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

Find the smallest natural number \( n \) for which \( (n+1)(n+2)(n+3)(n+4) \) is divisible by 1000.

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>.

In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\).

Answer: $25\pi$


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>.

The area of a right triangle is 1, and its hypotenuse is \(\sqrt{5}\). Find the cosine of the acute angle between the medians of the triangle that are drawn to its legs.

Answer: $\frac{5\sqrt{34}}{34}$


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>.

A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors?

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>.

Given the function
$$
f(x) = \left|8x^3 - 12x - a\right| + a
$$

The maximum value of this function on the interval \([0, 1]\) is 0. Find the maximum value of the real number \(a\).

Answer: $-2\sqrt{2}$


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

In $\triangle ABC$, where $AB = BC > AC$, let $AH$ and $AM$ be the altitude and median to side $BC$, respectively. Given $\frac{S_{\triangle AMH}}{S_{\triangle ABC}} = \frac{3}{8}$, determine the value of $\cos \angle BAC$.

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>.

A chocolate bar weighed 250 g and cost 50 rubles. Recently, for cost-saving purposes, the manufacturer reduced the weight of the bar to 200 g and increased its price to 52 rubles. By what percentage did the manufacturer's income increase?

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>.

The angle $A$ at the vertex of the isosceles triangle $ABC$ is $100^{\circ}$. On the ray $AB$, a segment $AM$ is laid off, equal to the base $BC$. Find the measure of the angle $BCM$.

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>.

For all real numbers \( r, s, t \) satisfying \( 1 \leq r \leq s \leq t \leq 4 \), find the minimum value of \( (r-1)^{2}+\left(\frac{s}{r}-1\right)^{2} +\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2} \).

Answer: $4(\sqrt{2} - 1)^2$


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

A cube with a side of 10 is divided into 1000 smaller cubes each with an edge of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (along any of the three directions) equals 0. In one of the cubes (denoted as A), the number 1 is written. There are three layers passing through cube A, and these layers are parallel to the faces of the cube (each layer has a thickness of 1). Find the sum of all the numbers in the cubes that do not lie in these layers.

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>.

For which natural number \( K \) does the expression \(\frac{n^{2}}{1.001^{n}}\) reach its maximum value?

Answer: $2001$


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

As a result of measuring the four sides and one of the diagonals of a certain quadrilateral, the following numbers were obtained: $1 ; 2 ; 2.8 ; 5 ; 7.5$. What is the length of the measured diagonal?

Answer: $2.8$


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

Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = \frac{1}{2}$ and $a_{n+1} = a_{n} + \frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$.

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>.

Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2, so he does not use it. Additionally, he dislikes when two identical digits stand next to each other. Furthermore, he wants the first digit to match the last. How many possible combinations must be tried to guarantee guessing Vasya's password?

Answer: $504$


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

Unlucky Emelya was given several metal balls. He broke the 3 largest ones (their mass was 35% of the total mass of all the balls), then lost the 3 smallest ones, and brought home the remaining balls (their mass was \( \frac{8}{13} \) of the unbroken ones). How many balls was Emelya given?

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>.

At an old estate, the house is surrounded in a circle by tall trees: spruces, pines, and birches. There are a total of 96 trees. These trees have a strange property: from any coniferous tree, if you take two trees skipping one tree in between, one of them is coniferous and the other is deciduous; and from any coniferous tree, if you take two trees skipping two trees in between, one of them is coniferous and the other is deciduous. How many birches are planted around the house?

Answer: $32$


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

A workshop produces items of types $A$ and $B$. One item of type $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while an item of type $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling an item of type $A$ is 80 thousand rubles, and for type $B$ it is 100 thousand rubles. The daily supply of steel is 700 kg, and non-ferrous metals is 642 kg. How many items of types $A$ and $B$ should be produced per shift to maximize profit from sales, given that resource consumption should not exceed the allocated supplies for the shift? State the maximum profit (in thousand rubles) that can be obtained under these conditions as a single number without specifying the unit.

Answer: $2180$


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

How many integer solutions does the inequality

$$
|x| + |y| < 1000
$$

have, where \( x \) and \( y \) are integers?

