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

A number composed of ten million, three hundred thousand, and fifty is written as \_\_\_\_\_\_, and this number is read as \_\_\_\_\_\_.

Answer: $10300050$


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

Given three people, A, B, and C, they stand in a row in any random order. What is the probability that A stands in front of B and C does not stand in front of A?

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>.

In the diagram, points \(B\), \(C\), and \(D\) lie on a line. Also, \(\angle ABC = 90^\circ\) and \(\angle ACD = 150^\circ\). The value of \(x\) is:

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>.

Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$.

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>.

Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point lies below the $x$-axis?

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


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

Given that  $a_1, a_2, a_3, . . . , a_{99}$  is a permutation of  $1, 2, 3, . . . , 99,$  find the maximum possible value of  $$ |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|. $$

Answer: $4900$


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

Let the function \( f(x) = x^3 + 3x^2 + 6x + 14 \), and \( f(a) = 1 \), \( f(b) = 19 \). Then \( a + b = \quad \).

Answer: $-2$


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

Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other four numbers in the same group, and a number in the other group that is not coprime with the other four numbers in that group.

Answer: $50$


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

Given that $α \in (0,π)$, if $\sin α + \cos α = \frac{\sqrt{3}}{3}$, find the value of $\cos^2 α - \sin^2 α$.

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


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

Given that $F(c,0)$ is the right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, a line perpendicular to one asymptote of the hyperbola passes through $F$ and intersects the two asymptotes at points $A$ and $B$. $O$ is the coordinate origin, and the area of $\triangle OAB$ is $\frac{12a^2}{7}$. Find the eccentricity of the hyperbola $e = \frac{c}{a}$.

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


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

Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones.

Answer: $\frac{40}{13}$


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

Given the function $f(x)={e}^{-x}+ \frac{nx}{mx+n}$.

(1) If $m=0$, $n=1$, find the minimum value of the function $f(x)$;

(2) If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac{m}{n}$.

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


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

Calculate the probability that four randomly chosen vertices of a cube form a tetrahedron (triangular pyramid).

Answer: $\frac{29}{35}$


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

Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct.

Answer: $19$


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

A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$.

Answer: $2\sqrt{2}$


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

Given $f(x)=\sin \omega x \cos \omega x - \sqrt{3} \cos^2 \omega x$, where $(\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\frac{2+\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\min }=\pi$. When $x\in[0,\frac{7\pi}{12}]$, the sum of the minimum and maximum values of $f(x)$ is ______.

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


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

What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?

Answer: $16.\overline{6}\%$


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

The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\overrightarrow{MF}⋅\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find the area of $\triangle PMN$.

Answer: $108$


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

For a complex number  $z \neq 3$ , $4$ , let  $F(z)$  denote the real part of  $\tfrac{1}{(3-z)(4-z)}$ . If \[
	\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers  $m$  and  $n$ , find  $100m+n$ .

*Proposed by Evan Chen*

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>.

Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$.

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


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

Determine the largest value of \(x\) that satisfies the equation \(\sqrt{3x} = 5x^2\). Express your answer in simplest fractional form.

Answer: $\left(\frac{3}{25}\right)^{\frac{1}{3}}$


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

Let  $ABCD$  be a trapezoid of bases  $AB$  and  $CD$  . Let  $O$  be the intersection point of the diagonals  $AC$  and  $BD$ . If the area of the triangle  $ABC$  is  $150$  and the area of the triangle  $ACD$  is  $120$ , calculate the area of the triangle  $BCO$ .

Answer: $\frac{200}{3}$


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

In the diagram, \( PR \) and \( QS \) meet at \( X \). Also, \(\triangle PQX\) is right-angled at \(Q\) with \(\angle QPX = 62^\circ\) and \(\triangle RXS\) is isosceles with \( RX = SX \) and \(\angle XSR = y^\circ\). The value of \( y \) is:

Answer: $76$


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

In the ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\overrightarrow{KM} \cdot \overrightarrow{KN} = 0$, find the minimum value of $\overrightarrow{KM} \cdot \overrightarrow{NM}$.

Answer: $\dfrac{23}{3}$


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 percentage of a seven-by-seven grid is shaded?
In a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the left) are left completely unshaded.

