You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 999 inclusive does the decimal representation of \( \frac{n}{1000} \) terminate?

Answer: $999$


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

In square  $ABCD$  with side length  $2$ , let  $M$  be the midpoint of  $AB$ . Let  $N$  be a point on  $AD$  such that  $AN = 2ND$ . Let point  $P$  be the intersection of segment  $MN$  and diagonal  $AC$ . Find the area of triangle  $BPM$ .

*Proposed by Jacob Xu*

Answer: $2/7$


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

Given that in rhombus $ABCD$, the diagonal $AC$ is twice as long as diagonal $BD$, and point $E$ is on segment $AC$ such that line segment $BE$ bisects $\angle ABC$, determine the ratio of the area of triangle $ABE$ to the area of rhombus $ABCD$.

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


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

Given the function $f(x)=2\sin (wx+\varphi+ \frac {\pi}{3})+1$ ($|\varphi| < \frac {\pi}{2},w > 0$) is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $\frac {\pi}{2}$.
$(1)$ Find the value of $f( \frac {\pi}{8})$.
$(2)$ When $x\in(- \frac {\pi}{2}, \frac {3\pi}{2})$, find the sum of the real roots of the equation $f(x)= \frac {5}{4}$.

Answer: $2\pi$


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

A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 576, and the sum of the lengths of its 12 edges is 168. Determine the radius $s$.

Answer: $3\sqrt{33}$


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

Given a configuration of four unit squares arranged in a 2x2 grid, find the area of triangle $\triangle ABC$, where $A$ is the midpoint of the top side of the top-left square, $B$ is the bottom-right corner of the bottom-right square, and $C$ is the midpoint of the right side of the bottom-right square.

Answer: $0.375$


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

Find the smallest constant \(C\) such that
\[ x^2 + y^2 + z^2 + 1 \ge C(x + y + z) \]
for all real numbers \(x, y,\) and \(z.\)

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


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

Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes.

Answer: $101.25$


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>.

Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*

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>.

The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 400 Newtons. What is the gravitational force (in Newtons) that the Earth exerts on her when she's standing on a space station, 360,000 miles from the center of the earth? Express your answer as a fraction.

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


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

From 5 students, 4 are to be selected to participate in competitions in four subjects: mathematics, physics, chemistry, and biology, with each competition having only 1 participant. If student A does not participate in the biology competition, then the total number of different selection schemes is \_\_\_\_\_\_.

Answer: $96$


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

In $\triangle ABC$, if $a= \sqrt {5}$, $b= \sqrt {15}$, $A=30^{\circ}$, then $c=$ \_\_\_\_\_\_.

Answer: $2 \sqrt {5}$


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

Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______.

Answer: $\dfrac{\sqrt{5}}{2}$


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>.

Sandy and Sam each selected a positive integer less than 250. Sandy's number is a multiple of 15, and Sam's number is a multiple of 20. What is the probability that they selected the same number? Express your answer as a common fraction.

Answer: $\frac{1}{48}$


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

In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \(r\), find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron. Express your answer as a simplified fraction.

Answer: $\frac{1}{27}$


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

What is the greatest common factor of 120, 180, and 300?

Answer: $60$


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

The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) is equal to:

Answer: $0.333$


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

In the cube ABCD-A1B1C1D1, the angle formed by the skew lines A1B and AC is \_\_\_\_\_\_°; the angle formed by the line A1B and the plane A1B1CD is \_\_\_\_\_\_\_\_\_°.

Answer: $30$


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

The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$.

Answer: $-1$


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

Given an ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(0,4)$, with an eccentricity of $\frac{3}{5}$
1. Find the equation of $C$;
2. Find the length of the line segment intercepted by $C$ on the line passing through point $(3,0)$ with a slope of $\frac{4}{5}$.

Answer: $\frac{41}{5}$


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>.

Crestview's school has expanded its official colors to include blue along with the original purple and gold. The students need to design a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes may be of the same color?

Answer: $27$


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

A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?

Answer: $\sqrt{244}$


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

Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.

