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

Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation?

Answer: $162$


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

In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written?

Answer: $19$


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

Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits.

Answer: $972538$


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

Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).

Answer: $14$


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

The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence.

Answer: $3028$


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

Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?

*Proposed by Michael Tang*

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>.

In parallelogram \(ABCD\), points \(A_{1}, A_{2}, A_{3}, A_{4}\) and \(C_{1}, C_{2}, C_{3}, C_{4}\) are respectively the quintisection points of \(AB\) and \(CD\). Points \(B_{1}, B_{2}\) and \(D_{1}, D_{2}\) are respectively the trisection points of \(BC\) and \(DA\). Given that the area of quadrilateral \(A_{4} B_{2} C_{4} D_{2}\) is 1, find the area of parallelogram \(ABCD\).

Answer: $15$


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

In triangle $ABC$, $\angle C=90^{\circ}, \angle B=30^{\circ}, AC=2$, $M$ is the midpoint of $AB$. Fold triangle $ACM$ along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Find the volume of the resulting triangular pyramid $A-BCM$.

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


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

The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to region $g$ (event $A$)?

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


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

Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer.

Answer: $5600$


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

Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]

If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the minimum value of the real number \(\lambda\).

Answer: $2 + \sqrt{5}$


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

Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the cells?

Answer: $101$


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

Find the greatest solution on the interval $[0 ; 10 \pi]$ of the equation $|2 \sin x - 1| + |2 \cos 2x - 1| = 0$. Round the answer to three significant digits according to rounding rules and enter it in the provided field.

Answer: $27.7$


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

Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere.

Answer: $\frac{\sqrt{41}}{8}$


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>.

Travel along the alley clockwise.

In 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.

In the first case, when moving along the alley, they would need to cover a distance of 6 km, and if they move towards each other, the required time is $\frac{6}{6+20}$ hours.

In the second case, when moving towards each other, after $\frac{2 \pi - 6}{6}$ hours, the pedestrian will reach point $B$, and the cyclist will still be riding along the road (since $\frac{4}{15} > \frac{2 \pi - 6}{6}$). Hence, the cyclist will continue to ride along the road until they meet, and the closing speed of the pedestrian and cyclist will be $15 + 6 = 21$ km/h. Therefore, they will meet in $\frac{2 \pi - 2}{21}$ hours.

Let's compare the numbers obtained in the 1st and 2nd cases:

$$
\frac{3}{13} > 0.23 > 0.21 > \frac{2 \cdot 3.15 - 2}{21} > \frac{2 \pi - 2}{21}
$$

(the first and third inequalities can be obtained, for example, by long division). Therefore, the answer is achieved in the 2nd case.

Answer: $0.21$


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>.

The first 14 terms of the sequence $\left\{a_{n}\right\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \ldots$. Following this pattern, what is $a_{18}$?

Answer: $51$


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

In the convex quadrilateral \(ABCD\), the length of side \(AD\) is 4, the length of side \(CD\) is 7, the cosine of angle \(ADC\) is \(\frac{1}{2}\), and the sine of angle \(BCA\) is \(\frac{1}{3}\). Find the length of side \(BC\) given that the circumcircle of triangle \(ABC\) also passes through point \(D\).

Answer: $\frac{\sqrt{37}}{3\sqrt{3}}(\sqrt{24} - 1)$


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

In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, "Not including yourself, are there an equal number of knights and liars on your island?" It turned out that on 7 islands, everyone answered "Yes," while on the rest, everyone answered "No." Then, each person was asked, "Is it true that, including yourself, people of your caste are less than half of the inhabitants of the island?" This time, on some 7 islands, everyone answered "No," while on the others, everyone answered "Yes." How many liars are there in the kingdom?

Answer: $1013$


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

Determine the largest positive integer \( n \) with \( n < 500 \) for which \( 6048 \cdot 28^n \) is a perfect cube (that is, it is equal to \( m^3 \) for some positive integer \( m \) ).

Answer: $497$


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

Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \frac{9}{2} = 0$ is true for all values of $z$.

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>.

Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number.

Answer: $520$


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

Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the product of the numbers did not change. Find the largest possible value of $k$.

Answer: $99$


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

Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:

- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the largest possible.

What is the measure of the angle \( \angle B A M \) in degrees?

Answer: $44$


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

In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form?

Answer: $23$


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

In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among the two trees that are three trees away from any coniferous tree, one is coniferous and the other is deciduous. How many birches have been 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>.

