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

How many distinct four letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements.

Answer: $72$


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

In the quadrilateral  $MARE$  inscribed in a unit circle  $\omega,$   $AM$  is a diameter of  $\omega,$  and  $E$  lies on the angle bisector of  $\angle RAM.$  Given that triangles  $RAM$  and  $REM$  have the same area, find the area of quadrilateral  $MARE.$

Answer: $\frac{8\sqrt{2}}{9}$


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>.

In triangle $PQR$, $PQ = 4$, $PR = 8$, and $\cos \angle P = \frac{1}{10}$. Find the length of angle bisector $\overline{PS}$.

Answer: $4.057$


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

Approximate the reading indicated by the arrow in the diagram of a measuring device.

Answer: $42.3$


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

The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the acute angle $ACB$, in degrees?

Answer: $120$


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

It is known that P and Q are two points on the unit circle centered at the origin O, and they are located in the first and fourth quadrants, respectively. The x-coordinate of point P is $\frac{4}{5}$, and the x-coordinate of point Q is $\frac{5}{13}$. Then, $\cos \angle POQ = \_\_\_\_\_\_$.

Answer: $\frac{56}{65}$


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>.

Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers?

Answer: $17$


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

A circle touches the extensions of two sides \(AB\) and \(AD\) of square \(ABCD\), and the point of tangency cuts off a segment of length \(2 + \sqrt{5 - \sqrt{5}}\) cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square, given that the angle between the tangents is \(72^\circ\), and it is known that \(\sin 36^\circ = \frac{\sqrt{5 - \sqrt{5}}}{2\sqrt{2}}\).

Answer: $\frac{\sqrt{\sqrt{5} - 1} \cdot \sqrt[4]{125}}{5}$


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

In a department store, they received 10 suitcases and 10 keys separately in an envelope. Each key opens only one suitcase, and every suitcase can be matched with a corresponding key.

A worker in the department store, who received the suitcases, sighed:

- So much hassle with matching keys! I know how stubborn inanimate objects can be!! You start matching the key to the first suitcase, and it always turns out that only the tenth key fits. You'll try the keys ten times because of one suitcase, and because of ten - a whole hundred times!

Let’s summarize the essence briefly. A salesperson said that the number of attempts is no more than \(10+9+8+\ldots+2+1=55\), and another employee proposed to reduce the number of attempts since if the key does not fit 9 suitcases, it will definitely fit the tenth one. Thus, the number of attempts is no more than \(9+8+\ldots+1=45\). Moreover, they stated that this will only occur in the most unfortunate scenario - when each time the key matches the last suitcase. It should be expected that in reality the number of attempts will be roughly

\[\frac{1}{2} \times \text{the maximum possible number of attempts} = 22.5.\]

Igor Fedorovich Akulich from Minsk wondered why the expected number of attempts is half the number 45. After all, the last attempt is not needed only if the key does not fit any suitcase except the last one, but in all other cases, the last successful attempt also takes place. Akulich assumed that the statement about 22.5 attempts is unfounded, and in reality, it is a bit different.

**Problem:** Find the expected value of the number of attempts (all attempts to open the suitcases are counted - unsuccessful and successful, in the case where there is no clarity).

Answer: $29.62$


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

The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers?

Answer: $26$


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

What is the smallest four-digit number that is divisible by $35$?

Answer: $1015$


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

Let \(a_{1}, a_{2}, \cdots, a_{10}\) be any 10 distinct positive integers such that \(a_{1} + a_{2} + \cdots + a_{10} = 1995\). Find the minimum value of \(a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{10} a_{1}\).

Answer: $6044$


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

Formulas for shortened multiplication (other).
Common fractions

Answer: $198719871987$


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

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

Answer: $\sqrt{2}$


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

Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $P$.

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


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

20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.

Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from village $B$. The stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter from village $A$ wins at least $k$ of the 400 fights.

It turns out that each village is stronger than its neighboring village in a clockwise direction. Determine the maximum value of $k$ so that this can be the case.