Answer: $\frac{1}{62}$


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

In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac {EQ}{EF} = \frac {23}{1005}$, and point $R$ is on $\overline{EH}$ such that $\frac {ER}{EH} = \frac {23}{2011}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find $\frac {EG}{ES}$.

Answer: $131$


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

Given that $|\vec{a}|=4$, and $\vec{e}$ is a unit vector. When the angle between $\vec{a}$ and $\vec{e}$ is $\frac{2\pi}{3}$, the projection of $\vec{a} + \vec{e}$ on $\vec{a} - \vec{e}$ is ______.

Answer: $\frac{5\sqrt{21}}{7}$


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 the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known that the line $l: \rho(2\cos\theta-\sin\theta)=6$.
(1) After stretching all the x-coordinates and y-coordinates of points on curve $C_1$ by $\sqrt{3}$ and 2 times respectively, curve $C_2$ is obtained. Please write down the Cartesian equation of line $l$ and the parametric equation of curve $C_2$;
(2) Find a point P on curve $C_2$ such that the distance from point P to line $l$ is maximized, and calculate this maximum value.

Answer: $2\sqrt{5}$


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

A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7?

Answer: $10\%$


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

Given the function $f(x)=\sin x\cos x-\sqrt{3}\cos^{2}x$.

(1) Find the smallest positive period of $f(x)$;

(2) Find the maximum and minimum values of $f(x)$ when $x\in[0,\frac{\pi }{2}]$.

Answer: $-\sqrt{3}$


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

Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \cdots + 10_6^2$.

Answer: $231_6$


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>.

For any real number  $x$ , we let  $\lfloor x \rfloor$  be the unique integer  $n$  such that  $n \leq x < n+1$ . For example.  $\lfloor 31.415 \rfloor = 31$ . Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right  \rfloor (2021).\]

*2021 CCA Math Bonanza Team Round #3*

Answer: $2020$


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

Convert the decimal number 2011 to a base-7 number.

Answer: $5602_7$


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 an arithmetic sequence $\{a_n\}$ with a non-zero common difference, $a_1$, $a_2$, and $a_5$ form a geometric sequence, and the sum of the first $10$ terms of this sequence is $100$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, and it satisfies $S_n=2b_n-1$.

$(I)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

$(II)$ Let $C_n=a_n+\log_{\sqrt{2}} b_n$. The sum of the first $n$ terms of the sequence $\{C_n\}$ is $T_n$. If the sequence $\{d_n\}$ is an arithmetic sequence, and $d_n= \frac{T_n}{n+c}$, where $c\neq 0$.

$(i)$ Find the non-zero constant $C$;

$(ii)$ If $f(n)=\frac{d_n}{(n+36)d_{n+1}}$ $(n\in \mathbb{N}^*)$, find the maximum value of the term in the sequence $\{f(n)\}$.

Answer: $\frac{1}{49}$


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

Determine the minimum value of the function $$y = \frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \(x > 1\).

Answer: $\frac{16 - 2\sqrt{7}}{3}$


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 Cartesian coordinate system, O is the origin. Given vector $\overrightarrow{a}=(2,1)$, point $A(1,0)$, and point $B(\cos\theta,t)$,
(1) If $\overrightarrow{a} \parallel \overrightarrow{AB}$ and $|\overrightarrow{AB}| = \sqrt{5}|\overrightarrow{OA}|$, find the coordinates of vector $\overrightarrow{OB}$.
(2) If $\overrightarrow{a} \perp \overrightarrow{AB}$, find the minimum value of $y=\cos^2\theta-\cos\theta+\left(\frac{t}{4}\right)^2$.

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


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

Eyes are the windows of the soul. In order to protect students' eyesight, Qihang High School conducts eye examinations for students every semester. The table below shows the results of the right eye vision examination for 39 students in a certain class at the school. In this set of vision data, the median is ______.

| Vision | $4.0$ | $4.1$ | $4.2$ | $4.3$ | $4.4$ | $4.5$ | $4.6$ | $4.7$ | $4.8$ | $4.9$ | $5.0$ |
|--------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| Number | $1$   | $2$   | $6$   | $3$   | $3$   | $4$   | $1$   | $2$   | $5$   | $7$   | $5$   |

Answer: $4.6$


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

The sum of all of the digits of the integers from 1 to 2008 is:

Answer: $28054$


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

Given an ellipse C: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\overrightarrow{BO}=2\overrightarrow{A_{1}A}$, determine the eccentricity of C.