The side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \ldots, C_{n+1} = C$). The triangles $C_i B_i C_{i+1}$ are shaded. What fraction of the area of the whole triangle is shaded?

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


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

Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).

Answer: $1014$


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

The side of rhombus \(ABCD\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.

Find the distance between the points where this circle touches the sides \(AB\) and \(BC\), if the diagonal \(AC\) is less than the diagonal \(BD\).

Answer: $3.84$


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

For the function $f(x)= \sqrt {2}(\sin x+\cos x)$, the following four propositions are given:
$(1)$ There exists $\alpha\in\left(- \frac {\pi}{2},0\right)$, such that $f(\alpha)= \sqrt {2}$;
$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \frac {3\pi}{4}$;
$(3)$ There exists $\phi\in\mathbb{R}$, such that the graph of the function $f(x+\phi)$ is centrally symmetric about the origin;
$(4)$ The graph of the function $f(x)$ can be obtained by shifting the graph of $y=-2\cos x$ to the left by $ \frac {\pi}{4}$.
Among these, the correct propositions are \_\_\_\_\_\_.

Answer: $(2)(3)$


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

What is the product of the solutions of the equation $-45 = -2x^2 + 6x?$

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>.

A natural number \( N \) ends with the digit 5. A ninth-grader Dima found all its divisors and discovered that the sum of the two largest proper divisors is not divisible by the sum of the two smallest proper divisors. Find the smallest possible value of the number \( N \). A divisor of a natural number is called proper if it is different from 1 and the number itself.

Answer: $725$


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

(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:
(1) If digits can be repeated, how many different five-digit even numbers can be formed?
(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not being 3, can be formed?
(3) If in the linear equation $ax + by = 0$, $a$ and $b$ can be any two different digits from the given six, how many different lines can be represented by the equation?

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>.

Calculate the smallest root \(x_0\) of the equation

$$
x^{2}-\sqrt{\lg x +100}=0
$$

(with a relative error of no more than \(10^{-390} \%\)).

Answer: $10^{-100}$


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>.

Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \).

Answer: $-3$


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

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


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

Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}} = 45 \times \overline{\text{deed}}$), find the largest possible value of the five-digit palindromic number.

Answer: $59895$


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

The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round are eliminated. Those who win in the first round are paired again. Those who lose in the second round are eliminated, and so on. In each round, a pair was found for every participant.)

Answer: $16$


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

Point \( K \) is the midpoint of edge \( A A_{1} \) of cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), and point \( L \) lies on edge \( B C \). Segment \( K L \) touches the sphere inscribed in the cube. In what ratio does the point of tangency divide segment \( K L \)?

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>.

The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. Find the area of this cross-section.

Answer: $\sqrt{6}$


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

A teacher has prepared three problems for the class to solve. In how many different ways can he present these problems to the students if there are 30 students in the class?

Answer: $24360$


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

Let the complex number \( z \) satisfy \( |z|=1 \). Given that the equation \( zx^2 + 2\bar{z}x + 2 = 0 \) in terms of \( x \) has a real root, find the sum of all such complex numbers \( z \).

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>.

Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\).

Answer: $2\sqrt{14}$


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

Given that $17^{-1} \equiv 11 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)

Answer: $42$


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

For which natural number \( n \) does the quantity \(\frac{n^2}{1.001^n}\) 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>.

Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest possible values of the mean of \(m_{1}, m_{2}, \ldots, m_{10}\).

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>.

In the acute triangle \( \triangle ABC \),
\[
\sin(A+B) = \frac{3}{5}, \quad \sin(A-B) = \frac{1}{5}, \quad AB = 3.
\]

Find the area of \( \triangle ABC \).

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


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

Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \cdots + |a_{10}|$.

Answer: $68$


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

Maria invested $10,000 for 3 years at an annual interest rate of 5 percent compounded annually. Liam invested $10,000 for the same period of time, at the same interest rate, but the interest was compounded semi-annually. Calculate the difference in the amount earned by Liam's investment compared to Maria's, to the nearest dollar.

Answer: $16$


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

Ponchik was having a snack at a roadside café when a bus passed by. Three pastries after the bus, a motorcycle passed by Ponchik, and three pastries after that, a car passed by. Syrupchik, who was snacking at another café on the same road, saw them in a different order: first the bus, after three pastries the car, and three pastries after that, the motorcycle. It is known that Ponchik and Syrupchik always eat pastries at a constant speed. Find the speed of the bus if the speed of the car is 60 km/h and the speed of the motorcycle is 30 km/h.

