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

Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\frac{1}{2}$ is $\frac{a - b \pi}{c}\left(a, b, c \in \mathbf{Z}_{+}, (a, b, c)=1\right)$. Find the value of $a + b + c$.

Answer: $59$


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

Given a finite sequence $S=(2, 2x, 2x^2,\ldots ,2x^{200})$ of $n=201$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{200}+a_{201}}{2}\right)$ of $n-1=200$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le 150$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(2, 2x, 2x^2,\ldots ,2x^{200})$. If $A^{150}(S)=(2 \cdot 2^{-75})$, then what is $x$?
A) $1 - \frac{\sqrt{2}}{2}$
B) $2^{3/8} - 1$
C) $\sqrt{2} - 2$
D) $2^{1/5} - 1$

Answer: $2^{3/8} - 1$


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

An organization has a structure where there is one president, two vice-presidents (VP1 and VP2), and each vice-president supervises two managers. If the organization currently has 12 members, in how many different ways can the leadership (president, vice-presidents, and managers) be chosen?

Answer: $554400$


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

A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads?

Answer: $\frac{3}{128}$


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

Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\).

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


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

The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a "lucky coin." Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is "Yes," she needs to pay a consultation fee of 20 yuan. If the answer is "No," she needs to pay a consultation fee of 10 yuan. She can also choose not to ask any questions and directly buy some coins. What is the minimum amount of money (in yuan) Feifei needs to pay to guarantee she gets the lucky coin?

Answer: $130$


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

Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$.

Answer: $343$


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

Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.

$(1)$ Find the equation of ellipse $C$ and its eccentricity;

$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why.

Answer: $\dfrac{8}{5}$


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>.

Inside a square $R_1$ with area 81, an equilateral triangle $T_1$ is inscribed such that each vertex of $T_1$ touches one side of $R_1$. Each midpoint of $T_1’s$ sides is connected to form a smaller triangle $T_2$. The process is repeated with $T_2$ to form $T_3$. Find the area of triangle $T_3$.

Answer: $\frac{81\sqrt{3}}{256}$


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

In the diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$.

Answer: $13:1$


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>.

Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \leqslant a \leqslant 4$ and $a \in R)$ units of the nutrient solution are released, its concentration $y (\text{g}/\text{L})$ changes over time $x (\text{days})$ according to the function $y = af(x)$, where $f(x)=\begin{cases} \frac{4+x}{4-x} & 0\leqslant x\leqslant 2 \\ 5-x & 2\prec x\leqslant 5 \end{cases}$. If the nutrient solution is released multiple times, the concentration at a given moment is the sum of the concentrations released at the corresponding times. According to experience, the nutrient solution is effective only when its concentration is not less than $4(\text{g}/\text{L})$.

(1) If $4$ units of the nutrient solution are released only once, how many days can it be effective?

(2) If $2$ units of the nutrient solution are released first, and then $b$ units are released after $3$ days. In order to keep the nutrient solution continuously effective in the next $2$ days, find the minimum value of $b$.

Answer: $24-16\sqrt{2}$


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

A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral?

Answer: $1/2$


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

Two rectangles, one $8 \times 10$ and the other $12 \times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction.

Answer: $65$


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

A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \(\frac{10}{3}\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of 1?

Answer: $56$


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

Let \( T = 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \cdots + 2023 + 2024 - 2025 - 2026 \). What is the residue of \( T \), modulo 2027?

Answer: $2026$


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

Given that $x$ and $y$ are real numbers, and they satisfy $xy + x + y = 17$, $x^2y + xy^2 = 66$, find the value of $x^4 + x^3y + x^2y^2 + xy^3 + y^4$.

Answer: $12499$


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

Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces?

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>.

Let $a^2 = \frac{9}{25}$ and $b^2 = \frac{(3+\sqrt{7})^2}{14}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a-b)^2$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$?

Answer: $22$


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

In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.

[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,linewidth(1));
draw((0,0)--(1,1)--(1,2)--(0,0)--cycle,linewidth(1));
dot((1,2));

label("A",(0,0),SW);
label("B",(2,0),SE);
label("C",(1,2),N);
label("D",(1,1),NE);
label("E",(1,0),S);
[/asy]

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>.

How many pairs of positive integers \((x, y)\) are there such that \(x < y\) and \(\frac{x^{2}+y^{2}}{x+y}\) is a divisor of 2835?