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


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

A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.
A) $40 + 2\pi$
B) $36 + 4\pi$
C) $40 + 4\pi$
D) $44 + 4\pi$
E) $48 + 2\pi$

Answer: $40 + 4\pi$


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

In the rectangular coordinate system xOy, the parametric equations of the curve C are $$\begin{cases} x=1+2\cos\theta, \\ y= \sqrt {3}+2\sin\theta\end{cases}$$ (where θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the straight line l₁ is $$θ=α(0＜α＜ \frac {π}{2})$$. Rotate the straight line l₁ counterclockwise around the pole O by $$\frac {π}{3}$$ units to obtain the straight line l₂.

1. Find the polar coordinate equations of C and l₂.
2. Suppose the straight line l₁ and the curve C intersect at O, A two points, and the straight line l₂ and the curve C intersect at O, B two points. Find the maximum value of |OA|+|OB|.

Answer: $4 \sqrt {3}$


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 triangle $\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $a\sin B=-\sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $S=2\sqrt{3}$, find the perimeter of $\triangle ABC$.

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


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>.

A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop?

Answer: $45$


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

$(1)$ If $A=\frac{a-1}{a+2}•\frac{{a}^{2}-4}{{a}^{2}-2a+1}÷\frac{1}{a-1}$, simplify $A$;<br/>$(2)$ If $a$ satisfies $a^{2}-a=0$, find the value of $A$.

Answer: $-2$


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

A point $P$ is randomly placed in the interior of right triangle $ABC$, with coordinates $A(0,6)$, $B(9,0)$, and $C(0,0)$. What is the probability that the area of triangle $APC$ is more than a third of the area of triangle $ABC$?

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


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

Remove all positive integers that are divisible by 7, and arrange the remaining numbers in ascending order to form a sequence $\{a_n\}$. Calculate the value of $a_{100}$.

Answer: $116$


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

A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.

Answer: $\frac{37}{61}$


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>.

To address the threat of AIDS to humanity, now three research institutes, designated as Institute A, Institute B, and Institute C, are independently developing an AIDS vaccine. The probabilities of successfully developing a vaccine are respectively $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$. Calculate:
(1) The probability that exactly one research institute is successful in developing the vaccine;
(2) In order to achieve at least a $\frac{99}{100}$ probability that the vaccine is successfully developed (meaning at least one research institute is successful), how many institutes similar to Institute B are minimally required? (Reference data: $\lg 2=0.3010$, $\lg 3=0.4771$)

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>.

For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$.

Answer: $504$


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

Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, the magnitude of $\overrightarrow {a}$ is the positive root of the equation x^2+x-2=0, $|\overrightarrow {b}|= \sqrt {2}$, and $(\overrightarrow {a}- \overrightarrow {b})\cdot \overrightarrow {a}=0$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.

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


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

Given the function $f(x)=\sin \frac {πx}{6}$, and the set $M={0,1,2,3,4,5,6,7,8}$, calculate the probability that $f(m)f(n)=0$ for randomly selected different elements $m$, $n$ from $M$.

Answer: $\frac{5}{12}$


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>.

Roll two dice consecutively. Let the number on the first die be $m$, and the number on the second die be $n$. Calculate the probability that:
(1) $m+n=7$;
(2) $m=n$;
(3) The point $P(m,n)$ is inside the circle $x^2+y^2=16$.

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


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

All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list?

Answer: $165$


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

Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ with distinct terms, given that $a_3a_5=3a_7$, and $S_3=9$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$, find the maximum value of $\frac{T_n}{a_{n+1}}$.

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


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

The operation $ \diamond $ is defined for positive integers $a$ and $b$ such that $a \diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \diamond x$ is a perfect square.

Answer: $19$


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

Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with positive area.