Answer: $290$


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

Distribute five students, namely A, B, C, D, and E, to Peking University, Tsinghua University, and Renmin University of China for recommendation, with the condition that each university gets at least one student and student A cannot be recommended to Peking University. How many different recommendation plans are there? (Answer with a number)

Answer: $100$


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

In the arithmetic sequence $\{a_n\}$, $a_3+a_6+a_9=54$. Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Then, determine the value of $S_{11}$.

Answer: $99$


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

Given that $\triangle ABC$ is an equilateral triangle with side length $s$, determine the value of $s$ when $AP = 2$, $BP = 2\sqrt{3}$, and $CP = 4$.

Answer: $\sqrt{14}$


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>.

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by at least $3$ from all multiples of $p$. Considering the prime numbers $5$, $7$, and $11$, find the number of positive integers less than or equal to $15,000$ which are simultaneously $5$-safe, $7$-safe, and $11$-safe.

Answer: $975$


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

Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer.

Answer: $15015$


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

Given the set of digits {1, 2, 3, 4, 5}, find the number of three-digit numbers that can be formed with the digits 2 and 3, where 2 is positioned before 3.

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>.

It is known that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, the number \( \log_{m} n \) is rational, and additionally,

$$
k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}}
$$

Find the minimum possible value of the sum \( k + 5m + n \).

Answer: $278$


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

Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$.

(I) When $a=3$, find the extreme values of the function $f(x)$.

(II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_1)$.

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


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

Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$.

Answer: $4\sqrt{3}$


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

Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?

Answer: $150$


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

For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$.

Answer: $5120$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, which may not be represented as  ${2^a-2^b\over 2^c-2^d}$ , where  $a,\,b,\,c,\,d$  are positive integers.

Answer: $11$


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

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

Answer: $15.5$


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>.

Determine the length of side $x$ in the following diagram:

[asy]
import olympiad;

draw((0,0)--(2,0)--(0,2*sqrt(2))--cycle);
draw((0,0)--(-2,0)--(0,2*sqrt(2))--cycle);
label("10",(-1, sqrt(2)),NW);
label("$x$",(sqrt(2),sqrt(2)),NE);
draw("$30^{\circ}$",(2.2,0),NW);
draw("$45^{\circ}$",(-1.8,0),NE);

draw(rightanglemark((0,2*sqrt(2)),(0,0),(2,0),4));
[/asy]

Answer: $\frac{20\sqrt{3}}{3}$


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>.

At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per batch. The other details remain the same. How many full recipes should they prepare?

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>.

Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$?

Answer: $128$


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

A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone.

Answer: $94.5$


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

Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. Additionally, the letter $C$ must appear at least once in the word. How many six-letter good words are there?

Answer: $94$


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

Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two tangent lines to circle $M$ are drawn, intersecting the x-axis at points $A$ and $B$.

$(1)$ Find the equation of the parabola $C$;
$(2)$ Find the minimum value of the area of $\triangle PAB$.

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


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

Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$),
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $g(m) = n^2$ or $g(n) = m^2.$

Find the sum of all possible values of $g(18).$

Answer: $104976$


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

4.	Find the biggest positive integer  $n$ , lesser thar  $2012$ , that has the following property:
If  $p$  is a prime divisor of  $n$ , then  $p^2 - 1$  is a divisor of  $n$ .

Answer: $1944$


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

There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and cut with a straight line, not passing through its vertices, into two sheets. After 60 moves, it turned out that all the sheets are triangles or hexagons. How many hexagons are there?

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>.

For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?

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>.

What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 45]=180$?

Answer: $292$


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

Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\]

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>.

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

Answer: $65$


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

Form a five-digit number with no repeated digits using the numbers 0, 1, 2, 3, and 4, where exactly one even number is sandwiched between two odd numbers. The total number of such five-digit numbers is

Answer: $28$


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

Find the modular inverse of \( 31 \), modulo \( 45 \).

Express your answer as an integer from \( 0 \) to \( 44 \), inclusive.

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 vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\cdot (a-b)=3$, find the angle between $a$ and $b$.

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


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

How many four-digit positive integers are multiples of 7?

Answer: $1286$


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

The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$?

Answer: $15$


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

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

Answer: $2880$


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

Calculate the product of all prime numbers between 1 and 20.

Answer: $9699690$


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

Given the set $A={m+2,2m^2+m}$, find the value of $m$ if $3\in A$.

