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

A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section?

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


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

Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.

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


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

Two workers were assigned to produce a batch of identical parts; after the first worked for \(a\) hours and the second for \(0.6a\) hours, it turned out that they had completed \(\frac{5}{n}\) of the entire job. After working together for another \(0.6a\) hours, they found that they still had \(\frac{1}{n}\) of the batch left to produce. How many hours will it take for each of them, working separately, to complete the whole job? The number \(n\) is a natural number; find it.

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>.

Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.

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


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

In how many ways can you arrange the digits of 11250 to get a five-digit number that is a multiple of 2?

Answer: $24$


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

In a math interest class, the teacher gave a problem for everyone to discuss: "Given real numbers $a$, $b$, $c$ not all equal to zero satisfying $a+b+c=0$, find the maximum value of $\frac{|a+2b+3c|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}."$ Jia quickly offered his opinion: Isn't this just the Cauchy inequality? We can directly solve it; Yi: I am not very clear about the Cauchy inequality, but I think we can solve the problem by constructing the dot product of vectors; Bing: I am willing to try elimination, to see if it will be easier with fewer variables; Ding: This is similar to the distance formula in analytic geometry, can we try to generalize it to space. Smart you can try to use their methods, or design your own approach to find the correct maximum value as ______.

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 triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, satisfying $\frac{a}{{2\cos A}}=\frac{b}{{3\cos B}}=\frac{c}{{6\cos C}}$, then $\sin 2A=$____.

Answer: $\frac{3\sqrt{11}}{10}$


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

Given the polar equation of curve $C$ is $\rho\sin^2\theta-8\cos\theta=0$, with the pole as the origin of the Cartesian coordinate system $xOy$, and the polar axis as the positive half-axis of $x$. In the Cartesian coordinate system, a line $l$ with an inclination angle $\alpha$ passes through point $P(2,0)$.
$(1)$ Write the Cartesian equation of curve $C$ and the parametric equation of line $l$;
$(2)$ Suppose the polar coordinates of points $Q$ and $G$ are $(2, \frac{3\pi}{2})$ and $(2,\pi)$, respectively. If line $l$ passes through point $Q$ and intersects curve $C$ at points $A$ and $B$, find the area of $\triangle GAB$.

Answer: $16\sqrt{2}$


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>.

Let  $ABC$  be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is  $10$ , what is the largest possible area of the triangle?

Answer: $25$


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

A cube with a side length of 10 is divided into 1000 smaller cubes with a side length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) is zero. In one of the small cubes (denoted as \( A \)), the number one is written. Three layers pass through cube \( A \), each parallel to the faces of the larger cube (with each layer having a thickness of 1). Find the sum of all the numbers in the cubes that are not in these layers.

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 there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor.

Answer: $89$


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

Given the set $A=\{x|1 < x < k\}$ and the set $B=\{y|y=2x-5, x \in A\}$, if $A \cap B = \{x|1 < x < 2\}$, find the value of the real number $k$.

Answer: $3.5$


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

Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units.

Answer: $\frac{11\pi}{24}$


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>.

In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \sqrt{2}, \frac{\pi}{4})$$, and the parametric equation of line $l$ is:
$$\begin{cases} x= \frac{3}{2} - \frac{\sqrt{2}}{2}t \\ y= \frac{1}{2} + \frac{\sqrt{2}}{2}t \end{cases}$$ (where $t$ is the parameter), and point A lies on line $l$.
(Ⅰ) Find the corresponding parameter $t$ of point A;
(Ⅱ) If the parametric equation of curve C is:
$$\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$$ (where $\theta$ is the parameter), and line $l$ intersects curve C at points M and N, find the length of line segment |MN|.

Answer: $\frac{4\sqrt{2}}{5}$


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

The positive integer \( N \) is the smallest one whose digits add to 41. What is the sum of the digits of \( N + 2021 \)?

A) 10

B) 12

C) 16

D) 2021

E) 4042

Answer: $10$


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

Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find:

(1) The probability that one of the groups A or B has exactly 2 weak teams.
(2) The probability that group A has at least 2 weak teams.

