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

Months of the year are usually labeled numerically by '01' for January, '02' for February, and so on, through to '12' for December. Lydia notices that during January, the number of letters in the name of the month is greater than the month's numerical label (i.e., $7>1$). For how many days during 2024 will the date have that property?

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>.

Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([1, 3]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{3+\sqrt{7}}{2}\right)) \ldots) \). If necessary, round your answer to two decimal places.

Answer: $0.18$


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>.

What is the sum and the average of the prime numbers between 20 and 40?

Answer: $30$


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

A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone.

Answer: $64$


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

In $\triangle ABC$, where $A > B > C$, if $2 \cos 2B - 8 \cos B + 5 = 0$, $\tan A + \tan C = 3 + \sqrt{3}$, and the height $CD$ from $C$ to $AB$ is $2\sqrt{3}$, then find the area of $\triangle ABC$.

Answer: $12 - 4\sqrt{3}$


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>.

The number of books issued from the library to readers constitutes $\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\frac{1}{15}$ of the number of books remaining on the shelves. How many books does the library have?

Answer: $544000$


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

A $2018 \times 2018$ square was cut into rectangles with integer side lengths. Some of these rectangles were used to form a $2000 \times 2000$ square, and the remaining rectangles were used to form a rectangle whose length differs from its width by less than 40. Find the perimeter of this rectangle.

Answer: $1076$


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

Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the train 30 seconds to completely pass Car A and 6 seconds to completely pass Car B, after how many more minutes will Car A and Car B meet once Car B has passed the train?

Answer: $1.25$


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>.

Let the width and length of the pan be $w$ and $l$ respectively. If the number of interior pieces is twice the number of perimeter pieces, then find the greatest possible value of $w \cdot l$.

Answer: $294$


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

Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots.

Answer: $\frac{4\sqrt{6}}{9}$


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>.

Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.
\]

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>.

Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B also doubles their speed and turns back. When Person A returns to point $A$, Person B is still 120 meters away from point $A$. What is the distance between points $A$ and $B$ in meters?

Answer: $420$


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

In triangle \(ABC\), side \(BC\) is equal to 5. A circle passes through vertices \(B\) and \(C\) and intersects side \(AC\) at point \(K\), where \(CK = 3\) and \(KA = 1\). It is known that the cosine of angle \(ACB\) is \(\frac{4}{5}\). Find the ratio of the radius of this circle to the radius of the circle inscribed in triangle \(ABK\).

Answer: $\frac{10\sqrt{10} + 25}{9}$


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

How many four-digit numbers, formed using the digits 0, 1, 2, 3, 4, 5 without repetition, are greater than 3410?

Answer: $132$


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

In parallelogram \(ABCD\), \(OE = EF = FD\). The area of the parallelogram is 240 square centimeters. The area of the shaded region is _______ square centimeters.

Answer: $20$


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

Given $x \gt 0$, $y \gt 0$, $x+2y=1$, calculate the minimum value of $\frac{{(x+1)(y+1)}}{{xy}}$.

Answer: $8+4\sqrt{3}$


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>.

Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer.

Answer: $5880$


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

How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros?

Answer: $7935$


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

What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other?

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>.

In the equation, $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\qquad$?

Answer: $31$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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-digit even numbers can be formed using the digits 0, 1, 2, 3?

Answer: $10$


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

A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field.

Answer: $16000$


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

Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get?

Answer: $28077$


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

Let $f(x) = |3\{x\} - 1.5|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation \[nf(xf(x)) = 2x\] has at least $1000$ real solutions.

Answer: $250$


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

Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( BO : OE = 5 : 6 \). Determine the ratio in which point \( E \) divides side \( AC \).

Answer: $21 : 44$


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

In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$?

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 real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\frac{1}{2}$. Determine the smallest possible value of $m$.

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>.

Given that \( p \) is a prime number, the decimal part of \( \sqrt{p} \) is \( x \). The decimal part of \( \frac{1}{x} \) is \( \frac{\sqrt{p} - 31}{75} \). Find all prime numbers \( p \) that satisfy these conditions.

Answer: $2011$


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

In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\).

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>.

Please write an irrational number that is smaller than $3$.

Answer: $\sqrt{2}$


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

The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure?