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


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 an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendicular to the x-axis.
(1) Find the equation of ellipse C;
(2) Find the minimum value of the slope of line MN, where line l, tangent to the parabola $y^2=4x$ in the first quadrant, intersects ellipse C at points A and B, intersects the x-axis at point M, and the perpendicular bisector of segment AB intersects the y-axis at point N.

Answer: $- \frac { \sqrt {3}}{12}$


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>.

Consider the set  $E$  of all positive integers  $n$  such that when divided by  $9,10,11$  respectively, the remainders(in that order) are all  $>1$  and form a non constant geometric progression. If  $N$  is the largest element of  $E$ , find the sum of digits of  $E$

Answer: $13$


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

9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost?

Answer: $1.11$


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

In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c= \sqrt {7}$, $C= \frac {\pi}{3}$.
(1) If $2\sin A=3\sin B$, find $a$ and $b$;
(2) If $\cos B= \frac {3 \sqrt {10}}{10}$, find the value of $\sin 2A$.

Answer: $\frac {3-4 \sqrt {3}}{10}$


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>.

During a year when Valentine's Day, February 14, falls on a Tuesday, what day of the week is Cinco de Mayo (May 5) and how many days are between February 14 and May 5 inclusively?

Answer: $81$


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>.

If the point corresponding to the complex number (2-i)z is in the second quadrant of the complex plane, calculate the value of z.

Answer: $-1$


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

Given the binary operations $\clubsuit$ and $\spadesuit$ defined as $\clubsuit$ : $a^{\log_5(b)}$ and $\spadesuit$ : $a^{\frac{1}{\log_5(b)}}$ for real numbers $a$ and $b$ where these expressions are defined, a sequence $(b_n)$ is defined recursively as $b_4 = 4 \spadesuit 2$ and $b_n = (n \spadesuit (n-2)) \clubsuit b_{n-1}$ for all integers $n \geq 5$. Find $\log_5(b_{2023})$ to the nearest integer.

Answer: $11$


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

In the polar coordinate system, the polar coordinate equation of the curve $\Gamma$ is $\rho= \frac {4\cos \theta}{\sin ^{2}\theta}$. Establish a rectangular coordinate system with the pole as the origin, the polar axis as the positive semi-axis of $x$, and the unit length unchanged. The lines $l_{1}$ and $l_{2}$ both pass through the point $F(1,0)$, and $l_{1} \perp l_{2}$. The slope angle of line $l_{1}$ is $\alpha$.
(1) Write the rectangular coordinate equation of the curve $\Gamma$; write the parameter equations of $l_{1}$ and $l_{2}$;
(2) Suppose lines $l_{1}$ and $l_{2}$ intersect curve $\Gamma$ at points $A$, $B$ and $C$, $D$ respectively. The midpoints of segments $AB$ and $CD$ are $M$ and $N$ respectively. Find the minimum value of $|MN|$.

Answer: $4 \sqrt {2}$


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

The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent?

Answer: $10.54$


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

Let  $k$  be the product of every third positive integer from  $2$  to  $2006$ , that is  $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$ . Find the number of zeros there are at the right end of the decimal representation for  $k$ .

Answer: $168$


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

Xiao Wang left home at 8:30 to visit a housing exhibition and returned home at 12:00, while his alarm clock at home was showing 11:46 when he got back. Calculate the time in minutes until the alarm clock will point to 12:00 exactly.

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 vectors $\overrightarrow{a}=m \overrightarrow{i}+5 \overrightarrow{j}- \overrightarrow{k}, \overrightarrow{b}=3 \overrightarrow{i}+ \overrightarrow{j}+r \overrightarrow{k}$, if $\overrightarrow{a}//\overrightarrow{b}$, find the values of real numbers $m=$ \_\_\_\_\_\_ and $r=$ \_\_\_\_\_\_.

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


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

Compute the product of the sums of the squares and the cubes of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 1 = 0,\] given that all roots are real and nonnegative.

Answer: $13754$


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

Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$,
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$,
(iv) $g(2) = 3$.

Find the sum of all possible values of $g(18)$.

Answer: $108$


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

Find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that
\[ f(x)f(-x) = f(x^3). \]

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>.