Answer: $73.47\%$


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

Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numbers on the middle three cards of this configuration?
A) 16
B) 17
C) 18
D) 19
E) 20

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>.

Leticia has a  $9\times 9$  board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has  $4$  friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:

 - If the square is green, write the number of red friends plus twice the number of blue friends.

 - If the square is red, write the number of blue friends plus twice the number of green friends.

 - If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Answer: $486$


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

The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola.

Answer: $4\sqrt{2}$


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

A kindergarten received cards for learning to read: some are labeled "МА", and the rest are labeled "НЯ".

Each child took three cards and started to form words from them. It turned out that 20 children could form the word "МАМА" from their cards, 30 children could form the word "НЯНЯ", and 40 children could form the word "МАНЯ". How many children have all three cards the same?

Answer: $10$


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

A right rectangular prism, with edge lengths $\log_{5}x, \log_{6}x,$ and $\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$
A) $24$
B) $36$
C) $120$
D) $\sqrt{240}$
E) $240$

Answer: $\sqrt{240}$


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

A chord $AB$ that makes an angle of $\frac{\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \frac{y^{2}}{3}=1$.
$(1)$ Find $|AB|$;
$(2)$ Find the perimeter of $\triangle F_{2}AB$ ($F_{2}$ is the right focus).

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


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>.

Let  $a$  be a real number. Find the minimum value of  $\int_0^1 |ax-x^3|dx$ .

How many solutions (including University Mathematics )are there for the problem?

Any advice would be appreciated.  :)

Answer: $1/8$


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

The volume of a rectangular prism is 360 cubic units where \(a\), \(b\), and \(c\) are integers with \(1 < c < b < a\). Determine the largest possible value of \(b\).

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>.

Bill can buy mags, migs, and mogs for $\$3$, $\$4$, and $\$8$ each, respectively. What is the largest number of mogs he can purchase if he must buy at least one of each item and will spend exactly $\$100$?

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 $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$.

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 $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.
(1) Find the general term formula of the sequence ${a_{n}}$;
(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfying $c_{k}⋅c_{k+1} < 0$ is called the signum change number of this sequence. If $c_{n}=1- \frac {4}{a_{n}}$ ($n$ is a positive integer), find the signum change number of the sequence ${c_{n}}$;
(3) Let $T_{n}$ be the sum of the first $n$ terms of the sequence ${\frac {1}{a_{n}}}$. If $T_{2n+1}-T_{n}\leqslant \frac {m}{15}$ holds for all positive integers $n$, find the minimum value of the positive integer $m$.

Answer: $23$


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

Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: $0.303$


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

Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(0,3)$, the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______.

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


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

Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$.

Answer: $\frac{50}{99}$


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

There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.

Answer: $60$


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

The intercept of the line $5x - 2y - 10 = 0$ on the x-axis is $a$, and on the y-axis is $b$. Find the values of $a$ and $b$.

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>.

In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G?

Answer: $1.5$


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

If $S$, $H$, and $E$ are all distinct non-zero digits (each less than $6$) and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$:
$$\begin{array}{c@{}c@{}c@{}c}
&S&H&E_6\\
&+&H&E_6\\
\cline{2-4}
&S&E&S_6\\
\end{array}$$

Answer: $11_6$


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

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

(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;

(2) If the sum of the maximum and minimum values of $f(x)$ in the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$ is $1$, find the value of $a$.

Answer: $a = -\frac{1}{4}$


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

Given a quadratic function $y=ax^{2}-4ax+3+b\left(a\neq 0\right)$.
$(1)$ Find the axis of symmetry of the graph of the quadratic function;
$(2)$ If the graph of the quadratic function passes through the point $\left(1,3\right)$, and the integers $a$ and $b$ satisfy $4 \lt a+|b| \lt 9$, find the expression of the quadratic function;
$(3)$ Under the conditions of $(2)$ and $a \gt 0$, when $t\leqslant x\leqslant t+1$ the function has a minimum value of $\frac{3}{2}$, find the value of $t$.

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


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 an isosceles triangle $PQR$ with $PQ=QR$ and the angle at the vertex $108^\circ$. Point $O$ is located inside the triangle $PQR$ such that $\angle ORP=30^\circ$ and $\angle OPR=24^\circ$. Find the measure of the angle $\angle QOR$.

Answer: $126$


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

An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle?

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


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>.