Answer: $1998001$


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

For a positive integer \( n \), consider the function

\[
f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}
\]

Calculate the value of

\[
f(1)+f(2)+f(3)+\cdots+f(40)
\]

Answer: $364$


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

Let \( N \) be the smallest positive integer such that \( \frac{N}{15} \) is a perfect square, \( \frac{N}{10} \) is a perfect cube, and \( \frac{N}{6} \) is a perfect fifth power. Find the number of positive divisors of \( \frac{N}{30} \).

Answer: $8400$


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

There are 20 chairs arranged in a circle. There are \(n\) people sitting in \(n\) different chairs. These \(n\) people stand, move \(k\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \((n, k)\) with \(1 \leq n \leq 20\) and \(1 \leq k \leq 20\) is this possible?

Answer: $72$


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

Suppose \( x_{1}, x_{2}, \ldots, x_{49} \) are real numbers such that

\[ x_{1}^{2} + 2 x_{2}^{2} + \cdots + 49 x_{49}^{2} = 1. \]

Find the maximum value of \( x_{1} + 2 x_{2} + \cdots + 49 x_{49} \).

Answer: $35$


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

The base of the pyramid \(SABC\) is an isosceles triangle \(ABC\) with \(AB = BC = 3\) and \(AC = 5\). The height of the pyramid \(SABC\) is the segment \(SO\), where \(O\) is the intersection point of the line passing through vertex \(B\) parallel to side \(AC\), and the line passing through \(C\) perpendicular to side \(AC\). Find the distance from the center of the inscribed circle of triangle \(ABC\) to the plane containing the lateral face \(BSC\), given that the height of the pyramid is \(7/12\).

Answer: $\frac{35 \sqrt{11}}{396}$


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

In a triangle, the larger angle at the base is $45^{\circ}$, and the altitude divides the base into segments of 20 and 21. Find the length of the larger lateral side.

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>.

In trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the diagonals intersect at point \(E\). The areas of \(\triangle ADE\) and \(\triangle BCE\) are given as 12 and 3, respectively. Find the area of the trapezoid.

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>.

Let \( a, b, c \) be positive integers such that \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} \) and \( \operatorname{gcd}(a, b, c) = 1 \). Suppose \( a + b \leq 2011 \). Determine the largest possible value of \( a + b \).

Answer: $1936$


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

Since the beginning of the school year, Andrey has been recording his math grades. Upon receiving a new grade (2, 3, 4, or 5), he called it unexpected if, up to that point, it had occurred less frequently than each of the other possible grades. (For example, if he received the grades 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 in order, the first 5 and the second 4 would be unexpected.) Over the entire school year, Andrey received 40 grades - exactly 10 fives, 10 fours, 10 threes, and 10 twos (the order is unknown). Can you definitively determine the number of grades that were unexpected for him?

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's define a number as complex if it has at least two different prime divisors. Find the greatest natural number that cannot be represented as the sum of two complex numbers.

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>.

On a rectangular sheet of paper, a picture in the shape of a "cross" made of two rectangles $ABCD$ and $EFGH$ was drawn, with sides parallel to the edges of the sheet. It is known that $AB = 9$, $BC = 5$, $EF = 3$, $FG = 10$. Find the area of the quadrilateral $AFCH$.

Answer: $52.5$


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

In a plane with a Cartesian coordinate system, there are 16 grid points \((i, j)\), where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, there will always exist 4 points among the \(n\) points that are the vertices of a square. Find the minimum value of \(n\).

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>.

All two-digit numbers divisible by 5, where the number of tens is greater than the number of units, were written on the board. There were \( A \) such numbers. Then, all two-digit numbers divisible by 5, where the number of tens is less than the number of units, were written on the board. There were \( B \) such numbers. What is the value of \( 100B + A \)?

Answer: $413$


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

Find the maximum value of the expression \((\sqrt{8-4 \sqrt{3}} \sin x - 3 \sqrt{2(1+\cos 2x)} - 2) \cdot (3 + 2 \sqrt{11 - \sqrt{3}} \cos y - \cos 2y)\). If the answer is a non-integer, round it to the nearest whole number.

Answer: $33$


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

Find the flux of the vector field

$$
\vec{a}=-x \vec{i}+2 y \vec{j}+z \vec{k}
$$

through the portion of the plane

$$
x+2 y+3 z=1
$$

located in the first octant (the normal forms an acute angle with the $OZ$ axis).

Answer: $\frac{1}{18}$


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>.