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 a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is \_\_\_\_\_\_.

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


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

Given that $\cos (α- \frac {π}{6})+ \sin α= \frac {4}{5} \sqrt {3}$, and $α \in (0, \frac {π}{3})$, find the value of $\sin (α+ \frac {5}{12}π)$.

Answer: $\frac{7 \sqrt{2}}{10}$


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

Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\frac{2S_n+8}{a_n+3}$ for all positive integers $n$.

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


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

Given $$\frac {1}{3} \leq a \leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$.
(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$);
(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be $N(a)$. Find the minimum value of $M(a) - N(a)$.

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


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

Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes?

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>.

Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first  $10$  customers has made their purchase, that there just enough cheese for the next  $10$  customers? If so, how much cheese will be left in the store after the first  $10$  customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)

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>.

How many ways are there to distribute 7 balls into 4 boxes if the balls are not distinguishable and neither are the boxes?

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 cube with a side length of \(4\), if a solid cube of side length \(1\) is removed from each corner, calculate the total number of edges of the resulting structure.

Answer: $36$


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

In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $ρsin^{2}\frac{θ}{2}=1$.
$(1)$ Find the rectangular coordinate equation of $C$ and the polar coordinates of the intersection points of $C$ with the $y$-axis.
$(2)$ If the line $l$ intersects $C$ at points $A$ and $B$, and intersects the $x$-axis at point $P$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

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


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

Calculate the values of:
(1) $8^{\frac{2}{3}} - (0.5)^{-3} + \left(\frac{1}{\sqrt{3}}\right)^{-2} \times \left(\frac{81}{16}\right)^{-\frac{1}{4}}$;
(2) $\log 5 \cdot \log 8000 + (\log 2^{\sqrt{3}})^2 + e^{\ln 1} + \ln(e \sqrt{e})$.

Answer: $\frac{11}{2}$


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 $\vec{m}=(2a\cos x,\sin x)$ and $\vec{n}=(\cos x,b\cos x)$, the function $f(x)=\vec{m}\cdot \vec{n}-\frac{\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\frac{\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\left(\frac{\pi}{12},1\right)$.

$(1)$ Find the values of $a$ and $b$;

$(2)$ Move the graph of the function $f(x)$ to the left by $\varphi (\varphi > 0)$ units, and then stretch the x-coordinates of the points on the graph by a factor of $2$ without changing the y-coordinates, to obtain the graph of the function $y=\sin x$. Find the minimum value of $\varphi$.

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


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

In the arithmetic sequence $\{a_{n}\}$, $d=-2$, $a_{1}+a_{4}+a_{7}+…+a_{31}=50$. Find the value of $a_{2}+a_{6}+a_{10}+…+a_{42}$.

Answer: $-82$


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

Given that $a$ and $b$ are coefficients, and the difference between $ax^2+2xy-x$ and $3x^2-2bxy+3y$ does not contain a quadratic term, find the value of $a^2-4b$.

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>.

What is the product of all real numbers that are tripled when added to their reciprocals?

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


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

Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - 2xy\]for all $x,$ $y.$  Find the sum of all possible values of $g(1).$

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>.

In △ABC, B = $$\frac{\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC.

Answer: $\frac{49\pi}{3}$


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

Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}.$

Answer: $\frac{476}{43}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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
\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\]

Answer: $424$


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

Consider the function $f(x) = x^3 + 3\sqrt{x}$. Evaluate $3f(3) - 2f(9)$.

Answer: $-1395 + 9\sqrt{3}$


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

The number $n$ is a four-digit positive integer and is the product of three distinct prime factors $x$, $y$ and $10y+x$, where $x$ and $y$ are each less than 10. What is the largest possible value of $n$?

Answer: $1533$


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

The isoelectric point of glycine is the pH at which it has zero charge. Its charge is  $-\frac13$  at pH  $3.55$ , while its charge is  $\frac12$  at pH  $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine?

Answer: $5.97$


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

Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that:
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + y) g(x, y) &= y g(x, x + y).
\end{align*}
Calculate $g(15, 33)$.

Answer: $165$


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

What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $71$?