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>.

Let \(x,\) \(y,\) \(z\) be real numbers such that \(9x^2 + 4y^2 + 25z^2 = 1.\) Find the maximum value of
\[8x + 3y + 10z.\]

Answer: $\sqrt{173}$


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

A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round.

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>.

Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an eight-sided die (numbered 1 to 8) is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
A) $\frac{1}{12}$
B) $\frac{1}{8}$
C) $\frac{1}{6}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$

Answer: $\frac{1}{6}$


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>.

Let the function $f(x)=2\tan \frac{x}{4}\cdot \cos^2 \frac{x}{4}-2\cos^2\left(\frac{x}{4}+\frac{\pi }{12}\right)+1$.

(Ⅰ) Find the smallest positive period and the domain of $f(x)$;

(Ⅱ) Find the intervals of monotonicity and the extremum of $f(x)$ in the interval $[-\pi,0]$;

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


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

Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .

*Proposed by Michael Ren*

Answer: $7500$


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

Given that there are 5 people standing in a row, calculate the number of ways for person A and person B to stand such that there is exactly one person between them.

Answer: $36$


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

A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in terms of $\pi$.

Answer: $9\pi\sqrt{15}$


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>.

Let  $n$  be a positive integer. All numbers  $m$  which are coprime to  $n$  all satisfy  $m^6\equiv 1\pmod n$ . Find the maximum possible value of  $n$ .

Answer: $504$


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

In a flood control emergency, a large oil tank drifting downstream from upstream needs to be exploded by shooting. It is known that there are only $5$ bullets. The first hit can only cause the oil to flow out, and the second hit can cause an explosion. Each shot is independent, and the probability of hitting each time is $\frac{2}{3}$.

$(1)$ Calculate the probability that the oil tank will explode;

$(2)$ If the oil tank explodes or the bullets run out, the shooting will stop. Let $X$ be the number of shots. Calculate the probability that $X$ is not less than $4$.

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


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>.

9. The real quartic  $P x^{4}+U x^{3}+M x^{2}+A x+C$  has four different positive real roots. Find the square of the smallest real number  $z$  for which the expression  $M^{2}-2 U A+z P C$  is always positive, regardless of what the roots of the quartic are.

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>.

Darryl has a six-sided die with faces  $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability  $1/2$  and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die  $5$  times and gets a  $1, 2, 3, 4$  and  $5$  in some unspecified
order. Compute the probability that his next roll is a  $6$ .

Answer: $3/26$


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

If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of "quasi-geometric progression." Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these $4$ numbers have the property of "quasi-geometric progression" is ____.

Answer: $\frac{13}{35}$


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>.

Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 chicks were sitting with their beaks open and 11 were growing feathers.

a) How many chicks were sitting with their beaks open two weeks ago?

b) How many chicks will be growing feathers next week?

Record the product of these numbers as the answer.

Answer: $165$


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

Let the function be $$f(x)=\sin(2\omega x+ \frac {\pi}{3})+ \frac { \sqrt {3}}{2}+a(\omega＞0)$$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $$\frac {\pi}{6}$$.
(1) Find the value of $\omega$;
(2) If the minimum value of $f(x)$ in the interval $$[- \frac {\pi}{3}, \frac {5\pi}{6}]$$ is $$\sqrt {3}$$, find the value of $a$;
(3) If $g(x)=f(x)-a$, what transformations are applied to the graph of $y=\sin x$ ($x\in\mathbb{R}$) to obtain the graph of $g(x)$? Also, write down the axis of symmetry and the center of symmetry for $g(x)$.

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


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

There are 6 balls of each of the four colors: red, blue, yellow, and green. Each set of 6 balls of the same color is numbered from 1 to 6. If 3 balls with different numbers are randomly selected, and these 3 balls have different colors and their numbers are not consecutive, the number of ways to do this is ______.

Answer: $96$


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

Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\).

Answer: $\frac{\ln 5}{2 \ln 2 - 3 \ln 5}$


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 a school's mentoring program, several first-grade students can befriend one sixth-grade student, while one sixth-grade student cannot befriend multiple first-grade students. It is known that $\frac{1}{3}$ of the sixth-grade students and $\frac{2}{5}$ of the first-grade students have become friends. What fraction of the total number of students in the first and sixth grades are these friends?

