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

Emma's telephone number is $548-1983$ and her apartment number contains different digits. The sum of the digits in her four-digit apartment number is the same as the sum of the digits in her phone number. What is the lowest possible value for Emma’s apartment number?

Answer: $9876$


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

In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than  $12$  employees have a more difficult work;
(ii) At least  $30$  employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

Answer: $42$


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

Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$.

Answer: $55$


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

In three sugar bowls, there is an equal number of sugar cubes, and the cups are empty. If each cup receives $\frac{1}{18}$ of the contents of each sugar bowl, then each sugar bowl will have 12 more sugar cubes than each cup.

How many sugar cubes were originally in each sugar bowl?

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 that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a} \perp ( \overrightarrow{a}- \overrightarrow{b})$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

Let \( p(x) \) be a monic quartic polynomial such that \( p(1) = 1, p(2) = 9, p(3) = 28, \) and \( p(4) = 65. \) Find \( p(5) \).

Answer: $126$


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

What is the largest number, with all different digits, whose digits add up to 19?

Answer: $982$


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

Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with points $A$, $A'$ on the smaller circle and $B$, $B'$ on the larger circle. If $PA=AB=5$ and the radius of the larger circle is 3 times the radius of the smaller circle, find the area of the smaller circle.

Answer: $5\pi$


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

A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$?

Answer: $\frac{41}{450}$


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

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is  $60$  centimetres and its base has radius  $30$  centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Answer: $108000$


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

Let $G$ be the centroid of quadrilateral $ABCD$. If $GA^2 + GB^2 + GC^2 + GD^2 = 116$, find the sum $AB^2 + AC^2 + AD^2 + BC^2 + BD^2 + CD^2$.

Answer: $464$


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

The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius  $12$  and circle B has radius  $42,$  find the radius of circle C.

Answer: $147$


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

My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?

Answer: $3/2$


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

What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares?

Answer: $272$


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

A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \leq x \leq 500$.

Answer: $201$


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

Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=4$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-10$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ (denoted as $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$).

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


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

In December 2022, $550$ cm of snow fell in Winterpark, Canada. What was the average snowfall in cm per minute during that month?
A) $\frac{550}{31\times 24 \times 60}$
B) $\frac{550 \times 31}{24 \times 60}$
C) $\frac{550 \times 24 \times 60}{31}$
D) $\frac{31 \times 24 \times 60}{550}$
E) $550 \times 31 \times 24 \times 60$

Answer: $\frac{550}{31\times 24 \times 60}$


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

Let \( m \) be the smallest positive integer that is a multiple of \( 100 \) and has exactly \( 100 \) positive integral divisors, including 1 and itself. Find \( \frac{m}{100} \).

Answer: $4050$


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

A triangle has sides of length  $48$ ,  $55$ , and  $73$ .  A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle.  If  $c$  and  $d$  are relatively prime positive integers such that  $c/d$  is the length of a side of the square, find the value of  $c+d$ .

Answer: $200689$


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

In triangle $XYZ$, $XY=25$ and $XZ=14$. The angle bisector of $\angle X$ intersects $YZ$ at point $E$, and point $N$ is the midpoint of $XE$. Let $Q$ be the point of the intersection of $XZ$ and $YN$. The ratio of $ZQ$ to $QX$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: $39$


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

Compute the number of ordered triples of integers  $(a,b,c)$  between  $1$  and  $12$ , inclusive, such that, if   $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$   then  $q$  is a positive rational number and, when  $q$  is written in lowest terms, the numerator is divisible by  $13$ .

*Proposed by Ankit Bisain*

Answer: $132$


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

Vertex E of equilateral triangle ∆ABE is inside square ABCD. F is the intersection point of diagonal BD and line segment AE. If AB has length √(1+√3), calculate the area of ∆ABF.

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>.

$ABCDEFGH$ is a cube. Find $\cos \angle GAC$.

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


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

Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Assume that each parabola is tangent to a line that forms a $45^\circ$ angle with the x-axis. Find the radius of the circle.

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>.

Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \(\frac{1}{99!}=0. \underbrace{00\ldots00}_{155 \text{ zeros}} 10715 \ldots \) printed on it). Sasha wants to cut a piece from one ribbon that contains \(N\) consecutive digits without any decimal points. For what largest \(N\) can he do this so that Dima cannot determine from this piece which ribbon Sasha spoiled?