Answer: $1139$


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

In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^2 = b(b + c)$. Find the value of $\frac{B}{A}$.

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


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

Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, the line $y=\frac{1}{2}$ intersects $C$ at points $A$ and $B$, where $|AB|=3\sqrt{3}$.
$(1)$ Find the equation of $C$;
$(2)$ Let the left and right foci of $C$ be $F_{1}$ and $F_{2}$ respectively. The line passing through $F_{1}$ with a slope of $1$ intersects $C$ at points $G$ and $H$. Find the perimeter of $\triangle F_{2}GH$.

Answer: $12$


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

The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point.

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>.

There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)?

Answer: $12$


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

Determine the smallest positive real number \(x\) such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 7.\]

Answer: $\frac{71}{8}$


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

Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$.

Answer: $\frac{85}{6}$


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

Determine the slope of line $\overline{CD}$ where circles given by $x^2 + y^2 - 6x + 4y - 5 = 0$ and $x^2 + y^2 - 10x + 16y + 24 = 0$ intersect at points $C$ and $D$.

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>.

Rectangle $ABCD$ has area $4032$. An ellipse with area $4032\pi$ passes through points $A$ and $C$ and has foci at points $B$ and $D$. Determine the perimeter of the rectangle.

Answer: $8\sqrt{2016}$


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

A certain type of rice must comply with the normal distribution of weight $(kg)$, denoted as $\xi ～N(10,{σ}^{2})$, according to national regulations. Based on inspection results, $P(9.9≤\xi≤10.1)=0.96$. A company purchases a bag of this packaged rice as a welfare gift for each employee. If the company has $2000$ employees, the approximate number of employees receiving rice weighing less than $9.9kg$ is \_\_\_\_\_\_\_\_\_\_\_.

Answer: $40$


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

In Montana, 500 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Soda'' sector of the graph is $200^\circ$, to the nearest whole degree. How many of the people surveyed chose ``Soda''? Express your answer as a whole number.

Answer: $278$


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

Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse.

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


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>.

Suppose $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 4y^2 = 12$. Find the maximum possible value of $x^2 + 3xy + 4y^2$.

Answer: $84$


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

Let $g(x) = 2x^7 - 3x^3 + 4x - 8.$ If $g(6) = 12,$ find $g(-6).$

Answer: $-28$


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

Let \(a\), \(b\), \(c\), and \(d\) be nonnegative numbers whose sum is 150. Find the largest possible value of
\[
ab + bc + cd.
\]

Answer: $5625$


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

Given the function $f(x)=e^{x}$, $g(x)=-x^{2}+2x+b(b\in\mathbb{R})$, denote $h(x)=f(x)- \frac {1}{f(x)}$
(I) Determine the parity of $h(x)$ and write down the monotonic interval of $h(x)$, no proof required;
(II) For any $x\in[1,2]$, there exist $x_{1}$, $x_{2}\in[1,2]$ such that $f(x)\leqslant f(x_{1})$, $g(x)\leqslant g(x_{2})$. If $f(x_{1})=g(x_{2})$, find the value of the real number $b$.

Answer: $e^{2}-1$


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

Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?

Answer: $-30$


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

Rotate a square around a line that lies on one of its sides to form a cylinder. If the volume of the cylinder is $27\pi \text{cm}^3$, then the lateral surface area of the cylinder is _________ $\text{cm}^2$.

Answer: $18\pi$


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 a function $f(x)$ defined on $\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$.

Answer: $-5$


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

Using systematic sampling method to select 32 people from 960 for a questionnaire survey, they are randomly numbered from 1 to 960. After grouping, the number drawn by simple random sampling in the first group is 9. Among the 32 people drawn, those with numbers in the interval [1,450] will fill out questionnaire A, those in the interval [451,750] will fill out questionnaire B, and the rest will fill out questionnaire C. How many of the drawn people will fill out questionnaire B?

