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

Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering free shipping and their own discount schemes:
Store A: Buy one basketball and get one jump rope for free;
Store B: Pay 90% of the original price for both the basketball and jump rope.
It is known that they want to buy 40 basketballs and $x$ jump ropes $\left(x \gt 40\right)$.
$(1)$ If they purchase from Store A, the payment will be ______ yuan; if they purchase from Store B, the payment will be ______ yuan; (express in algebraic expressions with $x$)
$(2)$ If $x=80$, through calculation, determine which store is more cost-effective to purchase from at this point.
$(3)$ If $x=80$, can you provide a more cost-effective purchasing plan? Write down your purchasing method and calculate the amount to be paid.

Answer: $5700$


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

In a bus station in the city, there are 10 waiting seats arranged in a row. Now, if 4 passengers randomly choose some seats to wait, the number of ways to arrange them so that there are exactly 5 consecutive empty seats is $\boxed{480}$.

Answer: $480$


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

Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form.

Answer: $12$


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

Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\]

Answer: $502$


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

Subtract $256.26$ from $512.52$ and then multiply the result by $3$. Express the final result as a decimal to the nearest hundredth.

Answer: $768.78$


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

Given two numbers, a and b, are randomly selected within the interval (-π, π), determine the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root.

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


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

Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?

Answer: $18$


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

James wrote a different integer from 1 to 9 in each cell of a table. He then calculated the sum of the integers in each of the rows and in each of the columns of the table. Five of his answers were 12, 13, 15, 16, and 17, in some order. What was his sixth answer?

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>.

Given vectors $\overrightarrow {m}$=(cosx, sinx) and $\overrightarrow {n}$=(cosx, $\sqrt {3}$cosx), where x∈R, define the function f(x) = $\overrightarrow {m}$$\cdot \overrightarrow {n}$+ $\frac {1}{2}$.
(1) Find the analytical expression and the interval where the function is strictly increasing;
(2) Let a, b, and c be the sides opposite to angles A, B, and C of △ABC, respectively. If f(A)=2, b+c=$2 \sqrt {2}$, and the area of △ABC is $\frac {1}{2}$, find the value of a.

Answer: $\sqrt {3}-1$


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>.

Observe the following equations:
\\(① \dfrac {1}{ \sqrt {2}+1}= \dfrac { \sqrt {2}-1}{( \sqrt {2}+1)( \sqrt {2}-1)}= \sqrt {2}-1\\);
\\(② \dfrac {1}{ \sqrt {3}+ \sqrt {2}}= \dfrac { \sqrt {3}- \sqrt {2}}{( \sqrt {3}+ \sqrt {2})( \sqrt {3}- \sqrt {2})}= \sqrt {3}- \sqrt {2}\\);
\\(③ \dfrac {1}{ \sqrt {4}+ \sqrt {3}}= \dfrac { \sqrt {4}- \sqrt {3}}{( \sqrt {4}+ \sqrt {3})( \sqrt {4}- \sqrt {3})}= \sqrt {4}- \sqrt {3}\\);\\(…\\)
Answer the following questions:
\\((1)\\) Following the pattern of the equations above, write the \\(n\\)th equation: \_\_\_\_\_\_ ;
\\((2)\\) Using the pattern you observed, simplify: \\( \dfrac {1}{ \sqrt {8}+ \sqrt {7}}\\);
\\((3)\\) Calculate: \\( \dfrac {1}{1+ \sqrt {2}}+ \dfrac {1}{ \sqrt {2}+ \sqrt {3}}+ \dfrac {1}{ \sqrt {3}+2}+…+ \dfrac {1}{3+ \sqrt {10}}\\).

Answer: $\sqrt {10}-1$


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

In \\(\Delta ABC\\), given that \\(a= \sqrt{3}, b= \sqrt{2}, B=45^{\circ}\\), find \\(A, C\\) and \\(c\\).

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


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

What's the largest number of elements that a set of positive integers between  $1$  and  $100$  inclusive can have if it has the property that none of them is divisible by another?

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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, perpendicular lines to the $x$-axis are drawn through points $F\_1$ and $F\_2$ intersecting the ellipse at four points to form a square, determine the eccentricity $e$ of the ellipse.

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


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

Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$.

Answer: $-4$


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

In triangle $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{PR}$. Let $PX = u \cdot PQ$, $QY = v \cdot QR$, and $RZ = w \cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$ and $u^2+v^2+w^2=1/2$. The ratio of the area of triangle $XYZ$ to the area of triangle $PQR$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: $41$


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

What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors?