Answer: $155$


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

Calculate the distance between the foci of the ellipse defined by the equation
\[\frac{x^2}{36} + \frac{y^2}{9} = 9.\]

Answer: $2\sqrt{3}$


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

Given the function $f(x) = ax^7 + bx - 2$, if $f(2008) = 10$, then the value of $f(-2008)$ is.

Answer: $-12$


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

The perimeter of a rectangle exceeds the perimeter of an equilateral triangle by 1950 cm. The length of each side of the rectangle exceeds the length of each side of the triangle by \( d \ \text{cm} \). All sides of the triangle are equal, and the rectangle is such that its length is triple that of its width. The triangle has a perimeter greater than 0. Determine how many positive integers are NOT possible values for \( d \).

Answer: $650$


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

A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remaining chocolates and filled five columns completely, except for one space. He again ate the chocolates from the incomplete column. In the end, one-third of the original number of chocolates remained in the box. Determine:

a) How many chocolates were there in the entire box originally?

b) How many chocolates did Míša eat before the first rearrangement?

Answer: $25$


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

It is given polygon with  $2013$  sides  $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any  $9$  consecutive vertices is constant and its value is  $300$ . If we know that  $A_{13}$  is marked with  $13$  and  $A_{20}$  is marked with  $20$ , determine with which number is marked  $A_{2013}$

Answer: $67$


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

Let the rational number  $p/q$  be closest to but not equal to  $22/7$  among all rational numbers with denominator  $< 100$ . What is the value of  $p - 3q$ ?

Answer: $14$


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

In a right triangle, instead of having one $90^{\circ}$ angle and two small angles sum to $90^{\circ}$, consider now the acute angles are $x^{\circ}$, $y^{\circ}$, and a smaller angle $z^{\circ}$ where $x$, $y$, and $z$ are all prime numbers, and $x^{\circ} + y^{\circ} + z^{\circ} = 90^{\circ}$. Determine the largest possible value of $y$ if $y < x$ and $y > z$.

Answer: $47$


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

Given Professor Chen has ten different language books, including two Arabic, four German, and four Spanish books, arranged on the shelf so that the Arabic books are together, the Spanish books are together, and two of the German books are treated as indistinguishable, calculate the number of ways to arrange the ten books.

Answer: $576$


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

Let  $k$  be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white.
	A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly  $k$  times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
	Let  $m$  be the minimal  $k$  value such that Vera can guarantee that she wins no matter what Marco does. For  $k=m$ , let  $t$  be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most  $t$  turns. Compute  $100m + t$ .

*Proposed by Ashwin Sah*

Answer: $203$


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

There is a certain regularity in the operation between rational numbers and irrational numbers. For example, if $a$ and $b$ are rational numbers, and $a(\pi +3)+b=0$, then $a=0$, $b=0$. Given that $m$ and $n$ are rational numbers:
$(1)$ If $(m-3)×\sqrt{6}+n-3=0$, then the square root of $mn$ is ______;
$(2)$ If $(2+\sqrt{3})m-(1-\sqrt{3})n=5$, where $m$ and $n$ are square roots of $x$, then the value of $x$ is ______.

Answer: $\frac{25}{9}$


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

Determine the number of six-digit palindromes.

Answer: $9000$


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

The extensions of sides \(AD\) and \(BC\) of a convex quadrilateral \(ABCD\) intersect at point \(M\), and the extensions of sides \(AB\) and \(CD\) intersect at point \(O\). Segment \(MO\) is perpendicular to the angle bisector of \(\angle AOD\). Find the ratio of the areas of triangle \(AOD\) and quadrilateral \(ABCD\), given that \(OA = 12\), \(OD = 8\), and \(CD = 2\).

Answer: $2:1$


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 an ellipse with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\angle PF_1O = 45^\circ$ (where O is the origin).
(1) Determine the values of $a$ and $b$;
(2) Given that line $l_1: y = kx + m_1$ intersects the ellipse at points A and B, and line $l_2: y = kx + m_2$ ($m_1 \neq m_2$) intersects the ellipse at points C and D, and $|AB| = |CD|$:
① Find the value of $m_1 + m_2$;
② Determine the maximum value of the area S of quadrilateral ABCD.

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>.

Let $(a_1,a_2,a_3,\ldots,a_{14})$ be a permutation of $(1,2,3,\ldots,14)$ where $a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14}$. An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14)$. Determine the number of such permutations.