Answer: $30$


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

A numerical sequence is defined by the conditions: \( a_{1} = 1 \), \( a_{n+1} = a_{n} + \left\lfloor \sqrt{a_{n}} \right\rfloor \).

How many perfect squares are there among the first terms of this sequence that do not exceed \( 1{,}000{,}000 \)?

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 a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$.
$(1)$ Find the analytical expression of the function $f\left(x\right)$;
$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\right)+2\left(1-m\right)x$ over $x\in \left[-2,5\right]$ is $13$, determine the value of the real number $m$.

Answer: $m = 2$


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 relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were randomly selected, with 5 students from each class participating in the survey. The scores of the 5 students in class A were: 5, 8, 9, 9, 9; the scores of the 5 students in class B were: 6, 7, 8, 9, 10.
(Ⅰ) Please estimate which class, A or B, has more stable questionnaire scores;
(Ⅱ) If we consider the scores of the 5 students in class B as a population and use a simple random sampling method to draw a sample with a size of 2, calculate the probability that the absolute difference between the sample mean and the population mean is not less than 1.

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


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

Given data: $2$, $5$, $7$, $9$, $11$, $8$, $7$, $8$, $10$, the $80$th percentile is ______.

Answer: $10$


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

Given the function $f(x)=\sin (2x+ \frac {\pi}{3})$.
$(1)$ If $x\in(- \frac {\pi}{6},0]$, find the minimum value of $4f(x)+ \frac {1}{f(x)}$ and determine the value of $x$ at this point;
$(2)$ If $(a\in(- \frac {\pi}{2},0),f( \frac {a}{2}+ \frac {\pi}{3})= \frac { \sqrt {5}}{5})$, find the value of $f(a)$.

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>.

Given the function g(n) = log27n if log27n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n).

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>.

The Greater Fourteen Basketball League has two divisions, each containing seven teams. Each team plays each of the other teams in its own division twice and every team in the other division twice. How many league games are scheduled?

Answer: $182$


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

Given that six balls are numbered 1, 2, 3, 4, 5, and 6, and the requirement that ball number 1 must be adjacent to ball number 2, and ball number 5 must not be adjacent to ball number 6, calculate the total number of different arrangements.

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 the function $f(x)=2\ln (3x)+8x+1$, find the value of $\underset{\Delta x\to 0}{{\lim }}\,\dfrac{f(1-2\Delta x)-f(1)}{\Delta x}$.

Answer: $-20$


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

Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.
A) $\frac{70}{7}$
B) $\frac{75}{7}$
C) $\frac{80}{7}$
D) $10$
E) $15$

Answer: $\frac{75}{7}$


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 coefficient of $x^2$ in the expansion of $(1+2x)^5$ is __________.

Answer: $40$


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

Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), what is the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \)?

Answer: $390$


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

Determine the constant term in the expansion of $(x-2+ \frac {1}{x})^{4}$.

Answer: $70$


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

The square \( STUV \) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \( 40 \text{ cm} \). What is the area, in \( \text{cm}^2 \), of the square \( STUV \)?
(a) 400
(b) 200
(c) 160
(d) 100
(e) 80

Answer: $400$


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

In $\triangle ABC$, $A=30^{\circ}$, $2 \overrightarrow{AB}\cdot \overrightarrow{AC}=3 \overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\triangle ABC$.

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>.

Suppose  $a_1, a_2, a_3, \dots$  is an increasing arithmetic progression of positive integers.  Given that  $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen*

Answer: $365$


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

The rapid growth of the Wuhan Economic and Technological Development Zone's economy cannot be separated from the support of the industrial industry. In 2022, the total industrial output value of the entire area reached 3462.23 billion yuan, ranking first in the city. Express the number 3462.23 in scientific notation as ______.

Answer: $3.46223 \times 10^{3}$


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

The parabola with the equation \( y = 25 - x^2 \) intersects the \( x \)-axis at points \( A \) and \( B \).

(a) Determine the length of \( AB \).

(b) Rectangle \( ABCD \) is formed such that \( C \) and \( D \) are below the \( x \)-axis and \( BD = 26 \). Determine the length of \( BC \).