A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.

Answer: $15$


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

Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions.

Answer: $\dfrac{19}{36}$


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>.

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let  $ N$  be the total number of intersection points of these circles. Determine the smallest possible values of  $ N$ .

Answer: $3012$


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

All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle.

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>.

The expression \( 29 \cdot 13 \) is a product of two integers greater than 1. When this product is divided by any integer \( k \) where \( 2 \leqslant k \leqslant 11 \), the remainder is 1. What is the difference between the two smallest such integers?

(29th Annual American High School Mathematics Examination, 1978)

Answer: $27720$


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

The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result.

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>.

In triangle  $ABC$ ,  $\angle ACB=50^{\circ}$  and  $\angle CBA=70^{\circ}$ . Let  $D$  be a foot of perpendicular from point  $A$  to side  $BC$ ,  $O$  circumcenter of  $ABC$  and  $E$  antipode of  $A$  in circumcircle  $ABC$ . Find  $\angle DAE$

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>.

Knowing the horizontal component of the tension force, find the minimum force acting on a buoy from the current.

Consider the forces acting on the buoy along the horizontal axis: the force from the current acts in one direction, and the horizontal component of the tension force acts in the opposite direction.

\[M_5 \cdot a_{\sigma_{-} x} = T \cdot \sin(a) - F_{\text{tey_rop}}\]

where \(M_{\text{Б}}\) is the mass of the buoy, \(a_x\) is its acceleration along the horizontal axis, and \(\mathrm{F}_{\text{теч_го}}\) is the force acting on the buoy from the current in the horizontal direction.

Therefore, at the moment the anchor is lifted, the following equality is satisfied:

\[\mathrm{F}_{\text{Tey_rop}} = T \cdot \sin(a)\]

Hence, the minimum value of the force acting on the buoy from the current is equal to its horizontal projection, which is

\[\mathrm{F}_{\text{теч}} = \mathrm{F}_{\text{теч_гор}} = 400 \text{ N}\]

Answer: $400$


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

When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point.

Answer: $14.5$


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

Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\).

Answer: $\frac{27}{7 \sqrt{130}}$


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

Given the function $y=f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+2)=\frac{1}{f(x)}$. When $x \in (0, 2]$, $f(x)=x$.
(1) Find the analytical expression of $f(x)$ when $x \in (2, 4]$;
(2) If $f(m)=1$, find the value of $m$;
(3) Calculate the sum: $f(1)+f(2)+f(3)+...+f(2015)$.

Answer: $\frac{4535}{2}$


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>.

Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.
What is the area of the rectangle?

A 16
B 18
C 20
D 22
E 24

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 corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid down. What could be the maximum number of uncovered sections (the width of the carpet strip is equal to the width of the corridor)?

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:
\\((1)y=x+ \\frac {4}{x}\\)
\\((2)y=\\sin x+ \\frac {4}{\\sin x}(0 < x < π)\\)
\\((3)y= \\frac {x^{2}+13}{ \\sqrt {x^{2}+9}}\\)
\\((4)y=4⋅2^{x}+2^{-x}\\)
\\((5)y=\\log \_{3}x+4\\log \_{x}3(0 < x < 1)\\)

Find the function(s) with a minimum value of $4$. (Fill in the correct question number)

Answer: $(4)$


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

Suppose point P is on the curve $y=x^2+1$ (where $x \geq 0$), and point Q is on the curve $y=\sqrt{x-1}$ (where $x \geq 1$). Then the minimum value of the distance $|PQ|$ is ______.

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


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

Consider all non-empty subsets of the set \( S = \{1, 2, \cdots, 10\} \). A subset is called a "good subset" if the number of even numbers in the subset is not less than the number of odd numbers. How many "good subsets" are there?

Answer: $637$


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

The diagonal $KM$ of trapezoid $KLMN$ is 3 times the length of segment $KP$ on this diagonal. The base $KN$ of the trapezoid is 3 times the length of the base $LM$. Find the ratio of the area of trapezoid $KLMN$ to the area of triangle $KPR$, where $R$ is the point of intersection of line $PN$ and side $KL$.

Answer: $32/3$


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

Given that $\{a_n\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$.

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>.

Two subsets of the set $T = \{w, x, y, z, v\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order?

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>.