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>.

In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the sizes of angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^2 - a^2 = ac \), what is the radian measure of angle \( B \)?

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


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

Let \( ABCD \) be a trapezoid such that \( (AB) \) is parallel to \( (CD) \), \( AB = 3 \), \( CD = 3 \), \( DA = 3 \) and \( \widehat{ADC} = 120^\circ \). Determine the angle \( \widehat{CBA} \) in degrees.

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>.

Inside rectangle \(ABCD\), points \(E\) and \(F\) are located such that segments \(EA, ED, EF, FB, FC\) are all congruent. The side \(AB\) is \(22 \text{ cm}\) long and the circumcircle of triangle \(AFD\) has a radius of \(10 \text{ cm}\).

Determine the length of side \(BC\).

Answer: $16$


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

On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's trail and the diameter of this trajectory was 10 meters. Estimate the mass of the ice floe, assuming the mass of the bear is 600 kg.

Answer: $11400$


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

Let the function \( y = f(x) \) satisfy: for all \( x \in \mathbb{R} \), \( y = f(x) \geqslant 0 \), and \( f(x+1) = \sqrt{9 - f(x)^2} \). When \( x \in [0,1) \),
$$
f(x) = \begin{cases}
2^x, & 0 \leqslant x < \frac{1}{2}, \\
\log_{10} (x + 31), & \frac{1}{2} \leqslant x < 1
\end{cases}
$$

Find \( f(\sqrt{1000}) \).

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


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

Let \( n \) be a positive integer. When \( n > 100 \), what are the first two decimal places of the fractional part of \( \sqrt{n^2 + 3n + 1} \)?

Answer: $49$


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>.

In a Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from point $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real numbers $x$ and $y$ satisfy $0 \leq x \leq 10$ and $0 \leq y \leq 10$, then the total length of the trajectory of all points $C$ that satisfy the condition is .

Answer: $5(\sqrt{2} + 1)$


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

A point moving in the positive direction of the $O x$ axis has the abscissa $x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \geqslant 24$ for all $t \geqslant 0$.

Answer: $2 \sqrt{\left( \frac{24}{7} \right)^7}$


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

What is the smallest number with three different prime factors, none of which can be less than 10?

Answer: $2431$


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

The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.)

Answer: $0.25$


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

A five-digit number is called a "pit" if its first three digits are in descending order and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number 22222?

Answer: $36$


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

Given $cos(\frac{π}{4}-α)=\frac{3}{5}$ and $sin(\frac{5π}{4}+β)=-\frac{12}{13}$, where $α∈(\frac{π}{4},\frac{3π}{4})$ and $β∈(0,\frac{π}{4})$, find $\frac{tanα}{tanβ}$.

Answer: $-17$


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

Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school, arriving 20 minutes late. By how many times does the speed of Uncle Vanya's car exceed Nikita's running speed?

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>.

Let \( O \) be an interior point of \( \triangle ABC \). Extend \( AO \) to meet \( BC \) at \( D \). Similarly, extend \( BO \) and \( CO \) to meet \( CA \) and \( AB \) respectively at \( E \) and \( F \). Given \( AO = 30 \), \( FO = 20 \), \( BO = 60 \), \( DO = 10 \) and \( CO = 20 \), find \( EO \).

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>.

One mole of an ideal monatomic gas is first heated isobarically, during which it performs 40 J of work. Then it is heated isothermally, receiving the same amount of heat as in the first case. What work does the gas perform (in Joules) in the second case?

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>.

Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots in total are needed for this?

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>.

Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be checked 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>.

In the cells of a $100 \times 100$ square, the numbers $1, 2, \ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in those cells differ exactly by 5000, were calculated. Let $S$ be the minimum of these distances. What is the maximum value that $S$ can take?

Answer: $50\sqrt{2}$


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>.

Someone, when asked for the number of their ticket, replied: "If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number." Determine the ticket number.

Answer: $198$


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

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at 1 meter per second. Suddenly someone turns on the turntable; it spins at $30 \mathrm{rpm}$. Consider the set $S$ of points the mouse can reach in his car within 1 second after the turntable is set in motion. What is the area of $S$, in square meters?

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


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

A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

Answer: $438$


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

Each face of a fair six-sided die is marked with one of the numbers $1, 2, \cdots, 6$. When two such identical dice are rolled, the sum of the numbers on the top faces of these dice is the score for that roll. What is the probability that the product of the scores from three such rolls is divisible by 14? Express your answer as a simplified fraction.