Answer: $1716$


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

On some cells of a 10x10 board, there is a flea. Every minute, the fleas jump simultaneously, each one to a neighboring cell (adjacent by side). Each flea jumps strictly in one of the four directions parallel to the board's sides and maintains this direction as long as possible; otherwise, it changes to the opposite direction. Barbos the dog observed the fleas for an hour and never saw two fleas on the same cell. What is the maximum number of fleas that could have been jumping on the board?

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>.

Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses?

Answer: $\frac{1}{12}$


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>.

Rhombus $PQRS$ has sides of length $4$ and $\angle Q = 150^\circ$. Region $T$ is defined as the area inside the rhombus that is closer to vertex $Q$ than to any of the other vertices $P$, $R$, or $S$. Calculate the area of region $T$.
A) $\frac{2\sqrt{3}}{3}$
B) $\frac{4\sqrt{3}}{3}$
C) $\frac{6\sqrt{3}}{3}$
D) $\frac{8\sqrt{3}}{9}$
E) $\frac{10\sqrt{3}}{3}$

Answer: $\frac{8\sqrt{3}}{9}$


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

Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this angle, rounded to two decimal places if necessary.

Answer: $0.04$


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

Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron where each edge is 1 meter. A bug starts at vertex $A$ and at each vertex chooses randomly among the three incident edges to move along. Compute the probability $p$ that the bug returns to vertex $A$ after exactly 10 meters, where $p = \frac{n}{59049}$.

Answer: $4921$


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

Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]

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


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

In the Cartesian coordinate system, the coordinates of the two foci of an ellipse are $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\sqrt{2}$.

$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose a line with a slope of $-2$ intersects curve $C$ at points $E$ and $F$. Find the equation of the trajectory of the midpoint $N$ of segment $EF$;
$(3)$ Suppose a line passing through point $F_{1}(-2\sqrt{2},0)$ intersects curve $C$ and forms a chord segment $PQ$. Find the maximum area of $\triangle PQO$ ($O$ is the origin).

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


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>.

$-2^{3}+|2-3|-2\times \left(-1\right)^{2023}$.

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>.

The graph of the function $f(x)=\sqrt{3}\cos 2x-\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\sin 2x$ by an unspecified distance.

Answer: $\dfrac{\pi}{6}$


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

In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played by each of the other teams was different. Find the number of matches already played by Team B from city $A$.

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>.

Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number).

Answer: $1080$


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

Suppose in a right triangle LMN, where angle M is the right angle, $\cos N = \frac{4}{5}$ and length LM is given by 20 units. What is the length of LN?

Answer: $25$


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

In a particular year, the price of a commodity increased by $30\%$ in January, decreased by $10\%$ in February, increased by $20\%$ in March, decreased by $y\%$ in April, and finally increased by $15\%$ in May. Given that the price of the commodity at the end of May was the same as it had been at the beginning of January, determine the value of $y$.

Answer: $38$


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

Let $g(x)$ be the function defined on $-2 \leq x \leq 2$ by the formula $$g(x) = 2 - \sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?

Answer: $2.28$


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

Given the function $f(x)=\ln x-xe^{x}+ax$ where $a\in \mathbb{R}$.
(Ⅰ) If the function $f(x)$ is monotonically decreasing on $\left[1,+\infty \right)$, find the range of real number $a$.
(Ⅱ) If $a=1$, find the maximum value of $f(x)$.

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>.

A certain unit decides to invest $3200$ yuan to build a warehouse (in the shape of a rectangular prism) with a constant height. The back wall will be built reusing the old wall at no cost, the front will be made of iron grilles at a cost of $40$ yuan per meter in length, and the two side walls will be built with bricks at a cost of $45$ yuan per meter in length. The top will have a cost of $20$ yuan per square meter. Let the length of the iron grilles be $x$ meters and the length of one brick wall be $y$ meters. Find:
$(1)$ Write down the relationship between $x$ and $y$;
$(2)$ Determine the maximum allowable value of the warehouse area $S$. In order to maximize $S$ without exceeding the budget, how long should the front iron grille be designed?

Answer: $15$


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

Given that $\cos (α-β)= \frac{3}{5}$, $\sin β=- \frac{5}{13}$, and $α∈\left( \left. 0, \frac{π}{2} \right. \right)$, $β∈\left( \left. - \frac{π}{2},0 \right. \right)$, find the value of $\sin α$.

Answer: $\frac{33}{65}$


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>.