Answer: $100$


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

Given \( x \in [0, 2\pi] \), determine the maximum value of the function
\[
f(x) = \sqrt{4 \cos^2 x + 4 \sqrt{6} \cos x + 6} + \sqrt{4 \cos^2 x - 8 \sqrt{6} \cos x + 4 \sqrt{2} \sin x + 22}.
\]

Answer: $2(\sqrt{6} + \sqrt{2})$


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

Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool.

Answer: $40$


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

A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle.

Answer: $1799$


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

Let \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \), and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \). Find \( \int_{0}^{\infty} f(x) g(x) \, dx \).

Answer: $\sqrt{e}$


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

A rod with a length of four meters has weights attached as follows: $20 \mathrm{~kg}$ at one end, and at distances of one, two, and three meters from that end, weights of $30, 40, 50$ $\mathrm{kg}$ respectively. Additionally, a weight of $60 \mathrm{~kg}$ is attached at the other end. Where should the rod be supported to remain balanced, assuming the rod's own weight is negligible?

Answer: $2.5$


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

Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misinformed, how many total people did Laura survey?

Answer: $78$


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

As shown in the diagram, in the square \(ABCD\), \(AB = 2\). Draw an arc with center \(C\) and radius equal to \(CD\), and another arc with center \(B\) and radius equal to \(BA\). The two arcs intersect at \(E\). What is the area of the sector \(BAE\)?

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>.

Let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct positive integers such that \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is some integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that satisfy the above conditions, find the maximum value of \( n \cdot d \).

Answer: $530$


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

Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the number of different scoring situations.

Answer: $44$


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

Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey "What is your name?" conducted among 7 Annas, 9 Olgas, 8 Julias, Vitya calculated 50% Julias.

Vitya conducted a survey in his school: what type of triangle has sides of \(3, 4, 5\)? According to his calculations, 5% answered "acute", 5% "obtuse", and 5% "such a triangle does not exist", 50% "right", and the remaining \(a \%\) answered "depends on the geometry". What is \(a\)?

Answer: $110$


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

Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), \( M \) is the center of the face \( A B B_{1} A_{1} \), \( N \) is a point on the edge \( B_{1} C_{1} \), \( L \) is the midpoint of \( A_{1} B_{1} \); \( K \) is the foot of the perpendicular dropped from \( N \) to \( BC_{1} \). In what ratio does point \( N \) divide the edge \( B_{1} C_{1} \) if \( \widehat{L M K} = \widehat{M K N} \)?

Answer: $\sqrt{2} + 1$


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

Given a rectangle \(ABCD\), a circle intersects the side \(AB\) at points \(K\) and \(L\), and the side \(CD\) at points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), and \(DN = 7\).

Answer: $23$


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

Given a point P $(x, y)$ on the circle $x^2 - 4x - 4 + y^2 = 0$, find the maximum value of $x^2 + y^2$.

Answer: $12 + 8\sqrt{2}$


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 the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$

Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$.

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


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

There are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability \(\frac{3}{5}\).

In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining in the tournament, and they are declared the winner.

Determine the expected value of the rank of the winner.

Answer: $103$


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

On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour?

Answer: $23$


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

In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\).

Answer: $10$


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

Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions.

Answer: $288$


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

There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \).

Answer: $52$


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

Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___.

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>.

In a trapezoid, the smaller base is 1 decimeter, and the angles adjacent to it are $135^{\circ}$. The angle between the diagonals, opposite to the base, is $150^{\circ}$. Find the area of the trapezoid.

Answer: $0.5$


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

Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD.

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


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

132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers.

Answer: $513$


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

The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables.

Answer: $79$


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

On the side \( AD \) of the rhombus \( ABCD \), a point \( M \) is taken such that \( MD = 0.3 \, AD \) and \( BM = MC = 11 \). Find the area of triangle \( BCM \).

Answer: $20\sqrt{6}$


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>.

Solve for $X$ if $\sqrt[4]{X^5} = 32\sqrt[16]{32}$.

Answer: $16\sqrt[4]{2}$


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>.

Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.

Billy Bones flipped the gold coin, and it landed heads immediately. Then Billy Bones started flipping the silver coin, and heads came up only on the second flip. Find the probability that the gold coin is the non-symmetric one.