Determine the number of ways to arrange the letters of the word "PERCEPTION".

Answer: $907,200$


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

The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \).

Answer: $984$


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

The device consists of three independently operating elements. The probabilities of failure-free operation of the elements (over time $t$) are respectively: $p_{1}=0.7$, $p_{2}=0.8$, $p_{3}=0.9$. Find the probabilities that over time $t$, the following will occur:
a) All elements operate without failure;
b) Two elements operate without failure;
c) One element operates without failure;
d) None of the elements operate without failure.

Answer: $0.006$


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

Given a hyperbola with eccentricity $2$ and equation $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola at two points $P$ and $Q$. $F_1$ is the left focus of the hyperbola. If $PF_1 \perp QF_1$, then find the slope of line $l$.

Answer: $\dfrac{3\sqrt{7}}{7}$


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>.

How many positive 3-digit numbers are multiples of 30, but not of 75?

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>.

After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. The number of table tennis rackets purchased does not exceed 150 sets. The purchase price and selling price are as follows:

|            | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $50$ |
| Badminton Racket | $b$ | $60$ |

It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $110$ yuan, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $260$ yuan.

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

$(2)$ Based on past sales experience, the store decided to purchase a number of table tennis rackets not less than half the number of badminton rackets. Let $x$ be the number of table tennis rackets purchased, and $y$ be the profit obtained after selling these sports equipment.

① Find the functional relationship between $y$ and $x$, and write down the range of values for $x$.

② When the store actually makes purchases, coinciding with the "Double Eleven" shopping festival, the purchase price of table tennis rackets is reduced by $a$ yuan per set $(0 < a < 10)$, while the purchase price of badminton rackets remains the same. Given that the selling price remains unchanged, and all the sports equipment can be sold out, how should the store make purchases to maximize profit?

Answer: $150$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\cos B$ in terms of $a$ and $r$.

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


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

Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engaged in walking activities and found that the body fat percentage of the survey respondents follows a normal distribution $N(0.2, \sigma^2)$. It is defined that people with a body fat percentage less than or equal to $0.17$ have a good body shape. If $16\%$ of the people participating in walking have a good body shape, then the value of $\sigma$ is approximately ____.<br/>Reference data: $P(\mu -\sigma \leq X \leq \mu + \sigma) \approx 0.6827$, $P(\mu -2\sigma \leq X \leq \mu + 2\sigma) \approx 0.9545$.

Answer: $0.03$


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

Consider the curve $y=x^{n+1}$ (where $n$ is a positive integer) and its tangent at the point (1,1). Let the x-coordinate of the intersection point between this tangent and the x-axis be $x_n$.
(Ⅰ) Let $a_n = \log{x_n}$. Find the value of $a_1 + a_2 + \ldots + a_9$.
(Ⅱ) Define $nf(n) = x_n$. Determine whether there exists a largest positive integer $m$ such that the inequality $f(n) + f(n+1) + \ldots + f(2n-1) > \frac{m}{24}$ holds for all positive integers $n$. If such an $m$ exists, find its value; if not, explain why.

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>.

In a right triangle $\triangle STU$, where $\angle S = 90^\circ$, suppose $\sin T = \frac{3}{5}$. If the length of $SU$ is 15, find the length of $ST$.

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>.

Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$.

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>.

Translate the graph of $y= \sqrt{2}\sin(2x+ \frac{\pi}{3})$ to the right by $\varphi$ ($0<\varphi<\pi$) units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Find the value of $\varphi$.

Answer: $\frac{13\pi}{24}$


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>.

Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

It is known that there is a four-person team consisting of boys A and B and girls C and D participating in a quiz activity organized by a TV station. The activity consists of four rounds. Let the probabilities for the boys to pass rounds one to four be $\frac{5}{6}, \frac{4}{5}, \frac{3}{4}, \frac{2}{3}$, respectively, and the probabilities for the girls to pass rounds one to four be $\frac{4}{5}, \frac{3}{4}, $\frac{2}{3}, \frac{1}{2}$, respectively.
(1) Find the probability that a boy passes all four rounds;
(2) Let $\xi$ represent the number of people in the four-person team who pass all four rounds. Find the distribution and the expectation of the random variable $\xi$.

Answer: $\frac{16}{15}$


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

Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.

(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.
(2) If $A={2}$ and $a\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(a)$.

