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

Suppose two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of their first $n$ terms as $S_n$ and $T_n$, respectively. Given that $\frac{S_n}{T_n} = \frac{7n}{n+3}$, find the value of $\frac{a_5}{b_5}$.

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


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 that the sequence {an} is a decreasing geometric sequence and satisfies the conditions $$a_{2}a_{7}= \frac {1}{2}$$ and $$a_{3}+a_{6}= \frac {9}{4}$$, find the maximum value of a1a2a3…a2n.

Answer: $64$


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

Given that $a+b+c=0$, calculate the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}+\frac{|ab|}{ab}+\frac{|ac|}{ac}+\frac{|bc|}{bc}+\frac{|abc|}{abc}$.

Answer: $-1$


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

Determine how many integer palindromes are between 200 and 700.

Answer: $50$


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

A student, Theo, needs to earn a total of 30 homework points. For the first six homework points, he has to do one assignment each; for the next six points, he needs to do two assignments each; and so on, such that for every subsequent set of six points, the number of assignments he needs to complete doubles the previous set. Calculate the minimum number of homework assignments necessary for Theo to earn all 30 points.

Answer: $186$


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

Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employees from department Y that will shake hands with the employees from department X also equals 20. Find the total number of handshakes that occur between employees of different departments.

Answer: $600$


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

Given the function \( y = y_1 + y_2 \), where \( y_1 \) is directly proportional to \( x^2 \) and \( y_2 \) is inversely proportional to \( x^2 \). When \( x = 1 \), \( y = 5 \); when \( x = \sqrt{3} \), \( y = 7 \). Find the value of \( x \) when \( y \) is minimized.

Answer: $\sqrt[4]{\frac{3}{2}}$


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

Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear.

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


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

How many three-digit whole numbers contain at least one 6 or at least one 8?

Answer: $452$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $\{a_n\}$, the sum $a_1 + a_2 + \ldots + a_5 = 30$, and the sum $a_6 + a_7 + \ldots + a_{10} = 80$. Calculate the sum $a_{11} + a_{12} + \ldots + a_{15}$.

Answer: $130$


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

A bag contains 3 red balls, 2 black balls, and 1 white ball. All 6 balls are identical in every aspect except for color and are well mixed. Balls are randomly drawn from the bag.
(1) With replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws;
(2) Without replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws.

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


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

In the geometric sequence $\{a_{n}\}$, $a_{20}$ and $a_{60}$ are the two roots of the equation $(x^{2}-10x+16=0)$. Find the value of $\frac{{{a}\_{30}}\cdot {{a}\_{40}}\cdot {{a}\_{50}}}{2}$.

Answer: $32$


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

In isosceles $\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\angle A$. If $B(1,3)$, find:
(I) The equation of line $BC$;
(II) The area of $\triangle ABC$.

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


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

A certain product in a shopping mall sells an average of 70 items per day, with a profit of $50 per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every $1 decrease in price, the mall can sell an additional 2 items per day. Let $x$ represent the price reduction per item. Based on this rule, please answer:
$(1)$ The daily sales volume of the mall increases by ______ items, and the profit per item is ______ dollars. (Express using algebraic expressions involving $x$)
$(2)$ With the above conditions unchanged, how much should the price of each item be reduced so that the mall's daily profit reaches $3572$ dollars.

Answer: $12$


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

Consider a kite-shaped field with the following measurements and angles: sides AB = 120 m, BC = CD = 80 m, DA = 120 m. The angle between sides AB and BC is 120°. The angle between sides CD and DA is also 120°. The wheat harvested from any location in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side, which in this case is side BC?

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


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

There is a tram with a starting station A and an ending station B. A tram departs from station A every 5 minutes towards station B, completing the journey in 15 minutes. A person starts cycling along the tram route from station B towards station A just as a tram arrives at station B. On his way, he encounters 10 trams coming towards him before reaching station A. At this moment, another tram is just departing from station A. How many minutes did it take for him to travel from station B to station A?

Answer: $50$


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

Given that the random variable $X$ follows a normal distribution $N(0,\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \leqslant X \leqslant 2)$.