Find the largest value of the expression   $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$  , where  $p,R, r$  is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.

Answer: $\frac{5\sqrt{3}}{2}$


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

Given the set $A={3,3^{2},3^{3},…,3^{n}}$ $(n\geqslant 3)$, choose three different numbers from it and arrange them in a certain order to form a geometric sequence. Denote the number of geometric sequences that satisfy this condition as $f(n)$.
(I) Find $f(5)=$ _______ ;
(II) If $f(n)=220$, find $n=$ _______ .

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>.

Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$?

Answer: $\frac{25}{32}$


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

In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(2:1\) from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CQ\) of triangle \(CED\) is equal to \(\frac{\sqrt{23}}{2}\), and \(DE = \frac{\sqrt{23}}{2}\). Find the radius of the circumcircle of triangle \(ABC\).

Answer: $12/5$


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>.

Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction.

Answer: $\frac{1}{80}$


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

The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined.

Answer: $\frac{460}{14}$


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>.

Determine the value of $b$, where $b$ is a positive number, such that the terms $10, b, \frac{10}{9}, \frac{10}{81}$ are the first four terms, respectively, of a geometric sequence.

Answer: $10$


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

When purchasing goods with a total amount of at least 1000 rubles, the store provides a 50% discount on subsequent purchases. With 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 rubles per kg, and sugar at a price of 30 rubles per kg. Realizing that she did not have enough money for the purchase, Dasha still managed to buy everything she planned. How did she do that?

Answer: $1200$


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

Given a square initially painted black, with $\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white.

Answer: $\frac{729}{8192}$


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

Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number.

Answer: $44$


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

Given that $x = \frac{3}{4}$ is a solution to the equation $108x^2 + 61 = 145x - 7,$ what is the other value of $x$ that solves the equation? Express your answer as a common fraction.

Answer: $\frac{68}{81}$


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

Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$?

Answer: $2400$


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

Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with its right focus at point $F$, and a point $P$ on the left branch of the hyperbola. Also given is that $PF$ is tangent to the circle $x^2 + y^2 = a^2$ at point $M$, where $M$ is precisely the midpoint of the line segment $PF$. Find the eccentricity of the hyperbola.

Answer: $\sqrt{5}$


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

Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A.

Answer: $3541.50$


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

There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number?

Answer: $83159$


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

Two \(10 \times 24\) rectangles are inscribed in a circle as shown. Find the shaded area.

Answer: $169\pi - 380$


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

An economist-cryptographer received a cryptogram from the ruler that contained a secret decree to introduce a per-unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that it was not possible to collect a higher amount of tax revenue in this market. Unfortunately, the economist-cryptographer deciphered the cryptogram incorrectly, rearranging the digits in the amount of tax revenue. Based on this incorrect data, a decision was made to introduce a per-unit tax on consumers of 30 monetary units per unit of the product. It is known that the market supply function is \( Q_s = 6P - 312 \), and the market demand is linear. In the situation with no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times the absolute value of the price elasticity of market demand. After the tax was introduced, the consumer price increased to 118 monetary units.

1. Restore the market demand function.
2. Determine the amount of tax revenue collected at the chosen tax rate.
3. Determine the per-unit tax rate that would meet the ruler’s decree.
4. What is the amount of tax revenue specified by the ruler?

Answer: $8640$


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

From the numbers $1, 2, \cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$.

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>.

A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with contents. How many are empty?

Answer: $18054$


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

What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9?

Answer: $45486$


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

Determine the numerical value of $k$ such that
\[\frac{12}{x + z} = \frac{k}{z - y} = \frac{5}{y - x}.\]

Answer: $17$


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

Find the greatest common divisor of $8!$ and $(6!)^2.$

Answer: $5760$


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

A cylindrical glass vessel is heated to a temperature of $t = 1200^{\circ}$C, causing part of the air to be expelled. After covering the vessel with a well-sealing glass plate, it is left to cool to room temperature at $t^{\prime}=16^{\circ}$C. The remaining air inside becomes less dense, and the external air pressure presses the glass plate against the top edge of the vessel. What is the force in kilograms due to this pressure? What is the degree of rarefaction (the ratio of the internal and external air densities)? The cylinder has a radius of $r = 8.00 \text{ cm}$, a height of $h = 24.16 \text{ cm}$, and the prevailing atmospheric pressure is $p = 745.6 \text{ mmHg}$ (with mercury density at room temperature $\sigma = 13.56$). The expansion of the glass vessel is negligible, and the coefficient of air expansion is $\alpha = \frac{1}{273}$.