$ABCDEFGH$ is a cube. Find $\sin \angle BAE$, where $E$ is the top vertex directly above $A$.

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


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

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

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>.

Tom adds up all the even integers from 2 to 600, inclusive. Lara adds up all the integers from 1 to 200, inclusive. What is Tom's sum divided by Lara's sum?

Answer: $4.5$


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

Let  $f:\mathbb{N} \longrightarrow \mathbb{N}$  be such that for every positive integer  $n$ , followings are satisfied.
i.  $f(n+1) > f(n)$ ii.  $f(f(n)) = 2n+2$ Find the value of  $f(2013)$ .
(Here,  $\mathbb{N}$  is the set of all positive integers.)

Answer: $4026$


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

How many positive integer divisors of $1800^{1800}$ are divisible by exactly 180 positive integers?

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>.

Given that the coordinates of a direction vector of line $l$ are $({-1,\sqrt{3}})$, the inclination angle of line $l$ is ____.

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 triangle $XYZ$ with right angle at $Z$, $\angle XYZ < 45^\circ$ and $XY = 6$. A point $Q$ on $\overline{XY}$ is chosen such that $\angle YQZ = 3\angle XQZ$ and $QZ = 2$. Determine the ratio $\frac{XQ}{YQ}$ in simplest form.

Answer: $\frac{7 + 3\sqrt{5}}{2}$


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

Given integers $x$ and $y$ satisfy the equation $2xy + x + y = 83$, find the values of $x + y$.

Answer: $-85$


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

First, factorize 42 and 30 into prime factors, then answer the following questions:
(1) 42=    , 30=    .
(2) The common prime factors of 42 and 30 are     .
(3) The unique prime factors of 42 and 30 are     .
(4) The greatest common divisor (GCD) of 42 and 30 is     .
(5) The least common multiple (LCM) of 42 and 30 is     .
(6) From the answers above, you can conclude that     .

Answer: $210$


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

Given the function $f\left(x\right)=\cos x+\left(x+1\right)\sin x+1$ on the interval $\left[0,2\pi \right]$, find the minimum and maximum values of $f(x)$.

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


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

Find the sum of the distinct prime factors of $7^7 - 7^4$.

Answer: $24$


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

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

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


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

The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list?

Answer: $720$


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

In trapezoid  $ABCD$ ,  $AD \parallel BC$  and  $\angle ABC + \angle CDA = 270^{\circ}$ .  Compute  $AB^2$  given that  $AB \cdot \tan(\angle BCD) = 20$  and  $CD = 13$ .

*Proposed by Lewis Chen*

Answer: $260$


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

Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is     .

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>.

A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square?

Answer: $\frac{3\pi}{32}$


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>.

There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points)
(Ⅰ) If the 3 girls must stand together, how many different arrangements are there?
(Ⅱ) If no two girls are next to each other, how many different arrangements are there?
(Ⅲ) If there are exactly three people between person A and person B, how many different arrangements are there?

Answer: $720$


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

The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?

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


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>.

Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\]

Answer: $7\sqrt{2}.$


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 numbers \( p_1, p_2, p_3, q_1, q_2, q_3, r_1, r_2, r_3 \) are equal to the numbers \( 1, 2, 3, \dots, 9 \) in some order. Find the smallest possible value of
\[
P = p_1 p_2 p_3 + q_1 q_2 q_3 + r_1 r_2 r_3.
\]

Answer: $214$


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

Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.

(I) Find the maximum and minimum values of $|MQ|$;

(II) Find the maximum and minimum values of $\frac{n-3}{m+2}$.

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>.

Problem 4. Angel has a warehouse, which initially contains  $100$  piles of  $100$  pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.

However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.

What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse?

Answer: $199$


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

How many positive integers less than $1000$ are either a perfect cube or a perfect square?

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>.

A sphere is cut into three equal wedges. The circumference of the sphere is $18\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\pi$.