Answer: $0.954$


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

When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \(a, b, c\) are obtained respectively where \(a\) and \(b\) are from the standard dice and \(c\) is from the 8-sided die. Find the probability that \((a-1)(b-1)(c-1) \neq 0\).

Answer: $\frac{175}{288}$


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

Given the function $f(x) = |2x+a| + |2x-2b| + 3$
(Ⅰ) If $a=1$, $b=1$, find the solution set of the inequality $f(x) > 8$;
(Ⅱ) When $a>0$, $b>0$, if the minimum value of $f(x)$ is $5$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$.

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


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

If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$?

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


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

Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + y^2 + 4}}\]
over all real numbers $x$ and $y$.

Answer: $\sqrt{29}$


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

If
\[
(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^m,
\]
then find $m.$

Answer: $45$


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

Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, then find $\tan \theta$.

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


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

Given the parabola $y^{2}=4x$, its focus intersects the parabola at two points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If $x_{1}+x_{2}=10$, find the length of the chord $AB$.

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>.

What is the sum of all the even integers between $200$ and $400$?

Answer: $30100$


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

Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.

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>.

What is the smallest number of rectangles, each measuring $2 \mathrm{~cm}$ by $3 \mathrm{~cm}$, which are needed to fit together without overlap to form a rectangle whose sides are in the ratio 5:4?

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 output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \_\_\_\_\_\_\_\_. (Keep one decimal place, take $1.1^{5} \approx 1.6$)

Answer: $6.6$


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

Given $x = \frac{2}{3}$ and $y = \frac{5}{2}$, find the value of $\frac{1}{3}x^8y^9$.

Answer: $\frac{5^9}{2 \cdot 3^9}$


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

In quadrilateral ABCD, m∠B = m∠C = 120°, AB = 4, BC = 6, and CD = 7. Diagonal BD = 8. Calculate the area of ABCD.

Answer: $16.5\sqrt{3}$


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

What are the last two digits in the sum of the factorials of the first 15 positive integers?

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>.

Among the following propositions, the true one is numbered \_\_\_\_\_\_.
(1) The negation of the proposition "For all $x>0$, $x^2-x\leq0$" is "There exists an $x>0$ such that $x^2-x>0$."
(2) If $A>B$, then $\sin A > \sin B$.
(3) Given a sequence $\{a_n\}$, "The sequence $a_n, a_{n+1}, a_{n+2}$ forms a geometric sequence" is a necessary and sufficient condition for $a_{n+1}^2=a_{n}a_{n+2}$.
(4) Given the function $f(x)=\lg x+ \frac{1}{\lg x}$, then the minimum value of $f(x)$ is 2.

Answer: $(1)$


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

Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \sqrt {2}\sin ωx-\cos ωx$ $(x\in\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω.

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>.

In a circle with center $O$, the measure of $\angle RIP$ is $45^\circ$ and $OR=15$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$.

Answer: $7.5\pi$


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

Given that the equation about $x$, $x^{2}-2a\ln x-2ax=0$ has a unique solution, find the value of the real number $a$.

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


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

Matrices  $A$ ,  $B$  are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0  1 & 2 & 0  0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0  2 & 4 & 0  0 & 0 & 12\end{pmatrix}\]
Find volume of  $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ .

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


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

Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$.

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>.

The magnitude of the vector $\overset{→}{a} +2 \overset{→}{b}$, where $\overset{→}{a} =(2,0)$, $\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\circ}$.

Answer: $2\sqrt{3}$


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

Given $m, n \in \mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$.

Answer: $2\sqrt{2}$


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>.

I have two 10-sided dice where each die has 3 gold sides, 4 silver sides, 2 diamond sides, and 1 rainbow side. If I roll both dice, what is the probability that they come up showing the same color or pattern?

Answer: $\frac{3}{10}$


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

Given the function $f(x)=\cos^{4}x-2\sin x\cos x-\sin^{4}x$.

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

(2) When $x\in\left[0,\frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values where the minimum value is obtained.

Answer: $\left\{\frac{3\pi}{8}\right\}$


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

In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$.