Answer: $0.6$


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>.

Factorize the number \( 989 \cdot 1001 \cdot 1007 + 320 \) into prime factors.

Answer: $991 * 997 * 1009$


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

Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?

Answer: $47/32$


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>.

Mr. Pipkins said, "I was walking along the road at a speed of $3 \frac{1}{2}$ km/h when suddenly a car sped past me, almost knocking me off my feet."

"What was its speed?" his friend asked.

"I can tell you now. From the moment it sped past me until it disappeared around the bend, I took 27 steps. Then I continued walking without stopping and covered another 135 steps until I reached the bend."

"We can easily determine the speed of the car if we assume that both your speeds were constant," his friend replied.

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>.

On the sides \( AB \) and \( AD \) of a square \( ABCD \) with side length 108, semicircles are constructed inward. Find the radius of a circle that touches one side of the square and the semicircles: one externally and the other internally.

Answer: $27$


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

Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?

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


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

Each of $b_1, b_2, \dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of
\[\sum_{1 \le i < j \le 150} b_i b_j.\]

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 sequence of numbers \(a_1, a_2, \cdots, a_n, \cdots\) is defined. Let \(S(a_i)\) be the sum of all the digits of \(a_i\). For example, \(S(22) = 2 + 2 = 4\). If \(a_1 = 2017\), \(a_2 = 22\), and \(a_n = S(a_{n-1}) + S(a_{n-2})\), what is the value of \(a_{2017}\)?

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>.

Triangle \(A B C\) has side lengths \(A B = 65\), \(B C = 33\), and \(A C = 56\). Find the radius of the circle tangent to sides \(A C\) and \(B C\) and to the circumcircle of triangle \(A B C\).

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>.

If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$

Answer: $-\frac{855 \sqrt{2}}{1024}$


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

Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square and $\overline{p q}$ is not divisible by 5, determine the four-digit number $\overline{a b c d}$ and explain the reasoning.

Answer: $1983$


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

Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households?

Answer: $180$


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

Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \( a_1, a_2, \ldots, a_{101} \), where
\[
a_1 < a_2 < a_3 < \cdots < a_{101} .
\]

Answer: $2500$


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

The bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\circ}$. Find the perimeter of the trapezoid.

Answer: $22$


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

Let $s$ be a table composed of positive integers (the table may contain the same number). Among the numbers in $s$ there is the number 68. The arithmetic mean of all numbers in $s$ is 56. However, if 68 is removed, the arithmetic mean of the remaining numbers drops to 55. What is the largest possible number that can appear in $s$?

Answer: $649$


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

$$
\begin{gathered}
\sqrt{2 x^{2}-2024 x+1023131} + \sqrt{3 x^{2}-2025 x+1023132} + \sqrt{4 x^{2}-2026 x+1023133} = \\
= \sqrt{x^{2}-x+1} + \sqrt{2 x^{2}-2 x+2} + \sqrt{3 x^{2}-3 x+3}
\end{gathered}
$$

Answer: $2023$


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

Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$.

Answer: $13213$


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

Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11.

Answer: $98765$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 and product of the distinct prime factors of 420?

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>.

Let  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*

Answer: $164$


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

In a round glass, the axial cross-section of which is the graph of the function \( y = x^4 \), a cherry (a sphere with radius \( r \)) is placed. For which maximum \( r \) will the sphere touch the bottom point of the glass? (In other words, what is the maximum radius \( r \) of the circle lying in the region \( y \geq x^4 \) and containing the origin?)

Answer: $\frac{3 \cdot 2^{1/3}}{4}$


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>.

There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere.

Answer: $6/11$


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>.

Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM : MC = 2 \). Line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1.

Answer: $11/30$


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

Given the function $f(x)=x- \frac {1}{x}+2a\ln x$ $(a\in\mathbb{R})$.

$(1)$ Discuss the monotonicity of $f(x)$;

$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\in[e,+\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$.

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


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\).

Answer: $45$


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

Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\triangle ACE$. Compute the ratio of the area of $\triangle ABJ$ to the area of $\triangle ACE$.

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


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

In the diagram, point \( D \) is on side \( BC \) of \( \triangle ABC \). If \( BD = CD = AD \) and \( \angle ACD = 40^\circ \), what is the measure of \( \angle BAC \)?