Answer: $19.6\%$


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

A solid right prism $ABCDEF$ has a height of 20, as illustrated. The bases are equilateral triangles with a side length of 14. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$.

Answer: $7 + 2\sqrt{149}$


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

Given a regular tetrahedron with vertices $A, B, C$, and $D$ and edge length 1, where point $P$ lies on edge $AC$ one third of the way from vertex $A$ to vertex $C$, and point $Q$ lies on edge $BD$ one third of the way from vertex $B$ to vertex $D$, calculate the least possible distance between points $P$ and $Q$.

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


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

A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

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>.

23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85.
Ian sighs, "I am even older than that number."
Determine Ian's age.

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 the general term of the sequence $\{a_n\}$ is $a_n=n^2\left(\cos^2 \frac{n\pi}{3}-\sin^2 \frac{n\pi}{3}\right)$, calculate the value of $S_{30}$.

Answer: $470$


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

Let point $P$ be a moving point on the curve $C\_1$: $(x-2)^2 + y^2 = 4$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Rotate point $P$ counterclockwise by $90^{{∘}}$ around the pole $O$ to obtain point $Q$. Denote the trajectory equation of point $Q$ as curve $C\_2$.

1. Find the polar coordinate equations of curves $C\_1$ and $C\_2$.
2. The ray $\theta = \frac{\pi}{3} (\rho > 0)$ intersects curves $C\_1$ and $C\_2$ at points $A$ and $B$, respectively. Let $M(2,0)$ be a fixed point. Calculate the area of $\triangle MAB$.

Answer: $3 - \sqrt{3}$


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

A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the total surface area of the cone to the lateral surface area of the cone, given that the side length of the pyramid's base is 4 and the angle between the pyramid's height and the plane of its lateral face is $30^{\circ}$.

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


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

Given a sequence $ \{a_n\} $ with the first term $ \dfrac{3}{5} $, and the sequence $ \{a_n\} $ satisfies $ a_{n+1} = 2 - \dfrac{1}{a_n} $, calculate $ a_{2018} $.

Answer: $\dfrac{4031}{4029}$


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

Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three times.

Answer: $\frac{1}{10}$


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>.

\[\log_{10} x + \log_{\sqrt{10}} x + \log_{\sqrt[3]{10}} x + \ldots + \log_{\sqrt[1]{10}} x = 5.5\]

Answer: $\sqrt[10]{10}$


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

Let  $A_1A_2A_3A_4A_5A_6A_7A_8$   be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of  $A_1A_2A_3A_4A_5A_6A_7A_8$  and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest  $n$  satisfying:
We can color n "button" such that for all  $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$  are the same where  $s(i,k)$  denote the number of the  "sub quadrilaterals" has  $A_i,A_k$  be the vertices and the intersection of two its diagonals is "button".

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 $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $(x+1)$, and $y_{2}$ is inversely proportional to $(x+1)$. When $x=0$, $y=-5$; when $x=2$, $y=-7$. Find the value of $x$ when $y=5$.

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


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

In a row with 120 seats, some of the seats are already occupied. If a new person arrives and must sit next to someone regardless of their choice of seat, what is the minimum number of people who were already seated?

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>.

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

A trapezoid  $ABCD$  lies on the  $xy$ -plane. The slopes of lines  $BC$  and  $AD$  are both  $\frac 13$ , and the slope of line  $AB$  is  $-\frac 23$ . Given that  $AB=CD$  and  $BC< AD$ , the absolute value of the slope of line  $CD$  can be expressed as  $\frac mn$ , where  $m,n$  are two relatively prime positive integers. Find  $100m+n$ .

*Proposed by Yannick Yao*

Answer: $1706$


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

Given three coplanar vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$, where $\overrightarrow{a}=(\sqrt{2}, 2)$, $|\overrightarrow{b}|=2\sqrt{3}$, $|\overrightarrow{c}|=2\sqrt{6}$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$.
1. Find $|\overrightarrow{c}-\overrightarrow{a}|$;
2. If $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $3\overrightarrow{a}+2\overrightarrow{b}$, find the value of $\overrightarrow{a}\cdot(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$.

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>.

Let positive integers \( a \) and \( b \) be such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value of the smaller of these two squares.