Answer: $\frac{4}{11}$


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>.

Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence, and calculate the 2003rd term of this new sequence.

Answer: $2047$


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

The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:

$\frac{4}{3}\pi r^3 = 50,000$

Simplifying, we get $r^3 = \frac{50,000 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.

Now, we want to find the volume of the top portion of Logan’s model. It is given that this sphere should have a volume of 0.2 liters. Using the same formula, we can solve for the radius:

$\frac{4}{3}\pi r^3 = 0.2$

Simplifying, we get $r^3 = \frac{0.2 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.

Now, since the sphere is a model of the top portion of the water tower, the radius of the model is proportional to the radius of the real tower. Therefore, the ratio of the radius of the model to the radius of the real tower is equal to the ratio of the volume of the model to the volume of the real tower.

$\frac{r_{model}}{r_{real}} = \frac{0.2}{50000}$

We can now equate the two expressions for the radius and solve for the height of the model:

 $\frac{r_{model}}{60} = \frac{0.2}{50000}$

Answer: $0.95$


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

Find the 20th term in the sequence: $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \ldots, \frac{1}{m+1}, \frac{2}{m+1}, \ldots, \frac{m}{m+1}, \ldots$

Answer: $\frac{6}{7}$


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

The area of square \(ABCD\) is 64. The midpoints of its sides are joined to form the square \(EFGH\). The midpoints of its sides are \(J, K, L,\) and \(M\). The area of the shaded region is:

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>.

For environmental protection, Wuyang Mineral Water recycles empty bottles. Consumers can exchange 4 empty bottles for 1 bottle of mineral water (if there are fewer than 4 empty bottles, they cannot be exchanged). Huacheng Middle School bought 1999 bottles of Wuyang brand mineral water. If they exchange the empty bottles for new bottles of water as much as possible, then the teachers and students of Huacheng Middle School can drink a total of     bottles of mineral water; conversely, if they can drink a total of 3126 bottles of mineral water, then they originally bought     bottles of mineral water.

Answer: $2345$


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

One angle of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid.

Answer: $1:3$


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>.

Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day before. If he is late for work, he changes his mode of transportation the next day. Given all this, how likely is it that Maurice will be late for work on his 467th trip?

Answer: $2/3$


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 lighting power increased by
\[ \Delta N = N_{\text {after}} - N_{\text {before}} = 300\, \text{BT} - 240\, \text{BT} = 60\, \text{BT} \]

Answer: $60$


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

If \(x + \frac{1}{y} = 3\) and \(y + \frac{1}{z} = 3\), what is the value of the product \(xyz\)?

Answer: $-1$


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

Given $\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\sin A= \sqrt {3}a\cos C$.

(Ⅰ) Find angle $C$;
(Ⅱ) If $c= \sqrt {21}$, and $\sin C+\sin (B-A)=5\sin 2A$, find the area of $\triangle ABC$.

Answer: $\frac {5 \sqrt {3}}{4}$


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>.

Four teams, including Quixajuba, are competing in a volleyball tournament where:

- Each team plays against every other team exactly once;
- Any match ends with one team winning;
- In any match, the teams have an equal probability of winning;
- At the end of the tournament, the teams are ranked by the number of victories.

a) Is it possible that, at the end of the tournament, all teams have the same number of victories? Why?

b) What is the probability that the tournament ends with Quixajuba alone in first place?

c) What is the probability that the tournament ends with three teams tied for first place?

Answer: $\frac{1}{8}$


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>.

Determine the largest positive integer  $n$  for which there exists a set  $S$  with exactly  $n$  numbers such that
- each member in  $S$  is a positive integer not exceeding  $2002$ ,
- if  $a,b\in S$  (not necessarily different), then  $ab\not\in S$ .

Answer: $1958$


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

There are  $27$  unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a  $3\times 3 \times 3$  cube with these  $27$  cubes. What is the least number of marked points on the faces of the new cube?

Answer: $90$


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

Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\]

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>.

Express the quotient $2213_4 \div 13_4$ in base 4.

Answer: $53_4$


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

A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$?

Answer: $0.8$


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>.

Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \).