Answer: $10$


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

Given that $O$ is the coordinate origin, and vectors $\overrightarrow{OA}=(\sin α,1)$, $\overrightarrow{OB}=(\cos α,0)$, $\overrightarrow{OC}=(-\sin α,2)$, and point $P$ satisfies $\overrightarrow{AB}=\overrightarrow{BP}$.
(I) Denote function $f(α)=\overrightarrow{PB} \cdot \overrightarrow{CA}$, find the minimum positive period of function $f(α)$;
(II) If points $O$, $P$, and $C$ are collinear, find the value of $| \overrightarrow{OA}+ \overrightarrow{OB}|$.

Answer: $\frac{\sqrt{74}}{5}$


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>.

Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?

Answer: $\frac{148}{590}$


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

Find out how many positive integers  $n$  not larger than  $2009$  exist such that the last digit of  $n^{20}$  is  $1$ .

Answer: $804$


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

In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$

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>.

What is the smallest number that can be written as a sum of  $2$  squares in  $3$  ways?

Answer: $325$


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

Given that $\sin(x+\pi) + \cos(x-\pi) = \frac{1}{2}, x \in (0, \pi)$.
(1) Find the value of $\sin x \cos x$;
(2) Find the value of $\sin x - \cos x$.

Answer: $\frac{\sqrt{7}}{2}$


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

Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$?

Answer: $\frac{4}{81}$


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

Given the set $A=\{x|ax^{2}-x+1=0,a\in\mathbb{R},x\in\mathbb{R}\}$. If the proposition "Set $A$ contains only one element" is true, find the value of $a$.

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>.

What would the 25th number be in a numeric system where the base is five?

Answer: $100$


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

Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two full rounds.

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 geometric sequence $\{a_n\}$, $(a_1 \cdot a_2 \cdot a_3 = 27)$, $(a_2 + a_4 = 30)$, $(q > 0)$

Find:
$(1)$ $a_1$ and the common ratio $q$;

$(2)$ The sum of the first $6$ terms $(S_6)$.

Answer: $364$


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

Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$.

Answer: $1800$


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

In plane $\alpha$, there are four points, and in plane $\beta$, there are five points. From these nine points, the maximum number of planes that can be determined by any three points is     ; the maximum number of tetrahedrons that can be determined by any four points is     . (Answer with numbers)

Answer: $120$


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

Let \( a > 0 \). The function \( f(x) = x + \frac{100}{x} \) attains its minimum value \( m_{1} \) on the interval \( (0, a] \), and its minimum value \( m_{2} \) on the interval \( [a, +\infty) \). If \( m_{1} m_{2} = 2020 \), find the value of \( a \).

Answer: $100$


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

The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Find the minimum value of \( m \).

Answer: $33$


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

The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?

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 frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$.

Answer: $12$


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

Given the function $f(x)=2\sqrt{3}\cos^2\left(\frac{\pi}{2}+x\right)-2\sin(\pi+x)\cos x-\sqrt{3}$.
$(1)$ Find the extreme values of $f(x)$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.
$(2)$ If $f(x_0-\frac{\pi}{6})=\frac{10}{13}$, where $x_0\in\left[\frac{3\pi}{4}, \pi\right]$, find the value of $\sin 2x_0$.

Answer: $-\frac{5+12\sqrt{3}}{26}$


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

Given a sequence: $2016$, $2017$, $1$, $-2016$, $-2017$, $…$, the characteristic of this sequence is that starting from the second term, each term is equal to the sum of the term before and after it. Find the sum of the first $2017$ terms of the sequence.

Answer: $2016$


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

Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \)

Determine the minimum value of the real number \( A \) such that the inequality \( \left| a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \right| \leqslant A ( a_{1} - a_{n} ) \) holds, and provide a justification for this value.

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>.

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=1+\dfrac{\sqrt{2}}{2}t \\ y=2+\dfrac{\sqrt{2}}{2}t \end{cases}$ ($t$ 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 circle $C$ is $\rho=6\sin\theta$.
- (I) Find the standard equation of circle $C$ in Cartesian coordinates;
- (II) If point $P(l,2)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA| + |PB|$.

Answer: $2\sqrt{7}$


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

Given the expression $\frac{810 \times 811 \times 812 \times \cdots \times 2010}{810^{n}}$ is an integer, find the maximum value of $n$.