Answer: $90$


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

Abby, Bernardo, Carl, and Debra play a revised game where each starts with five coins and there are five rounds. In each round, five balls are placed in an urn—two green, two red, and one blue. Each player draws a ball at random without replacement. If a player draws a green ball, they give one coin to a player who draws a red ball. If anyone draws a blue ball, no transaction occurs for them. What is the probability that at the end of the fifth round, each of the players has five coins?
**A)** $\frac{1}{120}$
**B)** $\frac{64}{15625}$
**C)** $\frac{32}{3125}$
**D)** $\frac{1}{625}$
**E)** $\frac{4}{125}$

Answer: $\frac{64}{15625}$


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

Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth  $4$  and tails is worth  $2.$  What is the probability Katie’s score is strictly greater than Allie’s?

Answer: $25/72$


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

A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in centimetres?

A) \(\frac{1}{2}\)

B) \(2 - \sqrt{2}\)

C) \(\frac{1}{2} \sqrt{2}\)

D) \(5 - 3\sqrt{2}\)

E) \(2\sqrt{2} - 2\)

Answer: $2 - \sqrt{2}$


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>.

Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5?

Answer: $\frac{40}{1053}$


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

Calculate:
$(1)\frac{\sqrt{20}+\sqrt{5}}{\sqrt{5}}-2$;
$(2)\sqrt[3]{-8}+5\sqrt{\frac{1}{10}}-\sqrt{10}+\sqrt{4}$;
$(3)(\sqrt{3}-\sqrt{2})^2•(5+2\sqrt{6})$;
$(4)(π-3.14)^0+\frac{1}{2+\sqrt{3}}+(-\frac{1}{3})^{-1}+|1-\sqrt{3}|$.

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>.

Calculate the amount of personal income tax (НДФЛ) paid by Sergey over the past year if he is a resident of the Russian Federation and had a stable income of 30,000 rubles per month and a one-time vacation bonus of 20,000 rubles during this period. Last year, Sergey sold his car, which he inherited two years ago, for 250,000 rubles and bought a plot of land for building a house for 300,000 rubles. Sergey applied all applicable tax deductions. (Provide the answer without spaces and units of measurement.)

Answer: $10400$


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

The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage?

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>.

A rectangular prism measuring 20 cm by 14 cm by 12 cm has a small cube of 4 cm on each side removed from each corner. What percent of the original volume is removed?

Answer: $15.24\%$


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>.

Calculate the integral $\int_{2}^{7}(x-3)^{2} d x$.

a) Using the substitution $z=x-3$.

b) Using the substitution $z=(x-3)^{2}$.

Answer: $\frac{65}{3}$


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

There are 6 boxes, each containing a key that cannot be used interchangeably. If one key is placed in each box and all the boxes are locked, and it is required that after breaking open one box, the remaining 5 boxes can still be opened with the keys, then the number of ways to place the keys is ______.

Answer: $120$


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

Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice.

Answer: $6.5$


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

Given that $f(x)$ has a derivative and satisfies $\lim_{\Delta x \to 0} \, \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.

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 a sequence $\{a_n\}$, the first $m(m\geqslant 4)$ terms form an arithmetic sequence with a common difference of $2$. Starting from the $(m-1)$-th term, $a_{m-1}$, $a_{m}$, $a_{m+1}$, ... form a geometric sequence with a common ratio of $2$. If $a_{1}=-2$, then $m=$ ______, and the sum of the first $6$ terms of $\{a_n\}$, $S_{6}=$ ______.

Answer: $28$


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

Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola.

Answer: $\sqrt{5}$


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

Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$

$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.

$(2)$ Find the sum of the first $20$ terms of the sequence $\{a_{n}\}$.

Answer: $300$


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

Given a sequence ${a_n}$ that satisfies $a_1=1$ and $a_n=a_1+ \frac {1}{2}a_2+ \frac {1}{3}a_3+…+ \frac {1}{n-1}a_{n-1}$ for $n\geqslant 2, n\in\mathbb{N}^*$, if $a_k=2017$, then $k=$ \_\_\_\_\_\_.

Answer: $4034$


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

Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$.

Answer: $064$


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

Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion.

Answer: $375$


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>.

There is an opaque bag containing 4 identical balls labeled with the numbers 1, 2, 3, and 4.
(Ⅰ) If balls are drawn one by one without replacement twice, calculate the probability that the first ball drawn has an even number and the sum of the two balls’ numbers is divisible by 3.
(Ⅱ) If a ball is randomly taken from the bag and labeled as a, then put back into the bag, followed by randomly taking another ball, labeled as b, calculate the probability that the line $ax + by + 1 = 0$ has no common points with the circle $x^2 + y^2 = \frac{1}{16}$.