(c) If \( CD \) is extended in both directions, it meets the parabola at points \( E \) and \( F \). Determine the length of \( EF \).

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>.

Al-Karhi's rule for approximating the square root. If \(a^{2}\) is the largest square contained in the given number \(N\), and \(r\) is the remainder, then

$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2a+1}, \text{ if } r<2a+1
$$

Explain how Al-Karhi might have derived this rule. Estimate the error by calculating \(\sqrt{415}\) in the usual way to an accuracy of \(0.001\).

Answer: $20.366$


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

Calculate the value of the expression \[(5^{1003}+6^{1004})^2-(5^{1003}-6^{1004})^2\] and express it in the form of $k\cdot30^{1003}$ for some positive 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 $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$

Answer: $-145$


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

Simplify first, then evaluate: $(1-\frac{1}{a+1})÷\frac{a^2-2a+1}{a^2-1}$, where $a=-2$.

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


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

Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points.

Answer: $\sqrt{9653} + \sqrt{5} + \sqrt{5220} + \sqrt{68}$


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

What is the sum of the odd positive integers less than 50?

Answer: $625$


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

Given the function $f(x)= \begin{cases}x-1,0 < x\leqslant 2 \\ -1,-2\leqslant x\leqslant 0 \end{cases}$, and $g(x)=f(x)+ax$, where $x\in[-2,2]$, if $g(x)$ is an even function, find the value of the real number $a$.

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


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

Calculate:
$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;
$(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$.

Answer: $11 - \sqrt{3}$


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 vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,2)$.
(1) If vector $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to vector $\overrightarrow{a}-3\overrightarrow{b}$, find the value of the real number $k$;
(2) For what value of $k$ are vectors $k\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-3\overrightarrow{b}$ parallel? And determine whether they are in the same or opposite direction.

Answer: $-\dfrac{1}{3}$


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

Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \).

Answer: $31$


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

In a table tennis singles match between players A and B, the match is played in a best-of-seven format (i.e., the first player to win four games wins the match, and the match ends). Assume that both players have an equal chance of winning each game.
(1) Calculate the probability that player B wins by a score of 4 to 1;
(2) Calculate the probability that player A wins and the number of games played is more than 5.

Answer: $\frac{5}{16}$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a\cos B=(3c-b)\cos A$.
$(1)$ If $a\sin B=2\sqrt{2}$, find $b$;
$(2)$ If $a=2\sqrt{2}$ and the area of $\triangle ABC$ is $\sqrt{2}$, find the perimeter of $\triangle ABC$.

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


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>.

Mark's cousin has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all of the stickers on the sheets of paper, given that each sheet must have at least one sticker, and only the number of stickers on each sheet matters?

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>.

Let $\{a_n\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n = \frac{1}{\log_{2}a_n \cdot \log_{2}a_{n+1}}$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the minimum value of $T_n$.

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>.

For how many integer values of $n$ between 1 and 990 inclusive does the decimal representation of $\frac{n}{1000}$ terminate?

Answer: $990$


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

Given the function $f(x)=4-x^{2}+a\ln x$, if $f(x)\leqslant 3$ for all $x > 0$, determine the range of the real number $a$.

Answer: $[2]$


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

Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum.

Answer: $12$


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

Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate the eccentricity of the hyperbola.

Answer: $\frac{\sqrt{5} + 1}{2}$


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

In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. What is the length of side $AB$?

Answer: $\sqrt{17}$


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

A mini soccer team has 12 members. We want to choose a starting lineup of 5 players, which includes one goalkeeper and four outfield players (order of outfield players does not matter). In how many ways can we choose this starting lineup?

Answer: $3960$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 a geometric sequence, with $a_1=2$, common ratio $q>0$, and $a_2$, $6$, $a_3$ forming an arithmetic sequence.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Let $b_n=\log_2{a_n}$, and $$T_{n}= \frac {1}{b_{1}b_{2}}+ \frac {1}{b_{2}b_{3}}+ \frac {1}{b_{3}b_{4}}+…+ \frac {1}{b_{n}b_{n+1}}$$, find the maximum value of $n$ such that $$T_{n}< \frac {99}{100}$$.