Answer: $149$


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

Let $P$ be a moving point on the line $3x+4y+3=0$, and through point $P$, draw two tangents to the circle $C$: $x^{2}+y^{2}-2x-2y+1=0$, with the points of tangency being $A$ and $B$, respectively. Find the minimum value of the area of quadrilateral $PACB$.

Answer: $\sqrt{3}$


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

12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is ______.

Answer: $28$


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

A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded?

Answer: $33.33\%$


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

For the set \( \{1, 2, 3, \ldots, 8\} \) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \( \{1, 3, 4, 7, 8\} \) would be \( 8-7+4-3+1=3 \) and for \( \{8\} \) it is \( 8 \). Find the sum of all such alternating sums for \( n=8 \).

Answer: $1024$


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

If $\lceil{\sqrt{x}}\rceil=20$, how many possible integer values of $x$ are there?

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>.

The equation of the line joining the complex numbers $-1 + 2i$ and $2 + 3i$ can be expressed in the form
\[az + b \overline{z} = d\]for some complex numbers $a$, $b$, and real number $d$. Find the product $ab$.

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>.

Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[ f(x)f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right) \]
for all $x, y > 0.$

Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s$.

Answer: $\frac{4047}{2}$


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 $\overrightarrow{a}=(-3,4)$, $\overrightarrow{b}=(5,2)$, find $|\overrightarrow{a}|$, $|\overrightarrow{b}|$, and $\overrightarrow{a}\cdot \overrightarrow{b}$.

Answer: $-7$


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

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 $y=f(x)$, evaluate 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>.

In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the x-axis as the polar axis. Given circle C: ρ = 2cosθ - 2sinθ, and the parametric equation of line l is x = t, y = -1 + 2√2t (t is the parameter). Line l intersects with circle C at points M and N, and point P is any point on circle C that is different from M and N.

1. Write the rectangular coordinate equation of C and the general equation of l.
2. Find the maximum area of triangle PMN.

Answer: $\frac{10\sqrt{5}}{9}$


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>.

Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd.

Answer: $\frac{7}{16}$


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

Given a sector with a central angle of 120° and an area of $3\pi$, which is used as the lateral surface area of a cone, find the surface area and volume of the cone.

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


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

Given that the ellipse $C\_2$ passes through the two foci and the two endpoints of the minor axis of the ellipse $C\_1$: $\frac{x^{2}}{14} + \frac{y^{2}}{9} = 1$, find the eccentricity of the ellipse $C\_2$.

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


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

What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$?

Answer: $110 + \sqrt{1818}$


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

Given vectors $\overrightarrow {a}=(\sqrt{3}\sin x,\sin x)$ and $\overrightarrow {b}=(\cos x,\sin x)$, where $x \in (0, \frac{\pi}{2})$,
1. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x$;
2. Let the function $f(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{b}$, determine the maximum value of $f(x)$.

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


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

The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$.

Answer: $136$


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>.

Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$.

Answer: $\sqrt{26}-3$


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>.

Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February.

Answer: $142.00$


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

If   $$ \sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c} $$  where  $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$  then what is the value of  $a+ b ?$

Answer: $80$


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

Patrick tosses four four-sided dice, each numbered  $1$  through  $4$ . What's the probability their product is a multiple of four?

Answer: $\frac{13}{16}$


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

A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$

Answer: $\frac{85}{7}$


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>.

Solve the following equations:
(1) $\frac {1}{2}x - 2 = 4 + \frac {1}{3}x$
(2) $\frac {x-1}{4} - 2 = \frac {2x-3}{6}$
(3) $\frac {1}{3}[x - \frac {1}{2}(x-1)] = \frac {2}{3}(x - \frac {1}{2})$
(4) $\frac {x}{0.7} - \frac {0.17-0.2x}{0.03} = 1$

Answer: $\frac {14}{17}$


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 $$f(x)= \begin{cases} f(x+1), x < 4 \\ ( \frac {1}{2})^{x}, x\geq4\end{cases}$$, find $f(\log_{2}3)$.