Answer: $90$


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

We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?

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


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>.

The real numbers \(a, b, c\) satisfy the following system of equations:

$$
\left\{\begin{array}{l}
\frac{a b}{a+b}=4 \\
\frac{b c}{b+c}=5 \\
\frac{c a}{c+a}=7
\end{array}\right.
$$

Find the value of the expression \(\frac{a b c}{a b + b c + c a}\).

Answer: $280/83$


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

Given \((n+1)^{\alpha+1}-n^{\alpha+1} < n^{\alpha}(\alpha+1) < n^{\alpha+1}-(n-1)^{\alpha+1}, -1 < \alpha < 0\). Let \(x = \sum_{k=4}^{10^{6}} \frac{1}{\sqrt[3]{k}}\), find the integer part of \(x\).

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>.

We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon.

Answer: $2:3$


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

Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.

At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."

At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."

At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."

Find the minimum number of gold coins Huanhuan had at the beginning.

Answer: $70$


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

Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$.

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


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

Given $a, b, c, d \in \mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \cdot 10^{3} + b \cdot 10^{2} + c \cdot 10 + d$.

Answer: $1949$


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

A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth?

Answer: $40$


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

Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$.

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


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

Compute \( t(0)-t\left(\frac{\pi}{5}\right)+t\left(\frac{2\pi}{5}\right)-t\left(\frac{3\pi}{5}\right)+\ldots+t\left(\frac{8\pi}{5}\right)-t\left(\frac{9\pi}{5}\right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + *^2 \cos x + * \).

The coefficients indicated by * are missing. A math student claimed that he could compute the sum without knowing the values of the missing coefficients. Is he right?

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>.

Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?

Answer: $21600$


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

Given $|\overrightarrow {a}|=\sqrt {2}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}-\overrightarrow {b})\bot \overrightarrow {a}$, determine the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.

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


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

How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)?

Answer: $19$


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

Three students solved the same problem. The first one said: "The answer is an irrational number. It represents the area of an equilateral triangle with a side length of 2 meters." The second one said: "The answer is divisible by 4 (without remainder). It represents the radius of a circle whose circumference is 2 meters." The third one said: "The answer is less than 3 and represents the diagonal of a square with a side length of 2 meters." Only one statement from each student is correct. What is the answer to this problem?

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


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>.

There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3\%$. Determine the initial percentage concentration of salt.

Answer: $27$


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

Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \).

Answer: $2006$


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

A bagel is cut into sectors. Ten cuts were made. How many pieces resulted?

Answer: $11$


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

Given a square \( ABCD \). Point \( N \) lies on side \( AD \) such that \( AN : ND = 2 : 3 \), point \( F \) lies on side \( CD \) such that \( DF : FC = 1 : 4 \), and point \( K \) lies on side \( AB \) such that \( AK : KB = 1 : 4 \). Find the angle \( \angle KNF \).

Answer: $135$


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

Let  $A$  be a set of numbers chosen from  $1,2,..., 2015$  with the property that any two distinct numbers, say  $x$  and  $y$ , in  $A$  determine a unique isosceles triangle (which is non equilateral) whose sides are of length  $x$  or  $y$ . What is the largest possible size of  $A$ ?

Answer: $10$


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

Let the function \( f:(0,1) \rightarrow \mathbf{R} \) be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text{ if } x \text{ is irrational; } \\
\frac{p+1}{q}, \text{ if } x=\frac{p}{q}, \text{ where } (p, q)=1 \text{ and } 0<p<q.
\end{array}\right.
$$

Find the maximum value of \( f(x) \) on the interval \( \left(\frac{7}{8}, \frac{8}{9}\right) \).

Answer: $\frac{16}{17}$


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>.

Alice and the White Rabbit left the Rabbit's house together at noon to go to the Duchess's reception. Halfway through, the Rabbit remembered that he forgot his gloves and fan, and ran back home at twice the speed he had been walking with Alice. Grabbing the gloves and fan, he then ran towards the Duchess (at the same speed he had been running home). As a result, Alice (who had been walking at a constant speed the whole time) arrived at the Duchess's on time, while the Rabbit was 10 minutes late. At what time was the reception with the Duchess scheduled?