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 $\triangle XYZ$, the medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\triangle XYZ$.

Answer: $288$


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

Given the function $f\left(x\right)=\cos \left(\omega x+\varphi \right)\left(\omega\ \ \gt 0,0 \lt \varphi\ \ \lt \pi \right)$, if $f\left(x\right)$ is an odd function and monotonically decreasing on $(-\frac{π}{3},\frac{π}{6})$, then the maximum value of $\omega$ is ______.

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


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

Let $T$ be a subset of $\{1,2,3,...,60\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$?

Answer: $25$


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

Given the function $f(x)= \sqrt {3}\sin ^{2}x+\sin x\cos x- \frac { \sqrt {3}}{2}$ $(x\in\mathbb{R})$.
$(1)$ If $x\in(0, \frac {\pi}{2})$, find the maximum value of $f(x)$;
$(2)$ In $\triangle ABC$, if $A < B$ and $f(A)=f(B)= \frac {1}{2}$, find the value of $\frac {BC}{AB}$.

Answer: $\sqrt {2}$


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

The product of two consecutive page numbers is $2,156.$ What are the sum of these two page numbers?

Answer: $93$


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

What is the largest number of positive, consecutive integers whose sum is 105?

Answer: $14$


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

In the diagram, three identical circles touch each other, and each circle has a circumference of 24. Calculate the perimeter of the shaded region within the triangle formed by the centers of the circles.

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>.

Three different numbers are chosen from the set $\{-4, -3, -1, 3, 5, 8\}$ and multiplied. Find the largest possible positive product.

Answer: $120$


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

Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$?

Answer: $\frac{500}{3}$


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>.

(For science students) In the expansion of $(x^2 - 3x + 2)^4$, the coefficient of the $x^2$ term is __________ (Answer with a number).

Answer: $248$


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

Given that $α$ is an angle in the third quadrant and $\cos 2α=-\frac{3}{5}$, find $\tan (\frac{π}{4}+2α)$.

Answer: $-\frac{1}{7}$


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>.

Find the value of $m + n$ where $m$ and $n$ are integers such that the positive difference between the roots of the equation $4x^2 - 12x - 9 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, with $m$ not divisible by the square of any prime number.

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>.

Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that vertex B is one of the bottom vertices? There are three bottom vertices in total in a dodecahedron.

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>.

Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{2},$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of:
\[
\sqrt{4x + 2} + \sqrt{4y + 8} + \sqrt{4z + 12}.
\]

Answer: $3\sqrt{10}$


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

Given that $\sin\alpha + \cos\alpha = \frac{1}{5}$, and $0 \leq \alpha < \pi$, find the value of $\tan\alpha$.

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>.

Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride.

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>.

Let $P$ be a point on the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$, and let $F_{1}$ and $F_{2}$ be the left and right foci, respectively. If $|PF_{1}| = 9$, then find $|PF_{2}|$.

Answer: $17$


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

The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of  $A + E$ .

\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B  \hline 5 & C & 13 \hline D & E & 3  \hline\end{tabular} \]

Answer: $32$


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

The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers?

Answer: $103$


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

Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x<0, \\
6-3x&\text{if }x\ge 0.
\end{cases}
\]Find $f(-2)$ and $f(4)$.

Answer: $-6$


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

Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $\frac{S_6}{S_3} = 3$, then calculate $\frac{S_{12}}{S_{9}}$.

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


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

Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$.

Answer: $\sqrt{166}$


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

Read the material first, then answer the question.
$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\sqrt{5-2\sqrt{6}}$.
After thinking about it, Xiao Zhang's process of solving this problem is as follows:
$\sqrt{5-2\sqrt{6}}=\sqrt{2-2\sqrt{2\times3}+3}$①
$=\sqrt{{(\sqrt{2})}^2}-2\sqrt{2}\times\sqrt{3}+{(\sqrt{3})}^2$②
$=\sqrt{{(\sqrt{2}-\sqrt{3})}^2}$③
$=\sqrt{2}-\sqrt{3}$④
In the above simplification process, an error occurred in step ____, and the correct result of the simplification is ____;
$(2)$ Please simplify $\sqrt{8+4\sqrt{3}}$ based on the inspiration you obtained from the above material.