Answer: $1/3$


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

Attempt to obtain one billion (1,000,000,000) by multiplying two integers, each of which contains no zeros.

Answer: $512 * 1953125$


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

In the rectangular prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(A B = A A_{1} = 2\), \(A D = 2 \sqrt{3}\). Point \(M\) lies within plane \(B A_{1} C_{1}\). Find the minimum value of \(\overrightarrow{M A} \cdot \overrightarrow{M C}\).

Answer: $-\frac{16}{7}$


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

Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers.

Answer: $11$


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

Let natural numbers \( k \) and \( n \) be coprime, where \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line formed by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting the same endpoints. For example, a \((5 ; 2)\)-star has 5 points of self-intersection, which are the bold points in the drawing. How many self-intersections does the \((2018 ; 25)\)-star have?

Answer: $48432$


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

Let $x, y, z$ be positive numbers satisfying the following system of equations:

$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$

Find the value of the expression $xy + yz + xz$.

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>.

What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number?

Answer: $505$


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

On a \(6 \times 6\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column?

Answer: $40/119$


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

Given a trapezoid \(ABCD\). A point \(M\) is chosen on its lateral side \(CD\) such that \( \frac{CM}{MD} = \frac{4}{3} \). It turns out that segment \( BM \) divides the diagonal \( AC \) into two segments, the ratio of the lengths of which is also \( \frac{4}{3} \). What possible values can the ratio \( \frac{AD}{BC} \) take? If necessary, round your answer to 0.01 or write it as a common fraction.

Answer: $7/12$


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>.

A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots?

Answer: $3/4$


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

Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each?

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>.

Petya's watch runs 5 minutes fast per hour, and Masha's watch runs 8 minutes slow per hour. At 12:00, they set their watches to the accurate school clock and agreed to meet at the skating rink at 6:30 PM according to their respective watches. How long will Petya wait for Masha if each arrives at the skating rink exactly at 6:30 PM according to their own watch?

Answer: $1.5$


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

Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1.

Answer: $11111111100000000$


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

Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:

\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\]

Answer: $312$


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

Find
\[
\sum_{n = 1}^\infty \frac{3^n}{1 + 3^n + 3^{n + 1} + 3^{2n + 1}}.
\]

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>.

Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2023} \right| = \frac{1}{2023}.\]

Answer: $\frac{4042}{2023}$


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>.

Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma in front of the store, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait for the store to open for no more than 4 minutes.

Answer: $0.75$


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>.

A certain unit has 160 young employees. The number of middle-aged employees is twice the number of elderly employees. The total number of elderly, middle-aged, and young employees is 430. In order to understand the physical condition of the employees, a stratified sampling method is used for the survey. In a sample of 32 young employees, the number of elderly employees in this sample is ____.

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>.

A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \mathrm{~m} /\mathrm{s}$?

Answer: $0.8$


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

Calculate the value of the expression

$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$

Answer: $4080401$


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

How many strictly positive integers less than or equal to 120 are not divisible by 3, 5, or 7?

Answer: $54$


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

While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minutes away from it. At some point, the first lion cub steps on the turtle, and after a while, the second lion cub does too. 28 minutes and 48 seconds after the second occurrence, the turtle reaches the water hole. How many minutes passed between the two occurrences, given that all three moved at constant speeds?

Answer: $2.4$


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

A certain number of books are distributed among children. If each child gets $m$ books, there are 14 books left. If each child gets 9 books, the last child only gets 6 books. How many children are there in total? And how many books are there?

Answer: $150$


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

An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles?

Answer: $112.5$


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>.

If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______.

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


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

In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). \(\angle A = \angle D = 45^\circ\), while \(\angle B = \angle C = 135^\circ\). If \(AB = 6\) and the area of \(ABCD\) is 30, find \(BC\).

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 kite \( ABCD \) is symmetric with respect to diagonal \( AC \). The length of \( AC \) is 12 cm, the length of \( BC \) is 6 cm, and the internal angle at vertex \( B \) is a right angle. Points \( E \) and \( F \) are given on sides \( AB \) and \( AD \) respectively, such that triangle \( ECF \) is equilateral.

Determine the length of segment \( EF \).

(K. Pazourek)

Answer: $4\sqrt{3}$


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

Given that the graph of the power function $y=(m^{2}-2m-2)x^{m^{2}+4m}$ is symmetric with respect to the origin and does not intersect the $x$-axis or $y$-axis, determine the value of the integer $m$.