Answer: $420$


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

Given that $\cos(75^\circ + \alpha) = \frac{1}{3}$, where $\alpha$ is an angle in the third quadrant, find the value of $\cos(105^\circ - \alpha) + \sin(\alpha - 105^\circ)$.

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


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>.

Find the minimum value of  $m$  such that any  $m$ -element subset of the set of integers  $\{1,2,...,2016\}$  contains at least two distinct numbers  $a$  and  $b$  which satisfy  $|a - b|\le  3$ .

Answer: $505$


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

Given that α and β are acute angles, and $\tan \alpha = \frac{2}{t}$, $\tan \beta = \frac{t}{15}$. When $10\tan \alpha + 3\tan \beta$ reaches its minimum value, the value of $\alpha + \beta$ is \_\_\_\_\_\_.

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


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

A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.

Answer: $\frac{1}{6561}$


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

Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized.

Answer: $11$


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

Given a moving point $E$ such that the product of the slopes of the lines from $E$ to points $A(2,0)$ and $B(-2,0)$ is $- \frac {1}{4}$, and the trajectory of point $E$ is curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Draw a line $l$ through point $D(1,0)$ that intersects curve $C$ at points $P$ and $Q$. Find the maximum value of $\overrightarrow{OP} \cdot \overrightarrow{OQ}$.

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


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

A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of drawing a winning pair.
A) $\frac{1}{2}$
B) $\frac{10}{33}$
C) $\frac{30}{66}$
D) $\frac{35}{66}$
E) $\frac{40}{66}$

Answer: $\frac{35}{66}$


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

Given the set of integers $\{1, 2, 3, \dots, 9\}$, from which three distinct numbers are arbitrarily selected as the coefficients of the quadratic function $f_{(x)} = ax^2 + bx + c$, determine the total number of functions $f_{(x)}$ that satisfy $\frac{f(1)}{2} \in \mathbb{Z}$.

Answer: $264$


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

What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$?

Answer: $-\frac{13}{9}$


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

Given $f\left(\alpha \right)=\frac{\mathrm{sin}\left(\pi -\alpha \right)\mathrm{cos}\left(2\pi -\alpha \right)\mathrm{cos}\left(-\alpha +\frac{3\pi }{2}\right)}{\mathrm{cos}\left(\frac{\pi }{2}-\alpha \right)\mathrm{sin}\left(-\pi -\alpha \right)}$.

(1) Simplify $f(\alpha )$.

(2) If $\alpha$ is an angle in the third quadrant and $\mathrm{cos}(\alpha -\frac{3\pi }{2})=\frac{1}{5}$, find the value of $f(\alpha )$.

Answer: $\frac{2\sqrt{6}}{5}$


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>.

A standard die is rolled consecutively two times. Calculate the probability that the face-up numbers are adjacent natural numbers.

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


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

Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\sin \angle PF_{1}F_{2} = \frac {1}{3}$, determine the eccentricity of the hyperbola $C$.

Answer: $\sqrt{2}$


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

Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\]

Answer: $\frac{131}{11}$


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>.

From the five numbers \\(1, 2, 3, 4, 5\\), select any \\(3\\) to form a three-digit number without repeating digits. When the three digits include both \\(2\\) and \\(3\\), \\(2\\) must be placed before \\(3\\) (not necessarily adjacent). How many such three-digit numbers are there?

Answer: $51$


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

When the numbers \(\sqrt{5}, 2.1, \frac{7}{3}, 2.0 \overline{5}, 2 \frac{1}{5}\) are arranged in order from smallest to largest, the middle number is:

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


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

Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively.  She creates a sequence by rolling the die and recording the number on its bottom face.  However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5.  Sabrina stops the moment that all four numbers appear in the sequence.  Find the expected (average) number of terms in Sabrina's sequence.

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>.

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=3+t\cos \alpha \\ y=1+t\sin \alpha\end{cases}$ (where $t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-negative half-axis of $x$ as the polar axis), the equation of curve $C$ is $\rho=4\cos \theta$.
$(1)$ Find the equation of curve $C$ in the Cartesian coordinate system;
$(2)$ If point $P(3,1)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the minimum value of $|PA|+|PB|$.

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>.

Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find the perimeter of triangle APR.