A natural number \( n \) is called "cubowat" if \( n^{3} + 13n - 273 \) is a cube of a natural number. Find the sum of all cubowat numbers.

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>.

Find a four-digit number that is a perfect square if its first two digits are the same as its last two digits.

Answer: $7744$


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

In a certain group of men, 70% have brown eyes, 70% have dark hair, 85% are taller than 5 feet 8 inches, and 90% weigh more than 140 pounds. What percentage of men definitely possess all four of these characteristics?

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>.

There are 100 points on a coordinate plane. Let \( N \) be the number of triplets \((A, B, C)\) that satisfy the following conditions: the vertices are chosen from these 100 points, \( A \) and \( B \) have the same y-coordinate, and \( B \) and \( C \) have the same x-coordinate. Find the maximum value of \( N \).

Answer: $8100$


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

One hundred number cards are laid out in a row in ascending order: \(00, 01, 02, 03, \ldots, 99\). Then, the cards are rearranged so that each subsequent card is obtained from the previous one by increasing or decreasing exactly one of the digits by 1 (for example, after 29 there can be 19, 39 or 28, but not 30 or 20). What is the maximum number of cards that could remain in their original positions?

Answer: $50$


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

Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)?

Answer: $189/65$


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

The quadratic equations \(x^{2} + px + q\) and \(x^{2} + ax + b\) each have one root. Among the numbers \(p, q, a, b\) there are 16, 64, and 1024. What can the fourth number be?

If there are multiple possible answers, input the larger one into the system, and specify all of them in the written solution.

Answer: $262144$


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

The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it?

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


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>.

Private Petrov took a bucket of unpeeled potatoes and peeled them in 1 hour. During this process, 25% of the potatoes went to peels. How long did it take him to collect half a bucket of peeled potatoes?

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>.

Compute the definite integral:

$$
\int_{-\pi}^{0} 2^{8} \sin ^{6} x \cos ^{2} x \, dx
$$

Answer: $10\pi$


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>.

The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors?

Answer: $247$


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

The numbers \(a, b, c, d\) belong to the interval \([-5.5, 5.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).

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>.

How many distinct right triangles exist with one leg equal to \( \sqrt{2016} \), and the other leg and hypotenuse expressed as natural numbers?

Answer: $12$


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

Find a number such that when it is multiplied by its reverse, the product is 78445.

Answer: $145$


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

Distribute 61 books to a class of students. If at least one person must receive at least 3 books, what is the maximum number of students in the class?

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>.

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.

Answer: $5/11$


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>.

By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number).

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>.

At each vertex of a cube with an edge length of 1, there is the center of a sphere. All the spheres are identical, and each touches three neighboring spheres. Find the length of the part of the space diagonal of the cube that lies outside the spheres.

Answer: $\sqrt{3} - 1$


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>.

For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field.

Answer: $0.667$


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

The diagonals of a trapezoid are mutually perpendicular, and one of them is equal to 17. Find the area of the trapezoid if its height is 15.

Answer: $4335/16$


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

A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles?

Answer: $503$


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

Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer.

Answer: $-2002$


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

Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( S \).

Answer: $2010$


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

The curve given by the equation \( y = 2^p x^2 + 5px - 2^{p^2} \) intersects the \( Ox \) axis at points \( A \) and \( B \), and the \( Oy \) axis at point \( C \). Find the sum of all values of the parameter \( p \) for which the center of the circle circumscribed around triangle \( ABC \) lies on the \( Ox \) axis.

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>.

On the board, two sums are written:

\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]

\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]

Determine which one is greater (or if they are equal).

Answer: $1097393685$


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

Let \( a \) and \( b \) be the roots of \( x^2 + 2000x + 1 = 0 \) and let \( c \) and \( d \) be the roots of \( x^2 - 2008x + 1 = 0 \). Find the value of \( (a+c)(b+c)(a-d)(b-d) \).

Answer: $32064$


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

Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \) is \( 3\sqrt{10} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( 3\sqrt{14} \). Find the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) \).

Answer: $3\sqrt{2}$


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

A team of fishermen planned to catch 1800 centners of fish within a certain timeframe. During one-third of this period, there was a storm, causing them to fall short of their daily target by 20 centners each day. However, on the remaining days, the team managed to catch 20 centners more than the daily norm and completed the planned goal one day ahead of schedule. How many centners of fish were they planning to catch daily?