Answer: $2/3$


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

Given the set $A=\{(x,y) \,|\, |x| \leq 1, |y| \leq 1, x, y \in \mathbb{R}\}$, and $B=\{(x,y) \,|\, (x-a)^2+(y-b)^2 \leq 1, x, y \in \mathbb{R}, (a,b) \in A\}$, then the area represented by set $B$ is \_\_\_\_\_\_.

Answer: $12 + \pi$


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>.

What is the least positive integer with exactly $12$ positive factors?

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>.

Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$.

Answer: $26$


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

The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$.  Find the maximum possible area of such a triangle.

Answer: $36\sqrt{3}$


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

The probability that a car driving on this road does not stop at point A and the probability that it does not stop at point B and the probability that it does not stop at point C are $\left(1-\frac{25}{60}\right)$, $\left(1-\frac{35}{60}\right)$, and $\left(1-\frac{45}{60}\right)$, respectively.

Answer: $\frac{35}{192}$


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

In the right trapezoid \(ABCD\), it is known that \(AB \perp BC\), \(BC \parallel AD\), \(AB = 12\), \(BC = 10\), and \(AD = 6\). Point \(F\) is a movable point on a circle centered at point \(C\) with radius 8, and point \(E\) is a point on \(AB\). When the value of \(DE + EF\) is minimized, what is the length of \(AE\)?

Answer: $4.5$


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

Given $X \sim N(\mu, \sigma^2)$, $P(\mu-\sigma < X \leq \mu+\sigma) = 0.68$, $P(\mu-2\sigma < X \leq \mu+2\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scored above 120?

Answer: $500$


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

In the set of all three-digit numbers composed of the digits 0, 1, 2, 3, 4, 5, without any repeating digits, how many such numbers have a digit-sum of 9?

Answer: $12$


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

Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$.

Answer: $\frac{3\sqrt{6}}{2} + 1$


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 that the line $x - 2y + 2k = 0$ encloses a triangle with an area of $1$ together with the two coordinate axes, find the value of the real number $k$.

Answer: $-1$


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

Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits.

Answer: $252$


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

Each segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of the squares of which is equal to 1, and the segments carry the products of the numbers at their ends. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the maximum number that could be obtained?

Answer: $-1$


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

The perpendicular to the side $AB$ of the trapezoid $ABCD$, passing through its midpoint $K$, intersects the side $CD$ at point $L$. It is known that the area of quadrilateral $AKLD$ is five times greater than the area of quadrilateral $BKLC$. Given $CL=3$, $DL=15$, and $KC=4$, find the length of segment $KD$.

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>.

Albert now decides to extend his list to the 2000th digit. He writes down positive integers in increasing order with a first digit of 1, such as $1, 10, 11, 12, \ldots$. Determine the three-digit number formed by the 1998th, 1999th, and 2000th digits.

Answer: $141$


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

What is ${-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}$?

Answer: $-199$


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

Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface),
if $x + y + z = s$, and $0 \leq x \leq y \leq z \leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\_\_\_\_\_\_\_\_\_\_$.

Answer: $\frac{1}{6}$


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>.

Let ellipse M be defined by the equation $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defined by $x^{2}+y^{2}=4$.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and there is a point $P(1, \sqrt {2})$ on ellipse M, find the maximum area of triangle PAB.

Answer: $\sqrt {2}$


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

The digits from 1 to 9 are each used exactly once to write three one-digit integers and three two-digit integers. The one-digit integers are equal to the length, width and height of a rectangular prism. The two-digit integers are equal to the areas of the faces of the same prism. What is the surface area of the rectangular prism?

Answer: $198$


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

A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there?

Answer: $36$


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

Given the function $f(x) = \begin{cases} \log_{10} x, & x > 0 \\ x^{-2}, & x < 0 \end{cases}$, if $f(x\_0) = 1$, find the value of $x\_0$.