Answer: $\frac {1}{24}$


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

The slope angle of the tangent line to the curve y=\frac{1}{3}x^{3} at x=1 is what value?

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


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

Given that 30 balls are put into four boxes A, B, C, and D, such that the sum of the number of balls in A and B is greater than the sum of the number of balls in C and D, find the total number of possible ways.

Answer: $2600$


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

The area enclosed by the curves $y=e^{x}$, $y=e^{-x}$, and the line $x=1$ is $e^{1}-e^{-1}$.

Answer: $e+e^{-1}-2$


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>.

If $x$ is a real number and $\lceil x \rceil = 9,$ how many possible values are there for $\lceil x^2 \rceil$?

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>.

A factory packs its products in cubic boxes. In one store, they put  $512$  of these cubic boxes together to make a large  $8\times 8 \times 8$  cube. When the temperature goes higher than a limit in the store, it is necessary to separate the  $512$  set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?

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>.

In the sequence $\{{a_{n}}\}$, the adjacent terms ${a_{n}}$ and ${a_{n+1}}$ are the roots of the equation ${x^{2}}+3nx+{{b_{n}}=0}$. Given that ${{a_{10}}=-17}$, find the value of ${{b_{51}}}$.

Answer: $5840$


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

Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(x+3) = -f(1-x)$. If $f(3) = 2$, then $f(2013) = \ $.

Answer: $-2$


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

You are standing at the edge of a river which is  $1$  km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is  $1$  km . You can swim at  $2$  km/hr and walk at  $3$  km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel).

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


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

On a standard dice, the sum of the numbers of pips on opposite faces is always 7. Four standard dice are glued together as shown. What is the minimum number of pips that could lie on the whole surface?
A) 52
B) 54
C) 56
D) 58
E) 60

Answer: $58$


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

Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:
1. The Tigers won more than the Eagles.
2. The Patriots won more than the Cubs, but fewer than the Mounties.
3. The Cubs won more than 15 games.
4. The Falcons won more games than the Eagles but fewer than the Mounties.

How many games did the Patriots win?

Graph data:
Teams (in unknown order): 10, 18, 22, 27, 33 games.

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>.

In right triangle $ABC$, $\angle C=90^{\circ}$, $AB=9$, $\cos B=\frac{2}{3}$, calculate the length of $AC$.

Answer: $3\sqrt{5}$


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

A box contains 4 cards, each with one of the following functions defined on \\(R\\): \\(f_{1}(x)={x}^{3}\\), \\(f_{2}(x)=|x|\\), \\(f_{3}(x)=\sin x\\), \\(f_{4}(x)=\cos x\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new function, the probability that the resulting function is an odd function is \_\_\_\_\_.

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


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>.

A basketball team has 20 players. The coach needs to choose a starting lineup consisting of one point guard and 7 other players (these players are interchangeable). How many different starting lineups can the coach choose?

Answer: $1007760$


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

A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone?

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>.

Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$

Answer: $128$


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>.

Let set $A=\{x \mid |x-2| \leq 2\}$, and $B=\{y \mid y=-x^2, -1 \leq x \leq 2\}$, then $A \cap B=$ ?

Answer: $\{0\}$


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

In the ancient Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art," there is a problem called "Division of Grains in a Granary": When a granary is opened to receive grain, a person brings 1534 stones of rice. Upon inspection, it is found that there are grains of wheat mixed in the rice. After taking a handful of rice as a sample, it is found that out of 254 grains in the sample, 28 are grains of wheat. What is the approximate amount of wheat mixed in this batch of rice in stones (rounded to one decimal place)?

Answer: $169.1$


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 school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible.

Answer: $90$


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

In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$.

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


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

Two different natural numbers are selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.

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


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

Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \(10 \, \text{m/s}\) and Betty runs at \(15 \, \text{m/s}\). Betty gives Alice a 40 meter head start before they both start running. After 15 minutes, how many times will they have passed each other?
(a) 9
(b) 10
(c) 11
(d) 12

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>.

The sequence $1, 4, 5, 16, 17, 20, 21 \cdots$ consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $150^{\mbox{th}}$ term of this sequence.

Answer: $16660$