Answer: $100$


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

In how many ways can two rooks be arranged on a chessboard such that one cannot capture the other? (A rook can capture another if it is on the same row or column of the chessboard).

Answer: $3136$


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

In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees.

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 that the side lengths of triangle \( \triangle ABC \) are 6, \( x \), and \( 2x \), find the maximum value of its area \( S \).

Answer: $12$


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

Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?

Answer: $12$


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

Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned to project $A$, and teams B and C work together on project $B$. After a few days, team C joins team A to complete project $A$. How many days did team C and team B work together?

Answer: $15$


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

Given that the positive real numbers \( u, v, \) and \( w \) are all not equal to 1, if \(\log _{u} (v w)+\log _{v} w=5\) and \(\log _{v} u+\log _{w} v=3\), then find the value of \(\log _{w} u\).

Answer: $4/5$


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

A triangular pyramid \( S-ABC \) has a base that is an equilateral triangle with a side length of 4. It is given that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \).

Answer: $\frac{268\pi}{11}$


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>.

Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:

- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.

And Dan remarks to Jack:

- If I give you 14 sheep for one horse, your herd will have three times as many heads as mine.

Jim, in turn, says to Dan:

- And if I give you 4 cows for one horse, your herd will become 6 times larger than mine.

The deals did not take place, but can you still say how many heads of livestock were in the three herds?

Answer: $39$


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

Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \( 100 \, \mathrm{ft/s} \). At the same time, Edward started running away with an acceleration of \( 1 \, \mathrm{ft/s^2} \). Assuming that air resistance causes the arrow to decelerate at \( 1 \, \mathrm{ft/s^2} \) and that it does hit Edward, how fast was the arrow moving at the moment of impact (in \( \mathrm{ft/s} \))?

Answer: $75$


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

Let \( a, b, c, d, e \) be positive integers whose sum is 2018. Let \( M = \max (a+b, b+c, c+d, d+e) \). Find the smallest possible value of \( M \).

Answer: $673$


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

A stalker throws a small nut from the Earth's surface at an angle of $\alpha=30^{\circ}$ to the horizontal with an initial speed $v_{0}=10 \, \mathrm{m}/\mathrm{s}$. The normal acceleration due to gravity is $g=10 \, \mathrm{m}/\mathrm{s}^{2}$. At the highest point of its trajectory, the nut enters a zone of gravitational anomaly (an area where the acceleration due to gravity changes its magnitude sharply), and continues to move within it. As a result, the nut falls to the ground at a distance $S=3 \sqrt{3} \, \mathrm{m}$ from the stalker. Determine the acceleration due to gravity inside the anomaly. (15 points)

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>.

Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]

Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \).

Answer: $13$


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

We traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 hours, and the driver said that if the breakdown had occurred 50 miles further, the train would have arrived 40 minutes earlier.

What is the distance from Anglchester to Klinkerton?

Answer: $200$


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

In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe winning the majority of sections wins the district; finally, the giraffe winning the majority of districts is declared the contest winner. The giraffe named Tall won. What is the minimum number of voters who could have voted for him?

Answer: $24$


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

In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.

Answer: $222$


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

On side \(BC\) and on the extension of side \(AB\) through vertex \(B\) of triangle \(ABC\), points \(M\) and \(K\) are located, respectively, such that \(BM: MC = 4: 5\) and \(BK: AB = 1: 5\). Line \(KM\) intersects side \(AC\) at point \(N\). Find the ratio \(CN: AN\).

Answer: $5/24$


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>.

The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$.

Answer: $12877$


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

Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$.

Answer: $2\sqrt{2}$


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

The price of an item is an integer number of yuan. With 100 yuan, you can buy up to 3 items. Person A and Person B each have a certain number of 100-yuan bills. The amount of money Person A has can buy at most 7 items, and the amount of money Person B has can buy at most 14 items. Together, they can buy 1 more item than the sum of what each can buy individually. What is the price of each item in yuan?

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>.

Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory "ViP." Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part of the shares. They agreed, and after the deal, each owned a third of the company's shares. Anastasia Alekseevna paid 1,100,000 rubles for her share. Determine who of the entrepreneurs is entitled to a larger portion of this money. In the answer, write the amount he will receive.

Answer: $1000000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \( \frac{1}{2} \).