Answer: $10$


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

In a factor tree, each value is the product of the two values below it, unless a value is a prime number or a preset integer product of primes. Using this structure, calculate the value of $X$ in the factor tree provided:

[asy]
draw((-2,-.3)--(0,0)--(2,-.3),linewidth(1));
draw((-3,-1.3)--(-2,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(2,-.8)--(3,-1.3),linewidth(1));
label("X",(0,0),N);
label("F",(-2,-.8),N);
label("7",(-3,-1.3),S);
label("G",(2,-.8),N);
label("4",(-1,-1.3),S);
label("11",(1,-1.3),S);
label("H",(3,-1.3),S);
draw((-2,-2.3)--(-1,-1.8)--(0,-2.3),linewidth(1));
draw((2,-2.3)--(3,-1.8)--(4,-2.3),linewidth(1));
label("7",(-2,-2.3),S);
label("2",(0,-2.3),S);
label("11",(2,-2.3),S);
label("2",(4,-2.3),S);
[/asy]

Answer: $6776$


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

The area of the ground plane of a truncated cone  $K$  is four times as large as the surface of the top surface. A sphere  $B$  is circumscribed in  $K$ , that is to say that  $B$  touches both the top surface and the base and the sides. Calculate ratio volume  $B :$  Volume  $K$ .

Answer: $9/14$


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>.

The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in the population and the sample.

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 the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\).

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


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

In $\triangle ABC$, the sides opposite to the three internal angles are $a$, $b$, and $c$, respectively. Given that $\cos A= \frac{ \sqrt {10}}{10}$ and $a\sin A+b\sin B-c\sin C= \frac{ 2 \sqrt {5}}{5}a\sin B$.
1. Find the value of $B$;
2. If $b=10$, find the area $S$ of $\triangle ABC$.

Answer: $60$


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

In the diagram, $ABCD$ and $EFGD$ are squares each with side lengths of 5 and 3 respectively, and $H$ is the midpoint of both $BC$ and $EF$. Calculate the total area of the polygon $ABHFGD$.

Answer: $25.5$


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>.

Define a new operation: \( x \odot y = 18 + x - a \times y \), where \( a \) is a constant. For example:
\[ 1 \odot 2 = 18 + 1 - a \times 2. \]

If \( 2 \odot 3 = 8 \), then what is \( 3 \odot 5 \) and \( 5 \odot 3 \)?

Answer: $11$


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

Given the system of equations for the positive numbers \(x, y, z\):

$$
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=108 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=124
\end{array}\right.
$$

Find the value of the expression \(xy + yz + xz\).

Answer: $48$


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \_\_\_\_\_

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>.

Two particles move along the edges of a square $ABCD$ with \[A \Rightarrow B \Rightarrow C \Rightarrow D \Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at vertex $A$, and the other starts at the midpoint of side $CD$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of square $ABCD$?
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{9}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$

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


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

In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is half the height \(BH\) of triangle \(ABC\). Find the ratio \(AH:HC\).

Answer: $1/3$


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>.

Vasya wrote consecutive natural numbers \( N \), \( N+1 \), \( N+2 \), and \( N+3 \) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.

The sum of the numbers in the first two circles turned out to be 200, and the sum of the numbers in the third and fourth circles turned out to be 105. What is the sum of the numbers in the second and third circles?

Answer: $103$


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

Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$.

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 a regular tetrahedron \( P-ABCD \), where each face is an equilateral triangle with side length 1, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( BC \), respectively. Find the distance between the skew lines \( MN \) and \( PC \).

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


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

Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$.

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


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

Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of
\[\frac{2}{a} + \frac{4}{b}.\]

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>.

Given that $AC$ and $CE$ are two diagonals of a regular hexagon $ABCDEF$, and points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$.

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


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>.

How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)?

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>.

Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B.

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


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

Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(T\).

Answer: $2\sqrt{3}$


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

29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?

Answer: $29$


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

A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\circ$?

Answer: $48$


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

Given that there is a geometric sequence $\{a_n\}$ with a common ratio $q > 1$ and the sum of the first $n$ terms is $S_n$, $S_3 = 7$, the sequence $a_1+3$, $3a_2$, $a_3+4$ forms an arithmetic sequence. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and $6T_n = (3n+1)b_n + 2$ for $n \in \mathbb{N}^*$.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Find the general term formula for the sequence $\{b_n\}$.
(3) Let $A = \{a_1, a_2, \ldots, a_{10}\}$, $B = \{b_1, b_2, \ldots, b_{40}\}$, and $C = A \cup B$. Calculate the sum of all elements in the set $C$.

Answer: $3318$


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

The equations $x^3 + Cx + 20 = 0$ and $x^3 + Dx^2 + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$

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>.