Answer: $98$


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

Given the function $f(x)=x^2+x+b\ (b\in\mathbb{R})$ with a value range of $[0,+\infty)$, the solution to the equation $f(x) < c$ is $m+8$. Determine the value of $c$.

Answer: $16$


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

Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:

$\sqrt{1+\frac{5}{4}}=\sqrt{\frac{9}{4}}=\sqrt{{(\frac{3}{2})}^{2}}=\frac{3}{2}$;
$\sqrt{1+\frac{7}{9}}=\sqrt{\frac{16}{9}}=\sqrt{{(\frac{4}{3})}^{2}}=\frac{4}{3}$;
$\sqrt{1+\frac{9}{16}}=\sqrt{\frac{25}{16}}=\sqrt{{(\frac{5}{4})}^{2}}=\frac{5}{4}$;
$\ldots$

Practice and exploration:
$(1)$ According to this pattern, calculate: $\sqrt{1+\frac{17}{64}}=$______;
$(2)$ Calculate: $\sqrt{1+\frac{5}{4}}×\sqrt{1+\frac{7}{9}}×\sqrt{1+\frac{9}{16}}×⋅⋅⋅×\sqrt{1+\frac{21}{100}}$;
Transfer and application:
$(3)$ If $\sqrt{1+\frac{2023}{{n}^{2}}}=x$ follows the above pattern, please directly write down the value of $x$.

Answer: $\frac{1012}{1011}$


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

Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\frac {\sin B}{\sin A} = \cos(A + B),$$ find the maximum value of $\tan B$.

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


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>.

Xiaoying goes home at noon to cook noodles by herself, which involves the following steps: ① Wash the pot and fill it with water, taking 2 minutes; ② Wash the vegetables, taking 3 minutes; ③ Prepare the noodles and seasonings, taking 2 minutes; ④ Boil the water in the pot, taking 7 minutes; ⑤ Use the boiling water to cook the noodles and vegetables, taking 3 minutes. Except for step ④, each step can only be performed one at a time. The minimum time Xiaoying needs to cook the noodles is     minutes.

Answer: $12$


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

Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\frac{20}{3}-x$.

Answer: $16$


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

Points $A$, $B$, $C$, $D$ are on the same sphere, with $AB=BC=\sqrt{2}$, $AC=2$. If the circumscribed sphere of tetrahedron $ABCD$ has its center exactly on edge $DA$, and $DC=2\sqrt{3}$, then the surface area of this sphere equals \_\_\_\_\_\_\_\_\_\_

Answer: $16\pi$


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>.

Find the greatest common factor of 8! and 9!.

Answer: $40320$


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

The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&32&43&58&&\\
1&05&10&12&15&20\\
2&25&30&55&&\\
3&00&15&30&35&45\\
4&10&12&&&\\
\end{tabular}

Answer: $147.5$


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>.

What is the smallest positive integer \( n \) such that all the roots of \( z^6 - z^3 + 1 = 0 \) are \( n^{\text{th}} \) roots of unity?

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 that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is

Answer: $-1$


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

Given that $f(x)$ is an odd function defined on $\mathbb{R}$ with a minimum positive period of 3, and for $x \in \left(-\frac{3}{2}, 0\right)$, $f(x)=\log_{2}(-3x+1)$. Find $f(2011)$.

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 $\theta∈(0,\frac{π}{2}$, $\sin \theta - \cos \theta = \frac{\sqrt{5}}{5}$, find the value of $\tan 2\theta$.

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


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

In the numbers from 100 to 999, how many numbers have digits in strictly increasing or strictly decreasing order?

(From the 41st American High School Mathematics Exam, 1990)

Answer: $204$


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

Given triangle $ABC$ with midpoint $D$ on side $BC$, and point $G$ satisfies $\overrightarrow{GA}+ \overrightarrow{BG}+ \overrightarrow{CG}= \overrightarrow{0}$, and $\overrightarrow{AG}=\lambda \overrightarrow{GD}$, determine the value of $\lambda$.

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 the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and $f(-1)=2$, and the period of the function is $4$, calculate the values of $f(2012)$ and $f(2013)$.

