha$.

Answer: $\dfrac{21}{10}$


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

Given the geometric sequence $\{a_{n}\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$.

Answer: $-64$


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

Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event A be the event that exactly 3 members are involved in these two handshakes. If the probability of event A happening, $P$, is less than $\frac{1}{10}$, then the minimum value of $n$ is \_\_\_\_\_\_.

Answer: $20$


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

What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 1980 = 0$ has integral solutions?

Answer: $290$


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

Suppose
\[\frac{1}{x^3-7x^2+11x+15} = \frac{A}{x-5} + \frac{B}{x+3} + \frac{C}{(x+3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?

Answer: $\frac{1}{64}$


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

If 2035 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: $50$


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

Let $a$ and $b$ be randomly selected numbers from the set $\{1,2,3,4\}$. Given the line $l:y=ax+b$ and the circle $O:x^{2}+y^{2}=1$, the probability that the line $l$ intersects the circle $O$ is ______. The expected number of intersection points between the line $l$ and the circle $O$ is ______.

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>.

In square \(ABCD\), \(P\) is the midpoint of \(DC\) and \(Q\) is the midpoint of \(AD\). If the area of the quadrilateral \(QBCP\) is 15, what is the area of square \(ABCD\)?

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>.

A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$?

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>.

The value of \(2 \frac{1}{10} + 3 \frac{11}{100} + 4 \frac{111}{1000}\) is

Answer: $9.321$


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

In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$.

Answer: $\frac {24+7 \sqrt {3}}{50}$


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

What is the coefficient of \(x^3y^5\) in the expansion of \(\left(\frac{2}{3}x - \frac{4}{5}y\right)^8\)?

Answer: $-\frac{458752}{84375}$


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

Given the singing scores 9.4, 8.4, 9.4, 9.9, 9.6, 9.4, 9.7, calculate the average and variance of the remaining data after removing the highest and lowest scores.

Answer: $0.016$


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

What is the maximum value of $\frac{(3^t-2t)t}{9^t}$ for real values of $t$?
A) $\frac{1}{10}$
B) $\frac{1}{12}$
C) $\frac{1}{8}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$

Answer: $\frac{1}{8}$


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

What is the base $2$ representation of $125_{10}$?

Answer: $1111101_2$


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 $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$.
(1) Find the period of $f(x)$;
(2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$.

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


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>.

In $\triangle ABC$, $\frac{1}{2}\cos 2A = \cos^2 A - \cos A$.
(I) Find the measure of angle $A$;
(II) If $a=3$, $\sin B = 2\sin C$, find the area of $\triangle ABC$.

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


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>.

A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could not be collected in that market. Unfortunately, the economist-cryptographer made an error in decrypting the cryptogram—the digits of the tax revenue amount were identified in the wrong order. Based on erroneous data, a decision was made to introduce an itemized tax on producers of 90 monetary units per unit of goods. It is known that the market demand is represented by \( Q_d = 688 - 4P \), and the market supply is linear. When there are no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times higher than the modulus of the price elasticity of the market demand function. After the tax was introduced, the producer price fell to 64 monetary units.

1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the itemized tax rate that would meet the ruler's decree.
4) What is the amount of tax revenue specified by the ruler to be collected?

Answer: $6480$


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

Given that $α∈(0, \dfrac {π}{2})$ and $β∈(0, \dfrac {π}{2})$, with $cosα= \dfrac {1}{7}$ and $cos(α+β)=- \dfrac {11}{14}$, find the value of $sinβ$.

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


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 that the coefficient of the $x^3$ term in the expansion of $\left(x+a\right)\left(x-2\right)^5$ is $-60$, find the value of $a$.

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 point P(x, y) satisfies the equation (x-4 cos θ)^{2} + (y-4 sin θ)^{2} = 4, where θ ∈ R, find the area of the region that point P occupies.

Answer: $32 \pi$


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

If five squares of a  $3 \times 3$  board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?

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


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

On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day.

Answer: $46656$


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

Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.)

Answer: $84$


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

Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles right triangle where one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?
A) $\frac{1}{\sqrt{2}}$
B) $1$
C) $\sqrt{2}$
D) $\frac{\sqrt{2}}{2}$

Answer: $\sqrt{2}$


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

Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza.

Answer: $65.31\%$


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

Given that the center of the hyperbola is at the origin and one focus is F1(-$$\sqrt{5}$$, 0), if point P is on the hyperbola and the midpoint of segment PF1 has coordinates (0, 2), then the equation of this hyperbola is _________ and its eccentricity is _________.

Answer: $\sqrt{5}$


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

Peter borrows $2000$ dollars from John, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Peter will owe more than three times as much as he borrowed?

Answer: $19$


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

The sum of the numbers 300, 2020, and 10001 is

Answer: $12321$


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

If a computer executes the following program:
1. Initial values \( x = 3 \), \( S = 0 \)
2. \( x = x + 2 \)
3. \( S = S + x \)
4. If \( S \geq 10000 \), go to step 5; otherwise, repeat from step 2
5. Print \( x \)
6. Stop

What value will be printed by step 5?

Answer: $201$