In an infinite increasing sequence of natural numbers, each number is divisible by at least one of the numbers 1005 and 1006, but none is divisible by 97. Additionally, any two consecutive numbers differ by no more than $k$. What is the smallest possible $k$ for this scenario?

Answer: $2011$


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

Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.
(I) Find the standard equation of the ellipse;
(II) If a line passing through $F_1$ intersects the ellipse at points $B$ and $D$, and another line passing through $F_2$ intersects the ellipse at points $A$ and $C$, with $AC \perp BD$, find the minimum value of $|AC| + |BD|$.

Answer: $\frac{16\sqrt{3}}{5}$


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

Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$

Answer: $227.052227052227$


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

In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are located on sides $YZ$, $XZ$, and $XY$, respectively. The cevians $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$. Given that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the value of $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$.

Answer: $98$


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

In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\).

Answer: $70$


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

An equilateral pentagon  $AMNPQ$  is inscribed in triangle  $ABC$  such that  $M\in\overline{AB}$ ,  $Q\in\overline{AC}$ , and  $N,P\in\overline{BC}$ .

Suppose that  $ABC$  is an equilateral triangle of side length  $2$ , and that  $AMNPQ$  has a line of symmetry perpendicular to  $BC$ . Then the area of  $AMNPQ$  is  $n-p\sqrt{q}$ , where  $n, p, q$  are positive integers and  $q$  is not divisible by the square of a prime. Compute  $100n+10p+q$ .

*Proposed by Michael Ren*

Answer: $5073$


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

The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what?

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


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>.

Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\).

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>.

Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $EFGH$, has its sides parallel to the corresponding sides of the larger square, and each side of $EFGH$ is tangent to one of the semicircles. Provide the area of square $EFGH$.
A) $36$
B) $64$
C) $100$
D) $144$
E) $256$

Answer: $144$


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

$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals:

Answer: $1000$


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

A grid consists of multiple squares, as shown below. Count the different squares possible using the lines in the grid.

[asy]
unitsize(0.5 cm);
int i, j;
for(i = 0; i <= 5; ++i) {
    draw((0,i)--(5,i));
    draw((i,0)--(i,5));
}
for(i = 1; i <= 4; ++i) {
    for (j = 1; j <= 4; ++j) {
        if ((i > 1 && i < 4) || (j > 1 && j < 4)) continue;
        draw((i,j)--(i,j+1)--(i+1,j+1)--(i+1,j)--cycle);
    }
}
[/asy]

Answer: $54$


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

A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

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>.

$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\cos \angle BNG$?

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>.

An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each pair of adjacent red dots counts as one fault, and each pair of blue dots exactly two apart (separated by one dot) counts as another fault. What is the smallest number of faults the critic could find?

Answer: $250$


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

Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a\cos C=c\sin A$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\triangle ABC$.

Answer: $9 + 6\sqrt{2}$


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

For a natural number  $b$ , let  $N(b)$  denote the number of natural numbers  $a$  for which the equation  $x^2 + ax + b = 0$  has integer roots. What is the smallest value of  $b$  for which  $N(b) = 20$ ?

Answer: $240$


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

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is to the left, right, up, or down, all four equally likely. Let $q$ be the probability that the object reaches $(3,3)$ in eight or fewer steps. Write $q$ in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

Answer: $4151$


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

Given the product \( S = \left(1+2^{-\frac{1}{32}}\right)\left(1+2^{-\frac{1}{16}}\right)\left(1+2^{-\frac{1}{8}}\right)\left(1+2^{-\frac{1}{4}}\right)\left(1+2^{-\frac{1}{2}}\right) \), calculate the value of \( S \).

Answer: $\frac{1}{2}\left(1 - 2^{-\frac{1}{32}}\right)^{-1}$


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>.

A capacitor with a capacitance of $C_{1} = 20 \mu$F is charged to a voltage $U_{1} = 20$ V. A second capacitor with a capacitance of $C_{2} = 5 \mu$F is charged to a voltage $U_{2} = 5$ V. The capacitors are connected with opposite-charged plates. Determine the voltage that will be established across the plates.

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>.

Let \( x \) and \( y \) be non-zero real numbers such that
\[ \frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}. \]

(1) Find the value of \(\frac{y}{x}\).

(2) In triangle \( \triangle ABC \), if \( \tan C = \frac{y}{x} \), find the maximum value of \( \sin 2A + 2 \cos B \).

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