Answer: $162\pi$


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 large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\%$ of the area of the large square region. Find the ratio $\frac{w}{t}$ in this scenario.
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{2}{9}$
D) $\frac{1}{10}$
E) $\frac{1}{11}$

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


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

a) Find all the divisors of the number 572 based on its prime factorization.

b) How many divisors does the number \(572 a^{3} b c\) have if:

   I. \(a, b\), and \(c\) are prime numbers greater than 20 and different from each other?

   II. \(a = 31\), \(b = 32\), and \(c = 33\)?

Answer: $384$


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

In a rectangular grid where grid lines are spaced $1$ unit apart, the acronym XYZ is depicted below. The X is formed by two diagonal lines crossing, the Y is represented with a 'V' shape starting from a bottom point going up to join two endpoints with horizontal lines, the Z is drawn with a top horizontal line, a diagonal from top right to bottom left and a bottom horizontal line. Calculate the sum of lengths of the line segments that form the acronym XYZ.
A) $6 + 3\sqrt{2}$
B) $4 + 5\sqrt{2}$
C) $3 + 6\sqrt{2}$
D) $5 + 4\sqrt{2}$

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


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

In the diagram, $JKLM$ and $NOPM$ are squares each of area 25. If $Q$ is the midpoint of both $KL$ and $NO$, find the total area of polygon $JMQPON$.

[asy]
unitsize(3 cm);

pair J, K, L, M, N, O, P, Q;

O = (0,0);
P = (1,0);
M = (1,1);
N = (0,1);
Q = (N + O)/2;
J = reflect(M,Q)*(P);
K = reflect(M,Q)*(O);
L = reflect(M,Q)*(N);

draw(J--K--L--M--cycle);
draw(M--N--O--P--cycle);

label("$J$", J, N);
label("$K$", K, W);
label("$L$", L, S);
label("$M$", M, NE);
label("$N$", N, NW);
label("$O$", O, SW);
label("$P$", P, SE);
label("$Q$", Q, SW);
[/asy]

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>.

The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \sin t$, where $t \in [0, 12]$. The relationship between the storage capacity of reservoir B (in hundred tons) and time $t$ (in hours) is: $g(t) = 5 - |t - 6|$, where $t \in [0, 12]$. The question is: When do the combined storage capacities of reservoirs A and B reach their maximum value? And what is this maximum value?
(Reference data: $\sin 6 \approx -0.279$).

Answer: $6.721$


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

In trapezoid \( KLMN \), diagonal \( KM \) is equal to 1 and is also its height. From points \( K \) and \( M \), perpendiculars \( KP \) and \( MQ \) are drawn to sides \( MN \) and \( KL \), respectively. Find \( LM \) if \( KN = MQ \) and \( LM = MP \).

Answer: $\sqrt{2}$


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

In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees?

Answer: $146$


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

Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn.

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>.

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

Answer: $\sqrt{3}$


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

A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers.

Answer: $432$


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>.

Calculate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 + n - 1}{(n+3)!}
\]

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>.

How many irreducible fractions with numerator 2015 exist that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)?

Answer: $1440$


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

A student research group at a school found that the attention index of students during class changes with the listening time. At the beginning of the lecture, students' interest surges; then, their interest remains in a relatively ideal state for a while, after which students' attention begins to disperse. Let $f(x)$ represent the student attention index, which changes with time $x$ (minutes) (the larger $f(x)$, the more concentrated the students' attention). The group discovered the following rule for $f(x)$ as time $x$ changes:
$$f(x)= \begin{cases} 100a^{ \frac {x}{10}}-60, & (0\leqslant x\leqslant 10) \\ 340, & (10 < x\leqslant 20) \\ 640-15x, & (20 < x\leqslant 40)\end{cases}$$
where $a > 0, a\neq 1$.
If the attention index at the 5th minute after class starts is 140, answer the following questions:
(Ⅰ) Find the value of $a$;
(Ⅱ) Compare the concentration of attention at the 5th minute after class starts and 5 minutes before class ends, and explain the reason.
(Ⅲ) During a class, how long can the student's attention index remain at least 140?

Answer: $\dfrac {85}{3}$


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

Let  $ABC$  be a triangle in which  $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio  $\frac{AB}{BC}$ .

Answer: $3/4$


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

Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes.

Answer: $34$


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

Let  $f_1(x)=x^2-1$ , and for each positive integer  $n \geq 2$  define  $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial  $f_{2004}$  have?