Answer: $48$


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

Given a function defined on $\mathbb{R}$, $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, and $|\varphi| \leqslant \frac {\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\frac {\pi}{2}$. After the graph of the function is shifted to the left by $\frac {\pi}{12}$ units, the resulting graph corresponds to an even function.
$(1)$ Find the expression for the function $f(x)$.
$(2)$ If $f\left( \frac {x_{0}}{2}\right)=- \frac {3}{8}$, and $x_{0}\in\left[ \frac {\pi}{2},\pi\right]$, find the value of $\cos \left(x_{0}+ \frac {\pi}{6}\right)$.

Answer: $- \frac { \sqrt {741}}{32}- \frac {3}{32}$


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 shortest distance from a point on the curve $y=\ln x$ to the line $y=x+2$ is what value?

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


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

A regular hexagon's center and vertices together make 7 points. Calculate the number of triangles that can be formed using any 3 of these points as vertices.

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>.

Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\)

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>.

If $(2,12)$ and $(8,3)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $x$-coordinates of the other two vertices?

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>.

Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

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>.

Given the function $f(x)=\sin({ωx+φ})$ $({ω＞0,|φ|≤\frac{π}{2}})$, $f(0)=\frac{{\sqrt{2}}}{2}$, and the function $f\left(x\right)$ is monotonically decreasing on the interval $({\frac{π}{{16}},\frac{π}{8}})$, then the maximum value of $\omega$ is ______.

Answer: $10$


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

Suppose $x$ and $y$ satisfy the system of inequalities $\begin{cases} & x-y \geqslant 0 \\ & x+y-2 \geqslant 0 \\ & x \leqslant 2 \end{cases}$, calculate the minimum value of $x^2+y^2-2x$.

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


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

Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, determine the value of $f(-\frac{{5π}}{{12}})$.

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


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

How many ways are there to allocate a group of 8 friends among a basketball team, a soccer team, a track team, and the option of not participating in any sports? Each team, including the non-participant group, could have anywhere from 0 to 8 members. Assume the friends are distinguishable.

Answer: $65536$


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

From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained?

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>.

Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is.

Answer: $960$


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

Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$.

Answer: $20$


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

The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases}  n-5 & \mbox{if }n\ge 1200 \\  g(g(n+7)) & \mbox{if }n<1200. \end{cases}\] Find $g(70)$.

Answer: $1195$


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

Given the function $f(x) = \sin x \cos x - \sqrt{3} \cos (x+\pi) \cos x, x \in \mathbb{R}$.
(Ⅰ) Find the minimal positive period of $f(x)$;
(Ⅱ) If the graph of the function $y = f(x)$ is translated by $\overrightarrow{b}=\left( \frac{\pi}{4}, \frac{\sqrt{3}}{2} \right)$ to obtain the graph of the function $y = g(x)$, find the maximum value of $y=g(x)$ on the interval $\left[0, \frac{\pi}{4}\right]$.

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


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

Given that $0 < a < \pi, \tan a=-2$,

(1) Find the value of $\cos a$;

(2) Find the value of $2\sin^{2}a - \sin a \cos a + \cos^{2}a$.

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


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

The number halfway between $\dfrac{1}{8}$ and $\dfrac{1}{3}$ is
A) $\dfrac{11}{48}$
B) $\dfrac{11}{24}$
C) $\dfrac{5}{24}$
D) $\dfrac{1}{4}$
E) $\dfrac{1}{5}$

Answer: $\dfrac{11}{48}$


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

A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical layer was $2$ feet and the thickness was $1$ foot. Calculate the total volume of sand delivered, expressing your answer in terms of $\pi$.

Answer: $98\pi$


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

In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\cos θ-ρ\sin θ-25=0$, and the curve $W$ is defined by the parametric equations $x=2t, y=t^{2}-1$.

1. Find the Cartesian equation of line $l$ and the general equation of curve $W$.
2. If point $P$ is on line $l$, and point $Q$ is on curve $W$, find the minimum value of $|PQ|$.

Answer: $\frac{8\sqrt{17}}{17}$


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>.

Given the function $f(x)=\sin (x+ \frac{7\pi}{4})+\cos (x- \frac{3\pi}{4})$, where $x\in R$.
(1) Find the smallest positive period and the minimum value of $f(x)$;
(2) Given that $f(\alpha)= \frac{6}{5}$, where $0 < \alpha < \frac{3\pi}{4}$, find the value of $f(2\alpha)$.

Answer: $\frac{31\sqrt{2}}{25}$


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 that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$.

Answer: $\sqrt{136}$


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

Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 100^\circ$ and $\stackrel \frown {BC} = 80^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$.