Answer: $\sqrt{\frac{\sqrt{17}-1}{8}}$


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

It is known that when 2008 is divided by certain natural numbers, the remainder is always 10. How many such natural numbers are there?

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>.

How many parallelograms with sides 1 and 2, and angles \(60^{\circ}\) and \(120^{\circ}\), can be placed inside a regular hexagon with side length 3?

Answer: $12$


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

Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.

$M_{1}(2, 3, 1)$

$M_{2}(4, 1, -2)$

$M_{3}(6, 3, 7)$

$M_{0}(-5, -4, 8)$

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>.

A bot is held in orbit by the gravitational force of attraction to the planet, which creates a centripetal acceleration equal to $4 \pi^{2}(2 R) / T^{2}$, where $R$ is the radius of the planet and $T$ is the period of revolution of the bot. From Newton's second law, we have:

$$
m \frac{4 \pi^{2}(2 R)}{T^{2}}=G \frac{m \rho \frac{4}{3} \pi R^{3}}{(2 R)^{2}}, \quad \rho=\frac{24 \pi}{G T^{2}}
$$

Substituting the approximate value of the gravitational constant $G$ and $T = 4 \cdot 60 \cdot 60$ seconds, we get $\rho \approx 6000 \mathrm{kг} / \mathrm{m}^{3}$.

Answer: $6000$


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

Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \).

Answer: $-1/2$


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

A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a side of the original square. The perimeter of the resulting octagon is $40\%$ greater than the perimeter of the original square. By what percentage is the area of the octagon less than the area of the original square?

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>.

Inside an angle of $60^{\circ}$, there is a point located at distances $\sqrt{7}$ and $2 \sqrt{7}$ from the sides of the angle. Find the distance of this point from the vertex of the angle.

Answer: $\frac{14 \sqrt{3}}{3}$


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

In the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\), and \(AB + AD = AC\). Additionally, it is known that \(\angle ACD = 23^\circ\). What is the measure of angle \(ABC\) in degrees?

Answer: $83$


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>.

In a $4 \times 4$ grid, place candies according to the following requirements: (1) Each cell must contain candies; (2) In adjacent cells, the left cell has 1 fewer candy than the right cell and the upper cell has 2 fewer candies than the lower cell; (3) The bottom-right cell contains 20 candies. How many candies are there in total on the grid? (Adjacent cells are those that share a common edge)

Answer: $248$


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

The numbers \(a, b, c, d\) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).

Answer: $240$


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

A line passing through the intersection point of the medians of triangle \(ABC\) intersects the sides \(BA\) and \(BC\) at points \(A^{\prime}\) and \(C_1\) respectively. Given that:

\(BA^{\prime} < BA = 3\), \(BC = 2\), and \(BA^{\prime} \cdot BC_1 = 3\). Find \(BA^{\prime}\).

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


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

On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute.

Answer: $1/2$


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

A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:

$$
0,2,3,4,4,7,8,8,9
$$

What is this number?

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>.

How many ordered pairs \((b, g)\) of positive integers with \(4 \leq b \leq g \leq 2007\) are there such that when \(b\) black balls and \(g\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \(\frac{1}{2}\)?

Answer: $59$


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

What is the repeating sequence? Determine what is the repeating sequence in the decimal expansion of the fraction \(\frac{1}{49}\).


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

Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of the hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of point \( P \).

Answer: $-\frac{64}{5}$


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>.

Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tetrahedron?

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>.

On graph paper, two right triangles are drawn. Find the sum of the angles BCA and \(\mathrm{B}_{1} \mathrm{C}_{1} \mathrm{~A}_{1}\).

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>.

The number of solutions to the equation $\sin |x| = |\cos x|$ in the closed interval $[-10\pi, 10\pi]$ is __.

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>.

At a tribal council meeting, 60 people spoke in turn. Each of them said only one phrase. The first three speakers all said the same thing: "I always tell the truth!" The next 57 speakers also said the same phrase: "Among the previous three speakers, exactly two of them told the truth." What is the maximum number of speakers who could have been telling the truth?

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>.

A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476?

Answer: $1989$


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

Petrov writes down odd numbers: \(1, 3, 5, \ldots, 2013\), and Vasechkin writes down even numbers: \(2, 4, \ldots, 2012\). Each of them calculates the sum of all the digits of all their numbers and tells it to the star student Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the outcome?