Answer: $\frac{63}{4}$


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 $f(x)$ is an odd function on $\mathbb{R}$, when $x\geqslant 0$, $f(x)= \begin{cases} \log _{\frac {1}{2}}(x+1),0\leqslant x < 1 \\ 1-|x-3|,x\geqslant 1\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\frac {1}{2}$.

Answer: $\sqrt {2}-1$


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>.

Given the hyperbola $C: \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$, with its right vertex at $P$.
(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;
(2) Let line $l$ pass through point $P$ with normal vector $\overrightarrow{n}=(1,-1)$. If there are exactly three points $P_{1}$, $P_{2}$, and $P_{3}$ on hyperbola $C$ with the same distance $d$ to line $l$, find the value of $d$.

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


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

In $\triangle ABC$, $AB = AC = 3.6$, point $D$ lies on $AB$ with $AD = 1.2$, and point $E$ lies on the extension of $AC$. The area of $\triangle AED$ is equal to the area of $\triangle ABC$. Find the length of $AE$.

Answer: $10.8$


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>.

Cyclic pentagon \( A B C D E \) has a right angle \( \angle A B C = 90^\circ \) and side lengths \( A B = 15 \) and \( B C = 20 \). Supposing that \( A B = D E = E A \), find \( C D \).

Answer: $20$


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

Given two lines $l_1: y=a$ and $l_2: y= \frac {18}{2a+1}$ (where $a>0$), $l_1$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points A and B, and $l_2$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points C and D. Let the projection lengths of line segments AC and BD on the x-axis be $m$ and $n$ respectively. When $a= \_\_\_\_\_\_$, $\frac {n}{m}$ reaches its minimum value.

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>.

In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \sqrt{2}$. Find the value of $x$.

Answer: $-6$


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

A right triangle $ABC$ is inscribed in a circle. From the vertex $C$ of the right angle, a chord $CM$ is drawn, intersecting the hypotenuse at point $K$. Find the area of triangle $ABM$ if $BK: AB = 3:4$, $BC=2\sqrt{2}$, $AC=4$.

Answer: $\frac{36}{19} \sqrt{2}$


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 $f(x) = \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2) \end{cases}$, determine the value of $x$ if $f(x)=3$.

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>.

What is the nearest integer to $(3+\sqrt2)^6$?

Answer: $7414$


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

A sports conference has 12 teams in two divisions of 6 each. How many games are in a complete season for the conference if each team must play every other team in its own division three times and every team in the other division twice?

Answer: $162$


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

Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as  $\frac{a}{b}$  for positive integers  $a,b$  with  $\gcd (a,b) = 1$ , then find  $a + b$ .

*Proposed by Vismay Sharan*

Answer: $831$


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

Read the following material and complete the corresponding tasks.

平衡多项式
定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子.
例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x^{2}+7x+10)=-4$,所以多项式$x+1$,$x+2$,$x+5$,$x+6$是一组平衡多项式,其平衡因子为$|-4|=4$.

任务:
$(1)$小明发现多项式$x+3$,$x+4$,$x+6$,$x+7$是一组平衡多项式,在求其平衡因子时,列式如下:$(x+3)(x+7)-(x+4)(x+6)$,根据他的思路求该组平衡多项式的平衡因子；
$A$.判断多项式$x-1$,$x-2$,$x-4$,$x-5$是否为一组平衡多项式,若是,求出其平衡因子；若不是,说明理由.
$B$.若多项式$x+2$,$x-4$,$x+1$,$x+m(m$是常数）是一组平衡多项式,求$m$的值.

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>.

Let $\triangle DEF$ be an isosceles triangle with $DE = DF$. Three circles are defined as follows: the circle $\Omega$ with its center at the centroid of $\triangle DEF$, and two circles $\Omega_1$ and $\Omega_2$, where $\Omega_1$ is tangent to $\overline{EF}$ and externally tangent to the other sides extended, while $\Omega_2$ is tangent to $\overline{DE}$ and $\overline{DF}$ while internally tangent to $\Omega$. Determine the smallest integer value for the perimeter of $\triangle DEF$ if $EF$ is the smallest integer side, and $\Omega_1$ and $\Omega_2$ intersect only once.

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 product of the solutions to the quadratic equation $49 = -2x^2 - 8x$.