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


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

Given that $x_{0}$ is a zero of the function $f(x)=2a\sqrt{x}+b-{e}^{\frac{x}{2}}$, and $x_{0}\in [\frac{1}{4}$,$e]$, find the minimum value of $a^{2}+b^{2}$.

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


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>.

What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?

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>.

Compute $9 \cdot (-5) - (7 \cdot -2) + (8 \cdot -6)$.

Answer: $-79$


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

$A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____.

Answer: $144$


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

"Modulo $m$ graph paper" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \equiv 3y + 1 \pmod{13}.$$ Find the sum of the $x$-intercept and the $y$-intercept, where the intercepts are represented as $(x_0,0)$ and $(0,y_0)$ with $0\le x_0,y_0<13$.

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>.

Simplify the expression $\dfrac {\cos 40 ^{\circ} }{\cos 25 ^{\circ} \sqrt {1-\sin 40 ^{\circ} }}$.

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>.

Given a $6 \times 6$ square of $36$ square blocks, find the number of different combinations of $4$ blocks that can be selected so that no two are in the same row or column.

Answer: $5400$


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

Given the complex number $z= \frac {(1+i)^{2}+2(5-i)}{3+i}$.
$(1)$ Find $|z|$;
$(2)$ If $z(z+a)=b+i$, find the values of the real numbers $a$ and $b$.

Answer: $-13$


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

Given that the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $S\_n$, and it satisfies $S\_n=(\frac{1}{2})^{n}-1$, find the limit as $n$ approaches infinity of $(a\_1+a\_3+...+a\_2n-1)$ .

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


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

Given that in the expansion of $\left(1+x\right)^{n}$, the coefficient of $x^{3}$ is the largest, then the sum of the coefficients of $\left(1+x\right)^{n}$ is ____.

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>.

How many integers between $200$ and $300$ have three different digits in increasing order?

Answer: $21$


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

The minimum positive period of the function $f(x)=\sin x$ is $\pi$.

Answer: $2\pi$


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

Given $$\frac {\pi}{2} < \alpha < \pi$$, $$0 < \beta < \frac {\pi}{2}$$, $$\tan\alpha = -\frac {3}{4}$$, and $$\cos(\beta-\alpha) = \frac {5}{13}$$, find the value of $\sin\beta$.

Answer: $\frac {63}{65}$


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

Given the function $f(x) = 2\sin(\frac{1}{3}x - \frac{π}{6})$, where $x \in \mathbb{R}$.

(1) Find the value of $f(\frac{5π}{4})$;

(2) Let $\alpha, \beta \in [0, \frac{π}{2}], f(3\alpha + \frac{π}{2}) = \frac{10}{13}, f(3\beta + 2π) = \frac{6}{5}$, find the value of $\cos(\alpha + \beta)$.

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


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

A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.

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


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

Given that Connie adds $3$ to a number and gets $45$ as her answer, but she should have subtracted $3$ from the number to get the correct answer, determine the correct number.

Answer: $39$


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

Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area?

Answer: $102\sqrt{3}$


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>.

From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk?

Answer: $0.3$


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>.

Find the positive value of $k$ such that the equation $4x^3 + 9x^2 + kx + 4 = 0$ has exactly one real solution in $x$.

Answer: $6.75$


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

Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$?

Answer: $275$


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

Let \( ABC \) be a triangle with \( AB = 5 \), \( AC = 4 \), \( BC = 6 \). The angle bisector of \( \angle C \) intersects side \( AB \) at \( X \). Points \( M \) and \( N \) are drawn on sides \( BC \) and \( AC \), respectively, such that \( \overline{XM} \parallel \overline{AC} \) and \( \overline{XN} \parallel \overline{BC} \). Compute the length \( MN \).

Answer: $\frac{3 \sqrt{14}}{5}$


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 $α$ and $β ∈ ( \frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \frac{3}{5}$.
(1) If $a = \frac{1}{3}$, find the value of $sinαcosα + tanα - \frac{1}{3cosα}$;
(2) If $a = \frac{7}{13}$, find the value of $sinβ$.

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


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

Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\overset{→}{MN} \cdot \overset{→}{MP} = 6|\overset{→}{PN}|$.

(I) Find the trajectory equation of point $P$;

(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $H$. If $\overset{→}{HA} = λ_1\overset{→}{AN}$ and $\overset{→}{HB} = λ_2\overset{→}{BN}$, determine whether $λ_1 + λ_2$ is a constant value. If it is, find this value; if not, explain the reason.