Answer: $1007$


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

If the centroid of the inscribed triangle \( ABC \) of the curve \( y^{2}=4 \sqrt{2} x \) is its focus \( F \), then
\[ |FA|^{2} + |FB|^{2} + |FC|^{2} = \]

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>.

A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer \( b \), the number of bi-digital positive integers.

Answer: $243$


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

A heavy concrete platform anchored to the seabed in the North Sea supported an oil rig that stood 40 m above the calm water surface. During a severe storm, the rig toppled over. The catastrophe was captured from a nearby platform, and it was observed that the top of the rig disappeared into the depths 84 m from the point where the rig originally stood.

What is the depth at this location? (Neglect the height of the waves.)

Answer: $68.2$


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>.

Compute the sum of all possible distinct values of \( m+n \) if \( m \) and \( n \) are positive integers such that
$$
\operatorname{lcm}(m, n) + \operatorname{gcd}(m, n) = 2(m+n) + 11
$$

Answer: $32$


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

Find the smallest multiple of 9 that does not contain any odd digits.

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>.

A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \( B \) (see the drawing). Find the measure of angle \( ABC \).

(A polygon is called regular if all its sides are equal and all its angles are equal.)

Answer: $22.5$


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

Let \( A, B, C \) be points on the same plane with \( \angle ACB = 120^\circ \). There is a sequence of circles \( \omega_0, \omega_1, \omega_2, \ldots \) on the same plane (with corresponding radii \( r_0, r_1, r_2, \ldots \) where \( r_0 > r_1 > r_2 > \cdots \)) such that each circle is tangent to both segments \( CA \) and \( CB \). Furthermore, \( \omega_i \) is tangent to \( \omega_{i-1} \) for all \( i \geq 1 \). If \( r_0 = 3 \), find the value of \( r_0 + r_1 + r_2 + \cdots \).

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


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

There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportation cost for a ton of coal from Mine A to city $N$ is 8 yuan. What should the transportation cost per ton of coal from Mine B to city $N$ be for it to be more economical than transporting coal from Mine A?

Answer: $18$


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

In triangle \(ABC\), side \(AC = 42\). The angle bisector \(CL\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \(2:1\) from the vertex. Find the length of side \(AB\) if the radius of the circle inscribed in triangle \(ABC\) is 14.

Answer: $56$


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

Let \( x_{1}, x_{2}, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In a \( 100 \times 100 \) table, numbers are placed as follows: at the intersection of the \( i \)-th row and the \( k \)-th column, the number \( \log _{x_{k}} \frac{x_{i}}{4} \) is written. Find the smallest possible value of the sum of all the numbers in the table.

Answer: $-10000$


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

The numbers \(a, b, c, d\) belong to the interval \([-8.5, 8.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).

Answer: $306$


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

In $\triangle ABC$, $\angle ABC = \angle ACB = 40^\circ$, and $P$ is a point inside the triangle such that $\angle PAC = 20^\circ$ and $\angle PCB = 30^\circ$. Find the measure of $\angle PBC$.

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>.

What is the greatest common divisor of all the numbers $7^{n+2} + 8^{2n+1}$ for $n \in \mathbb{N}$?

Answer: $57$


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

A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters), and all digits except for the combination 000. Tanya considers a license plate happy if the first letter is a consonant, the second letter is also a consonant, and the third digit is odd (there are no restrictions on the other characters). How many license plates exist that Tanya considers happy?

Answer: $384000$


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

Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$.

Answer: $28$


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

Given \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \), their root mean square is defined as \(\left(\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right)^{\frac{1}{2}} \). Let \( M \) be the set of values of \( n \) (for \( n > 1 \)) such that the root mean square of the first \( n \) positive integers is an integer. Find the smallest number in the set \( M \).

Answer: $337$


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

Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins?

Answer: $120$


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

Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum.

Answer: $700000000$


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

Starting with 1, alternately add 4 and 3 to obtain the sequence $1, 5, 8, 12, 15, 19, 22, \ldots \ldots$. In this sequence, the number closest to 2013 is $\qquad$ .

Answer: $2014$


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

Someone says that 7 times their birth year divided by 13 gives a remainder of 11, and 13 times their birth year divided by 11 gives a remainder of 7. How old will this person be in the year 1954?

Answer: $86$