Answer: $\frac{49}{2}$


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 sides of rectangle $ABCD$ have lengths $12$ and $5$. A right triangle is drawn so that no point of the triangle lies outside $ABCD$, and one of its angles is $30^\circ$. The maximum possible area of such a triangle can be written in the form $p \sqrt{q} - r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.

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>.

Consider a list of numbers \[8, 3, x, 3, 7, 3, y\]. When the mean, median, and mode of this list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible values of \(y\) if \(x=6\)?
A) $\frac{51}{13}$
B) $33$
C) $\frac{480}{13}$
D) $40$
E) $34$

Answer: $\frac{480}{13}$


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

Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_.

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


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

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

Answer: $\frac{11}{14}$


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

There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ integers $b_k$ ($1\le k\le s$), with each $b_k$ either $1$ or $-1$, such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1007.\] Find $m_1 + m_2 + \cdots + m_s$.

Answer: $15$


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

The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______".

Answer: $-2$


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

How many integers between 10000 and 100000 include the block of digits 178?

Answer: $280$


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

Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.

Answer: $16$


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

There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this?

Answer: $761$


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

Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$.

Answer: $-52$


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

Let \( n \geq 1 \) be a positive integer. We say that an integer \( k \) is a fan of \( n \) if \( 0 \leq k \leq n-1 \) and there exist integers \( x, y, z \in \mathbb{Z} \) such that
\[
\begin{aligned}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{aligned}
\]
Let \( f(n) \) be the number of fans of \( n \). Determine \( f(2020) \).

Answer: $101$


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

The sequence $\{a\_n\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ .

Answer: $\frac {2017}{2}$


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 a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes. Each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same). The number of situations where both A and B grab a red envelope is \_\_\_\_\_\_ (Answer in digits).

Answer: $18$


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

In a circle, parallel chords of lengths 8, 15, and 17 determine central angles of $\gamma$, $\delta$, and $\gamma + \delta$ radians, respectively, where $\gamma + \delta < \pi$. If $\cos \gamma$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Answer: $32$


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

Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\).

Answer: $24\pi$


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

At a family outing to a theme park, the Thomas family, comprising three generations, plans to purchase tickets. The two youngest members, categorized as children, get a 40% discount. The two oldest members, recognized as seniors, enjoy a 30% discount. The middle generation no longer enjoys any discount. Grandmother Thomas, whose senior ticket costs \$7.50, has taken the responsibility to pay for everyone. Calculate the total amount Grandmother Thomas must pay.
A) $46.00$
B) $47.88$
C) $49.27$
D) $51.36$
E) $53.14$

Answer: $49.27$


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>.

Let  $A$ ,  $B$ , and  $C$  be distinct points on a line with  $AB=AC=1$ . Square  $ABDE$  and equilateral triangle  $ACF$  are drawn on the same side of line  $BC$ . What is the degree measure of the acute angle formed by lines  $EC$  and  $BF$ ?

*Ray Li*

Answer: $75$


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

Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$  Find the value of $p + q + r + s.$

Answer: $2028$


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

Given the function $f(x)=2x^{3}-ax^{2}+1$ $(a\in\mathbb{R})$ has exactly one zero in the interval $(0,+\infty)$, find the sum of the maximum and minimum values of $f(x)$ on the interval $[-1,1]$.

Answer: $-3$


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

Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy?

Answer: $406$


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

Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).
(1) Find the trajectory equation of the moving circle P's center.
(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve C. The perpendicular bisector of line segment AB passes through point D(0, 1/2). Find the maximum area of △OAB (O is the coordinate origin).

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>.

Let \(n = 2^{20}3^{25}\). How many positive integer divisors of \(n^2\) are less than \(n\) but do not divide \(n\)?

Answer: $499$


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

Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration.

Answer: $14$


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

Let $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation
\[ (3+i)^n = p_n + q_n i\] for all integers $n \geq 0$, where $i = \sqrt{-1}$. Determine the value of
\[ \sum_{n=0}^\infty \frac{p_n q_n}{9^n}.\]
A) $\frac{5}{8}$
B) $\frac{7}{8}$
C) $\frac{3}{4}$
D) $\frac{15}{16}$
E) $\frac{9}{20}$

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>.

Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution.

Answer: $1012$