Answer: $12:40$


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

A triangle  $ABC$  with  $AC=20$  is inscribed in a circle  $\omega$ . A tangent  $t$  to  $\omega$  is drawn through  $B$ . The distance  $t$  from  $A$  is  $25$  and that from  $C$  is  $16$ .If  $S$  denotes the area of the triangle  $ABC$ , find the largest integer not exceeding  $\frac{S}{20}$

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>.

On the lateral side \( C D \) of the trapezoid \( A B C D \) (\( A D \parallel B C \)), a point \( M \) is marked. From the vertex \( A \), a perpendicular \( A H \) is dropped onto the segment \( B M \). It turns out that \( A D = H D \). Find the length of the segment \( A D \), given that \( B C = 16 \), \( C M = 8 \), and \( M D = 9 \).

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>.

Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. Find the length of segment $PT$ which can be expressed in the form $\sqrt{k}-m$ units. What is the integer value of $k+m$?

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>.

Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs "+", "-", "x" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \((4-3) \times (2+5) + 1\). Can he get the number 123?

Forming numbers from multiple others is not allowed (for example, from the numbers 1 and 2, the number 12 cannot be formed).

Answer: $123$


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

In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \).

Answer: $60$


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

Find and describe the pattern by which the sequence of numbers is formed. Determine the next number in this sequence.

$$
112, 224, 448, 8816, 6612
$$

Answer: $224$


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

Given \( f(x) \) is a function defined on \(\mathbf{R}\), for any \( x, y \in \mathbf{R} \), it always holds that
\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\]

Find \( f(x) \) and calculate the value of \( f(\sqrt{2014}) \).

Answer: $-1006$


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

Let  $n \ge 2$  be an integer. Alex writes the numbers  $1, 2, ..., n$  in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment  $s$  we denote by  $d_s$  the difference of the numbers written in its extremities and by  $p_s$  the number of all other drawn segments which intersect  $s$  in its interior.
Find the greatest  $n$  for which Alex can write the numbers on the circle such that  $p_s \le |d_s|$ , for each drawn segment  $s$ .

Answer: $11$


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

The lengths of the sides of a triangle with positive area are $\log_{2}9$, $\log_{2}50$, and $\log_{2}n$, where $n$ is a positive integer. Find the number of possible values for $n$.

Answer: $445$


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

Given a set $\{4,6,8,12,14,18\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process.

Answer: $52$


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

A circle is tangent to sides \( AB \) and \( AD \) of rectangle \( ABCD \) and intersects side \( DC \) at a single point \( F \) and side \( BC \) at a single point \( E \).

Find the area of trapezoid \( AFCB \) if \( AB = 32 \), \( AD = 40 \), and \( BE = 1 \).

Answer: $1180$


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

Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry?

Answer: $1956$


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

A tram ticket is called "lucky in Leningrad style" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called "lucky in Moscow style" if the sum of its digits in even positions is equal to the sum of its digits in odd positions. How many tickets are there that are both lucky in Leningrad style and lucky in Moscow style, including the ticket 000000?

Answer: $6700$


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

There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible?

Answer: $180$


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

For all triples \((x, y, z)\) that satisfy the system

$$
\left\{\begin{array}{l}
2 \cos x = \operatorname{ctg} y \\
2 \sin y = \operatorname{tg} z \\
\cos z = \operatorname{ctg} x
\end{array}\right.
$$

find the minimum value of the expression \(\sin x + \cos z\).

Answer: $-\frac{5 \sqrt{3}}{6}$


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

A fair six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls is equal to the number rolled on the third roll, what is the probability that at least one of the numbers rolled is 2?

Answer: $\frac{8}{15}$


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

In this subtraction problem, \( P, Q, R, S, T \) represent single digits. What is the value of \( P + Q + R + S + T \)?

\[
\begin{array}{rrrrr}
7 & Q & 2 & S & T \\
-P & 3 & R & 9 & 6 \\
\hline
2 & 2 & 2 & 2 & 2
\end{array}
\]

Answer: $29$


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

Let $x_0$ be a zero of the function $f(x) = \sin \pi x$, and it satisfies $|x_{0}| + f(x_{0} + \frac {1}{2}) < 11$. Determine the number of such zeros.

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>.

There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \).

Answer: $12$