Answer: $2005$


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

22. Let the function  $f : \mathbb{Z} \to \mathbb{Z}$  take only integer inputs and have integer outputs. For any integers  $x$  and  $y$ , f satisfies  $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$  If  $f(2016) = 6102$  and  $f(6102) = 2016$ , what is  $f(1)$ ?

23. Let  $d$  be a randomly chosen divisor of  $2016$ . Find the expected value of  $$ \frac{d^2}{d^2 + 2016} $$  24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of  $N$  colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of  $N$ ?

Answer: $8117$


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

For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$?

Answer: $121$


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

Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number.

Answer: $21$


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

From 1000 to 9999, a four-digit number is randomly chosen. The probability that all the digits in this number are different and the units digit is an odd number is ( ).

Answer: $\frac{56}{225}$


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

Find the positive solution to
\[\sqrt{x + 2 + \sqrt{x + 2 + \dotsb}} = \sqrt{x \sqrt{x \dotsm}}.\]

Answer: $1 + \sqrt{3}$


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

A factory produced an original calculator that performs two operations:

(a) the usual addition, denoted by \( + \)

(b) an operation denoted by \( \circledast \).

We know that, for any natural number \( a \), the following hold:

\[
(i) \quad a \circledast a = a \quad \text{ and } \quad (ii) \quad a \circledast 0 = 2a
\]

and, for any four natural numbers \( a, b, c, \) and \( d \), the following holds:

\[
(iii) \quad (a \circledast b) + (c \circledast d) = (a+c) \circledast(b+d)
\]

What are the results of the operations \( (2+3) \circledast (0+3) \) and \( 1024 \circledast 48 \)?

Answer: $2000$


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

Suppose that $x$ is an integer that satisfies the following congruences:
\[
4 + x \equiv 3^2 \pmod{2^3}, \\
6 + x \equiv 2^3 \pmod{3^3}, \\
8 + x \equiv 7^2 \pmod{5^3}.
\]
What is the remainder when $x$ is divided by $30$?

Answer: $17$


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

Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\).

Answer: $\frac{9\pi}{16}$


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>.

Side  $AB$  of triangle  $ABC$  was divided into  $n$  equal parts (dividing points  $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side  $AC$  of this triangle was divided into  $(n + 1)$  equal parts (dividing points  $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangles  $C_iB_iC_{i+1}$  (where  $i = 1,2, ..., n$ ). What part of the area of the triangle is painted over?

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


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

The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes.

Answer: $428$


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

Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$.

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


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

For any 2016 complex numbers \( z_1, z_2, \ldots, z_{2016} \), it holds that
\[
\sum_{k=1}^{2016} |z_k|^2 \geq \lambda \min_{1 \leq k \leq 2016} \{ |z_{k+1} - z_k|^2 \},
\]
where \( z_{2017} = z_1 \). Find the maximum value of \( \lambda \).

Answer: $504$


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

Estimate the population of the island of Thalassa in the year 2050, knowing that its population doubles every 20 years and increases by an additional 500 people every decade thereafter, given that the population in the year 2000 was 250.

Answer: $1500$


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

$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards.
A) 600
B) 720
C) 840
D) 960
E) 1080

Answer: $720$


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

Given the sequence $\{a\_n\}$, where $a\_n= \sqrt {5n-1}$, $n\in\mathbb{N}^*$, arrange the integer terms of the sequence $\{a\_n\}$ in their original order to form a new sequence $\{b\_n\}$. Find the value of $b_{2015}$.

Answer: $5037$


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

Given real numbers $x$ and $y$ satisfy the equation $x^2+y^2-4x+1=0$.
(1) Find the maximum and minimum value of $\frac {y}{x}$.
(2) Find the maximum and minimum value of $y-x$.
(3) Find the maximum and minimum value of $x^2+y^2$.

Answer: $7-4\sqrt{3}$


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

Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty?

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>.

For what smallest natural $k$ is the number \( 2016 \cdot 20162016 \cdot 201620162016 \cdot \ldots \cdot 20162016\ldots2016 \) (with $k$ factors) divisible by \(3^{67}\)?

Answer: $34$


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

Given that $x$ and $y$ satisfy the equation $(x-1)^{2}+(y+2)^{2}=4$, find the maximum and minimum values of $S=3x-y$.

Answer: $5 - 2\sqrt{10}$