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>.

Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \), given that the volume of the prism is 35.

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>.

Hooligan Vasily tore out an entire chapter from a book, with the first page numbered 231, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book?

Answer: $41$


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

Given that odd prime numbers \( x, y, z \) satisfy
\[ x \mid (y^5 + 1), \quad y \mid (z^5 + 1), \quad z \mid (x^5 + 1). \]

Find the minimum value of the product \( xyz \).

Answer: $2013$


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

How many solutions in natural numbers does the equation $\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1$ have?

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>.

Find the area of a trapezoid with bases 11 and 4 and diagonals 9 and 12.

Answer: $54$


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

The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$?

Answer: $1120$


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

The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \).

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


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

The Euler family consists of four girls, each aged 8 years, and three boys, one aged 10 and twins both aged 12. Calculate the mean age of the children after one year passes.

Answer: $10.43$


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

If a positive integer \( N \) can be expressed as \( \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor \) for some real number \( x \), then we say that \( N \) is "visible"; otherwise, we say that \( N \) is "invisible". For example, 8 is visible since \( 8 = \lfloor 1.5 \rfloor + \lfloor 2(1.5) \rfloor + \lfloor 3(1.5) \rfloor \), whereas 10 is invisible. If we arrange all the "invisible" positive integers in increasing order, find the \( 2011^{\text{th}} \) "invisible" integer.

Answer: $6034$


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

Given that $\dfrac {π}{2} < β < α < \dfrac {3}{4}π$, $\cos (α+β)=- \dfrac {3}{5}$, and $\sin (α-β)= \dfrac {5}{13}$, find $\cos 2β$.

Answer: $- \dfrac {56}{65}$


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

A seven-digit natural number \( N \) is called interesting if:

- It consists of non-zero digits;
- It is divisible by 4;
- Any number obtained from \( N \) by permuting its digits is also divisible by 4.

How many interesting numbers exist?

Answer: $128$


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

Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \).

Answer: $\sqrt{3}$


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

A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.

How many Tim numbers are there?

Answer: $16$


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

The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\).

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 plane  $S$  in a space, given are unit circle  $C$  with radius 1 and the line  $L$ . Find the volume of the solid bounded by the curved surface formed by the point  $P$  satifying the following condition  $(a),\ (b)$ . $(a)$  The point of intersection  $Q$  of the line passing through  $P$  and perpendicular to  $S$  are on the perimeter or the inside of  $C$ . $(b)$  If  $A,\ B$  are the points of intersection of the line passing through  $Q$  and pararell to  $L$ , then  $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$ .

Answer: $\pi$


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>.

Compute the definite integral:

$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$

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


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 \(ABC\), side \(AB = 6\), \(\angle BAC = 30^\circ\), and the radius of the circumscribed circle is 5. Find side \(AC\).

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


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

Let $M=\{1,2, \cdots, 2005\}$. Subset $A$ of $M$ satisfies the condition: if $x \in A$, then $15x \notin A$. What is the maximum number of elements in $A$?

Answer: $1880$


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

A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the third prize is more than that for the second prize, and the total prize money for the second prize is more than that for the first prize, with the total number of winners not exceeding 20, then what is the minimum amount of the first prize?

Answer: $2700$


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

Find at the point \( M_{0}(1,1,1) \) the direction of the greatest change of the scalar field \( u = xy + yz + xz \) and the magnitude of this greatest change at this point.

Answer: $2\sqrt{3}$


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

A massive vertical plate is fixed to a car moving at a speed of $5 \, \text{m/s}$. A ball is flying towards it at a speed of $6 \, \text{m/s}$ with respect to the ground. Determine the speed of the ball with respect to the ground after a perfectly elastic normal collision.

Answer: $16$


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

$ABC$ is an isosceles triangle with base $AC$. $CD$ is the bisector of angle $C$, and $\angle ADC = 150^\circ$. Find $\angle B$.

Answer: $140$


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

In the coordinate plane, a parallelogram $O A B C$ is drawn such that its center is at the point $\left(\frac{19}{2}, \frac{15}{2}\right)$, and the points $A, B,$ and $C$ have natural number coordinates. Find the number of such parallelograms. (Here, $O$ denotes the origin - the point $(0,0)$; two parallelograms with the same set of vertices are considered the same, i.e., $OABC$ and $OCBA$ are considered the same parallelogram.)

Answer: $126$