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


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>.

Solve the following equations using appropriate methods:
(1) $x^2=49$;
(2) $(2x+3)^2=4(2x+3)$;
(3) $2x^2+4x-3=0$ (using the formula method);
(4) $(x+8)(x+1)=-12$.

Answer: $-5$


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

Given the function $f(x)=\frac{1}{2}x^{2}-a\ln x+b$ where $a\in R$.
(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.
(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real number $a$.
(III) If $-2\leqslant a < 0$, for any $x_{1}$, $x_{2}\in(0,2]$, the inequality $|f(x_{1})-f(x_{2})|\leqslant m| \frac{1}{x_{1}}- \frac{1}{x_{2}}|$ always holds. Find the minimum value of $m$.

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 $\tan (\theta-\pi)=2$, then $\sin ^{2}\theta+\sin \theta\cos \theta-2\cos ^{2}\theta=$ \_\_\_\_\_\_ .

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


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

Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored?

Answer: $150$


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

How many positive integer multiples of $77$ (product of $7$ and $11$) can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$?

Answer: $784$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 the arithmetic series $1-3+5-7+9-11+\cdots +2021-2023+2025$.

Answer: $1013$


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

Given a hyperbola $C$ that shares the same foci with the ellipse $\frac{x^{2}}{27}+ \frac{y^{2}}{36}=1$ and passes through the point $(\sqrt{15},4)$.

(I) Find the equation of the hyperbola $C$.

(II) If $F\_1$ and $F\_2$ are the two foci of the hyperbola $C$, and point $P$ is on the hyperbola $C$ such that $\angle F\_1 P F\_2 = 120^{\circ}$, find the area of $\triangle F\_1 P F\_2$.

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


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>.

In triangle ABC, BR = RC, CS = 3SA, and (AT)/(TB) = p/q. If the area of △RST is twice the area of △TBR, determine the value of p/q.

Answer: $\frac{7}{3}$


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>.

Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\]

Answer: $\frac{97}{40}$


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 that the random variable X follows a normal distribution N(2, σ²) and P(X≤4)=0.88, find P(0<X<4).

Answer: $0.76$


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

Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______.

Answer: $-3$


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

Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$.

Answer: $676$


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

Let the function $f(x)=\ln x- \frac{1}{2}ax^{2}-bx$.

$(1)$ When $a=b= \frac{1}{2}$, find the maximum value of the function $f(x)$;

$(2)$ Let $F(x)=f(x)+ \frac{1}{2}ax^{2}+bx+ \frac{a}{x}$, $(0 < x\leqslant 3)$, the slope of the tangent line at any point $P(x_{0},y_{0})$ on its graph is $k\leqslant \frac{1}{2}$ always holds, find the range of the real number $a$;

$(3)$ When $a=0$, $b=-1$, the equation $2mf(x)=x^{2}$ has a unique real solution, find the value of the positive number $m$.

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>.

Use the bisection method to find an approximate solution for $f(x)=0$. Given that $f(1)=-2$, $f(3)=0.625$, and $f(2)=-0.984$, calculate the next $f(m)$, where $m =$ __________.

Answer: $2.5$


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>.

(1) If 7 students stand in a row, and students A and B must stand next to each other, how many different arrangements are there?
(2) If 7 students stand in a row, and students A, B, and C must not stand next to each other, how many different arrangements are there?
(3) If 7 students stand in a row, with student A not standing at the head and student B not standing at the tail, how many different arrangements are there?

Answer: $3720$


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

Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy?

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 repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?

Answer: $11$


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

To solve the problem, we need to find the value of $\log_{4}{\frac{1}{8}}$.
A) $-\frac{1}{2}$
B) $-\frac{3}{2}$
C) $\frac{1}{2}$
D) $\frac{3}{2}$

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


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

Find the sum of $642_8$ and $157_8$ in base $8$.