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


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

Given two points $A(-2, 0)$ and $B(0, 2)$, point $C$ is any point on the circle $x^2 + y^2 - 2x = 0$, the minimum value of the area of $\triangle ABC$ is \_\_\_\_\_\_.

Answer: $3 - \sqrt{2}$


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

How many even divisors does \(10!\) have?

Answer: $240$


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

A circle is inscribed in trapezoid \( PQRS \).

If \( PS = QR = 25 \) cm, \( PQ = 18 \) cm, and \( SR = 32 \) cm, what is the length of the diameter of the circle?

Answer: $24$


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

Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$

Answer: $2\sqrt{5}$


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

Let  $ABCDE$  be a regular pentagon. The star  $ACEBD$  has area 1.  $AC$  and  $BE$  meet at  $P$ , while  $BD$  and  $CE$  meet at  $Q$ . Find the area of  $APQD$ .

Answer: $1/2$


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 vectors $\overrightarrow{m}=(\sin x, -1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x, -\frac{1}{2})$, let $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;
(2) Given that $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ in triangle $\triangle ABC$, respectively, and $A$ is an acute angle with $a=2\sqrt{3}$ and $c=4$. If $f(A)$ is the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.

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 the function $f(x)= \frac {2}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$, vector $ \overrightarrow{j}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{j}$, determine the value of $\frac {cos θ_{1}}{sin θ_{1}}+ \frac {cos θ_{2}}{sin θ_{2}}+ \frac {cos θ_{1}}{sin θ_{1}}+…+ \frac {cos θ_{2016}}{sin θ_{2016}}$.

Answer: $\frac{4032}{2017}$


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

Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice.

Answer: $800$


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

Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$.

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>.

What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors?

Answer: $3000$


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

Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty?

Answer: $2012$


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

Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix}$.

Answer: $6\sqrt{30}$


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

Consider the function $g(x) = \frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$.

Answer: $-4$


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

What is the first year after 2000 for which the sum of the digits is 15?

Answer: $2049$


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

In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\begin{cases}x=2\cos \alpha \\ y=\sin \alpha \\ \end{cases}$ (where $\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.

1. Find the polar coordinate equation of curve $C$.
2. If two points $M$ and $N$ on curve $C$ have $OM\bot ON$, find the minimum value of the area of triangle $OMN$.

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


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

Given the function $f(x)=2\sin (\pi-x)\cos x$.
- (I) Find the smallest positive period of $f(x)$;
- (II) Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac {\pi}{6}, \frac {\pi}{2}\right]$.

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


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

In the interval $[0,\pi]$, a number $x$ is randomly selected. The probability that $\sin x$ falls between $0$ and $\frac{1}{2}$ is ______.

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


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

Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.

(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;

(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$.

Answer: $2-4\ln 2$


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

Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?

Answer: $60$


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

Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)?

Answer: $670$


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

28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms.

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


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>.

Perform the calculations.
$(54+38) \times 15$
$1500-32 \times 45$
$157 \times (70 \div 35)$

Answer: $314$


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

A rectangle has dimensions $4$ and $2\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?
A) $2\sqrt{3}$
B) $4\sqrt{3}$
C) $6$
D) $8\sqrt{3}$

Answer: $4\sqrt{3}$


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

Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \]

Answer: $1/128$


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

Given a large circle with a radius of 11 and small circles with a radius of 1, determine the maximum number of small circles that can be placed inside the large circle, such that each small circle is internally tangent to the large circle and the small circles do not overlap.

Answer: $31$


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

From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9?

Answer: $\frac{19}{125}$


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

Among the following functions, identify which pairs represent the same function.

1. $f(x) = |x|, g(x) = \sqrt{x^2}$;
2. $f(x) = \sqrt{x^2}, g(x) = (\sqrt{x})^2$;
3. $f(x) = \frac{x^2 - 1}{x - 1}, g(x) = x + 1$;
4. $f(x) = \sqrt{x + 1} \cdot \sqrt{x - 1}, g(x) = \sqrt{x^2 - 1}$.

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>.

What is the difference between the maximum value and the minimum value of the sum  $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$  where  $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$ ?

Answer: $20$


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

What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?

Answer: $16.67\%$


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

Given that $l$ is the incenter of $\triangle ABC$, with $AC=2$, $BC=3$, and $AB=4$. If $\overrightarrow{AI}=x \overrightarrow{AB}+y \overrightarrow{AC}$, then $x+y=$ ______.

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


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

Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$.

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>.