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>.

One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?

A) 62

B) 152

C) 158

D) 162

E) 172

Answer: $162$


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

There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$.

Answer: $\frac{429}{1024}$


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

Given that the year 2010 corresponds to the Geng-Yin year, determine the year of the previous Geng-Yin year.

Answer: $1950$


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

There are two lathes processing parts of the same model. The yield rate of the first lathe is $15\%$, and the yield rate of the second lathe is $10\%$. Assuming that the yield rates of the two lathes do not affect each other, the probability of both lathes producing excellent parts simultaneously is ______; if the processed parts are mixed together, knowing that the number of parts processed by the first lathe accounts for $60\%$ of the total, and the number of parts processed by the second lathe accounts for $40\%$, then randomly selecting a part, the probability of it being an excellent part is ______.

Answer: $13\%$


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

Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}=k+2( \frac {1}{3})^{n}$, find the value of the constant $k$.

Answer: $-2$


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

Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + x + x/3 + 1$. Find the sum of all possible values of $g(49)$.

Answer: $\frac{116}{5}$


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>.

Triangle  $ABC$  has sidelengths  $AB=1$ ,  $BC=\sqrt{3}$ , and  $AC=2$ . Points  $D,E$ , and  $F$  are chosen on  $AB, BC$ , and  $AC$  respectively, such that  $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of  $[DEF]^2$  can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $\gcd (a, b) = 1$ , find  $a + b$ . (Here  $[DEF]$  denotes the area of triangle  $DEF$ .)

*Proposed by Vismay Sharan*

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>.

Compute $\sin 12^\circ \sin 36^\circ \sin 72^\circ \sin 84^\circ.$

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


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

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

Answer: $3507840$


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

In recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a total of 2920 yuan. The unit price of type A helmets is 11 yuan higher than the unit price of type B helmets.
$(1)$ What are the unit prices of type A and type B helmets, respectively?
$(2)$ The store decides to purchase 40 more helmets of type A and type B. They coincide with a promotional event by the manufacturer: type A helmets are sold at 20% off the unit price, and type B helmets are discounted by 6 yuan each. If the number of type A helmets purchased this time is not less than half the number of type B helmets, how many type A helmets should be purchased to minimize the total cost of this purchase? What is the minimum cost?

Answer: $1976$


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

Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of this octahedron.

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


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

Given $sinα+cosα=-\frac{{\sqrt{10}}}{5}, α∈(-\frac{π}{2},\frac{π}{2})$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $2\sin ^{2}\alpha +\sin \alpha \cos \alpha -1$.

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


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

In daily life, specific times are usually expressed using the 24-hour clock system. There are a total of 24 time zones globally, with adjacent time zones differing by 1 hour. With the Prime Meridian located in Greenwich, England as the reference point, in areas east of Greenwich, the time difference is marked with a "+", while in areas west of Greenwich, the time difference is marked with a "-". The table below shows the time differences of various cities with respect to Greenwich:

| City   | Beijing | New York | Sydney | Moscow |
|--------|---------|----------|--------|--------|
| Time Difference with Greenwich (hours) | +8 | -4 | +11 | +3 |

For example, when it is 12:00 in Greenwich, it is 20:00 in Beijing and 15:00 in Moscow.

$(1)$ What is the time difference between Beijing and New York?
$(2)$ If Xiao Ming in Sydney calls Xiao Liang in New York at 21:00, what time is it in New York?
$(3)$ Xiao Ming takes a direct flight from Beijing to Sydney at 23:00 on October 27th. After 12 hours, he arrives. What is the local time in Sydney when he arrives?
$(4)$ Xiao Hong went on a study tour to Moscow. After arriving in Moscow, he calls his father in Beijing at an exact hour. At that moment, his father's time in Beijing is exactly twice his time in Moscow. What is the specific time in Beijing when the call is connected?

Answer: $10:00$


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

For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci.

Answer: $\frac{2\sqrt{462}}{5}$


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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt {3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt {13}$ and $c=3$, find the area of $\triangle ABC$.

Answer: $3 \sqrt {3}$