Answer: $1021_8$


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

Given that $f(x)$ is a periodic function defined on $\mathbb{R}$ with a period of $2$, when $x \in (-1, 1]$, $f(x)=\begin{cases} -4x^{2}+ \frac{9}{8},-1 < x\leqslant 0, \\ \log _{2}x,0 < x\leqslant 1, \end{cases}$, find the value of $f(f( \frac{7}{2}))=\_\_\_\_\_\_\_\_$.

Answer: $-3$


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

Charlie is planning to construct a boundary around a rectangular playground using exactly 380 feet of fencing. Regulations specify that the length of the boundary must be at least 100 feet and the width must be at least 60 feet. Charlie wants to maximize the area enclosed by the fence for more play equipment and sitting area. What are the optimal dimensions and the maximum possible area of the playground?

Answer: $9000$


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

Given vectors $\overrightarrow {a}$=（sinx，cosx）, $\overrightarrow {b}$=（1，$\sqrt {3}$）.
(1) If $\overrightarrow {a}$$∥ \overrightarrow {b}$, find the value of tanx;
(2) Let f(x) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$, stretch the horizontal coordinates of each point on the graph of f(x) to twice their original length (vertical coordinates remain unchanged), then shift all points to the left by φ units (0 < φ < π), obtaining the graph of function g(x). If the graph of g(x) is symmetric about the y-axis, find the value of φ.

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


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

The distance from the point $(3,0)$ to one of the asymptotes of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ is $\frac{9}{5}$.

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


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

Given the function $f(x)=ax + a^{-x}$ ($a>0$ and $a\neq1$), and $f(1)=3$, find the value of $f(0)+f(1)+f(2)$.

Answer: $12$


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

If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company.

Answer: $8.1$


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>.

Find $x$ such that $\log_x 49 = \log_2 32$.

Answer: $7^{2/5}$


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 the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$.

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


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

John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property?

Answer: $456976$


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

On June 14, 2018, the 21st FIFA World Cup will kick off in Russia. A local sports channel organized fans to guess the outcomes of the matches for the top four popular teams: Germany, Spain, Argentina, and Brazil. Each fan can choose one team from the four, and currently, three people are participating in the guessing game.
$(1)$ If each of the three people can choose any team and the selection of each team is equally likely, find the probability that exactly two teams are chosen by people.
$(2)$ If one of the three people is a female fan, assuming the probability of the female fan choosing the German team is $\frac{1}{3}$ and the probability of a male fan choosing the German team is $\frac{2}{5}$, let $\xi$ be the number of people choosing the German team among the three. Find the probability distribution and the expected value of $\xi$.

Answer: $\frac{17}{15}$


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

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

Answer: $495$


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

Let \( x, y, z \) be complex numbers such that:
\[
xy + 3y = -9,
yz + 3z = -9,
zx + 3x = -9.
\]
Determine all possible values of \( xyz \).

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>.

Find the smallest positive integer that is both an integer power of 7 and is not a palindrome.

Answer: $2401$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 squares of the solutions to the equation
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\]

Answer: $\frac{2008}{1005}$


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

Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$.

Answer: $\frac{100}{101}$


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

If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$.

Answer: $-7$


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

$ABC$  is triangle.  $l_1$ - line passes through  $A$  and parallel to  $BC$ ,  $l_2$  -  line passes through  $C$  and parallel to  $AB$ . Bisector of  $\angle B$  intersect  $l_1$  and  $l_2$  at  $X,Y$ .  $XY=AC$ . What  value can take  $\angle A- \angle C$  ?

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>.

If one can find a student with at least  $k$  friends in any class which has  $21$  students such that at least two of  any three of these students are friends, what is the largest possible value of  $k$ ?

Answer: $10$


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

A student named Zhang has a set of 6 questions to choose from, with 4 categorized as type A and 2 as type B. Zhang randomly selects 2 questions to solve.
(1) What is the probability that both selected questions are type A?
(2) What is the probability that the selected questions are not of the same type?

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


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

We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively?

Answer: $\dfrac{1}{1024}$


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 Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot.

Answer: $49$


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

Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes.

Answer: $50$


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

We need to arrange the performance order for 4 singing programs and 2 skit programs. The requirement is that there must be exactly 3 singing programs between the 2 skit programs. The number of possible arrangements is \_\_\_\_\_\_ . (Answer with a number)

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>.

Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?

Answer: $5.5$