Given the function $f(x)=kx+b$, whose graph intersects the $x$ and $y$ axes at points A and B respectively, with $\overrightarrow{AB}=2\overrightarrow{i}+2\overrightarrow{j}$ ($\overrightarrow{i}$, $\overrightarrow{j}$ are unit vectors in the positive direction of the $x$ and $y$ axes). The function $g(x)=x^{2}-x-6$ is also given.

1. Find the values of $k$ and $b$.
2. When $x$ satisfies $f(x) > g(x)$, find the minimum value of the function $\frac{g(x)+1}{f(x)}$.

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>.

The difference between the maximum and minimum values of the function $f(x)= \frac{2}{x-1}$ on the interval $[-2,0]$ is $\boxed{\frac{8}{3}}$.

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>.

Given an odd function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(2-x) - f(x) = 0$, and $f(-1) = 1$, calculate the value of $f(1) + f(2) + f(3) + \ldots + f(2010)$.

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>.

Consider the arithmetic sequence defined by the set $\{2, 5, 8, 11, 14, 17, 20\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set.

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>.

In a geometric progression with a common ratio of 4, denoted as $\{b_n\}$, where $T_n$ represents the product of the first $n$ terms of $\{b_n\}$, the fractions $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, and $\frac{T_{40}}{T_{30}}$ form another geometric sequence with a common ratio of $4^{100}$. Analogously, for an arithmetic sequence $\{a_n\}$ with a common difference of 3, if $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$, then _____ also form an arithmetic sequence, with a common difference of _____.

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>.

In a right triangle JKL, the hypotenuse KL measures 13 units, and side JK measures 5 units. Determine $\tan L$ and $\sin L$.

Answer: $\frac{5}{13}$


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>.

A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931?

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>.

Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common.

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>.

In the rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinates of point $A$ are $\left( 4\sqrt{2}, \frac{\pi}{4} \right)$, and the polar equation of line $l$ is $\rho \cos \left( \theta - \frac{\pi}{4} \right) = a$, which passes through point $A$. The parametric equations of curve $C_1$ are given by $\begin{cases} x = 2 \cos \theta \\ y = \sqrt{3} \sin \theta \end{cases}$ ($\theta$ is the parameter).

(1) Find the maximum and minimum distances from points on curve $C_1$ to line $l$.
(2) Line $l_1$, which is parallel to line $l$ and passes through point $B(-2, 2)$, intersects curve $C_1$ at points $M$ and $N$. Compute $|BM| \cdot |BN|$.

Answer: $\frac{32}{7}$


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

The three sides of a triangle are 25, 39, and 40. Find the diameter of its circumscribed circle.

Answer: $\frac{125}{3}$


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 the function $y=f(x)$ that satisfies $f(-x)=-f(x)$ and $f(1+x)=f(1-x)$ for $x \in [-1,1]$ with $f(x)=x^{3}$, find the value of $f(2015)$.

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 $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$.

Answer: $4(1 + \sqrt{2})$


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

Given the function $f(x) = x^{3} + ax^{2} - 2x + 1$ has an extremum at $x=1$.
$(1)$ Find the value of $a$;
$(2)$ Determine the monotonic intervals and extremum of $f(x)$.

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 parametric equation of curve $C\_1$ as $\begin{cases} x=a\cos \theta \\ y=b\sin \theta \end{cases}$ $(a > b > 0, \theta$ is the parameter$)$, and the point $M(1, \frac{ \sqrt{3}}{2})$ on curve $C\_1$ corresponds to the parameter $\theta= \frac{ \pi}{3}$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C\_2$ is $ρ=2\sin θ$.

1. Write the polar coordinate equation of curve $C\_1$ and the rectangular coordinate equation of curve $C\_2$;
2. Given points $M\_1$ and $M\_2$ with polar coordinates $(1, \frac{ \pi}{2})$ and $(2,0)$, respectively. The line $M\_1M\_2$ intersects curve $C\_2$ at points $P$ and $Q$. The ray $OP$ intersects curve $C\_1$ at point $A$, and the ray $OQ$ intersects curve $C\_1$ at point $B$. Find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}$.

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


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

A school has 1200 students, and each student participates in exactly \( k \) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \( k \).

Answer: $23$


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

Given the "ratio arithmetic sequence" $\{a_{n}\}$ with $a_{1}=a_{2}=1$, $a_{3}=3$, determine the value of $\frac{{a_{2019}}}{{a_{2017}}}$.

Answer: $4\times 2017^{2}-1$