Answer: $481^2$


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

Let  $n$  be a positive integer and let  $d_{1},d_{2},,\ldots ,d_{k}$  be its divisors, such that  $1=d_{1}<d_{2}<\ldots <d_{k}=n$ . Find all values of  $n$  for which  $k\geq 4$  and  $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ .

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>.

John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green?

Answer: $110$


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

In the interval [1, 6], three different integers are randomly selected. The probability that these three numbers are the side lengths of an obtuse triangle is ___.

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 a regular tetrahedron $P-ABC$, where points $P$, $A$, $B$, and $C$ are all on the surface of a sphere with radius $\sqrt{3}$. If $PA$, $PB$, and $PC$ are mutually perpendicular, calculate the distance from the center of the sphere to the plane $ABC$.

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


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

Please write down an irrational number whose absolute value is less than $3: \_\_\_\_\_\_.$

Answer: $\sqrt{3}$


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

Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an "Audi" and a "Jetta". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determine the number of different seating arrangements possible.

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>.

Let ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ have an eccentricity of $\frac{\sqrt{3}}{2}$. A line with slope $k (k > 0)$ passes through the left focus $F$ and intersects the ellipse $\Gamma$ at points $A$ and $B$. If $\overrightarrow{A F}=3 \overrightarrow{F B}$, find $k$.

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 quadrilateral \(ABCD\) is inscribed in a circle. \(I\) is the incenter of triangle \(ABD\). Find the minimum value of \(BD\) given that \(AI = BC = CD = 2\).

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>.

How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive?

Answer: $6788$


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

A \(3 \times 3 \times 3\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \(3 \times 3 \times 1\) block with the property that the line joining the centers of the two cubes makes a \(45^{\circ}\) angle with the horizontal plane.

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>.

Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer.

Answer: $950$


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

In triangle $\triangle ABC$, it is known that $\overrightarrow{CD}=2\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\overrightarrow{CP}=\frac{1}{2}\overrightarrow{CA}+m\overrightarrow{CB}$. If the area of $\triangle ABC$ is $\sqrt{3}$ and $∠ACB=\frac{π}{3}$, then the minimum value of the length of segment $CP$ is ______.

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>.

Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$.

Answer: $384\pi$


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

Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes.

Answer: $66$


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

Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\left[\frac{1}{15}\right] + \left[\frac{2}{15}\right] + \left[\frac{3}{15}\right] + \cdots + \left[\frac{n-1}{15}\right] + \left[\frac{n}{15}\right] > 2011$, what is the smallest value of $n$?

Answer: $253$


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

A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\circ}$ right turn and sprints upwards for 3 meters. Calculate the average of the shortest distances to each side of the rectangle.

Answer: $5.75$


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

Four fair dice are tossed at random. What is the probability that the four numbers on the dice can be arranged to form an arithmetic progression with a common difference of one?

Answer: $\frac{1}{432}$


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

The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:
   1. The domain of function (2) is from $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ It is graphed as a set of discrete points.
   2. Function (4) is defined by the equation $$y = x^3$$ and is graphed from $$x = -3$$ to $$x = 3$$.
   3. Function (5) is a rational function defined by $$y = \frac{5}{x}$$, excluding the origin from its domain.

Determine the product of the labels of the functions that are invertible.

Answer: $20$


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

Given the function $f(x)=\ln x-mx (m\in R)$.
(I) Discuss the monotonic intervals of the function $f(x)$;
(II) When $m\geqslant \frac{ 3 \sqrt {2}}{2}$, let $g(x)=2f(x)+x^{2}$ and its two extreme points $x\_1$, $x\_2 (x\_1 < x\_2)$ are exactly the zeros of $h(x)=\ln x-cx^{2}-bx$. Find the minimum value of $y=(x\_1-x\_2)h′( \frac {x\_1+x\_2}{2})$.

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


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

We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?

Answer: $47$


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

Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$ to town $Z$, a distance of $25$ km all downhill, and then back to town $X$, a distance of $30$ km on the flat. Benny goes the other way around using the same route. What is the time in minutes that it takes Alvin to complete the $70$-km ride, minus the time in minutes it takes Benny to complete the same ride?

Answer: $33$


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

Two circles touch in  $M$ , and lie inside a rectangle  $ABCD$ . One of them touches the sides  $AB$  and  $AD$ , and the other one touches  $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in  $M$  divides  $AB$  and  $CD$ .

Answer: $1:1$