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

The three-tiered "pyramid" shown in the image is built from $1 \mathrm{~cm}^{3}$ cubes and has a surface area of $42 \mathrm{~cm}^{2}$. We made a larger "pyramid" based on this model, which has a surface area of $2352 \mathrm{~cm}^{2}$. How many tiers does it have?

Answer: $24$


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

Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \).

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>.

The sequence $\{a_n\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$.

Answer: $5150$


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

We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone?

Answer: $8\pi$


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>.

Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$.

Answer: $-\frac{\pi}{8}$


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

Given that point $P(-15a, 8a)$ is on the terminal side of angle $\alpha$, where $a \in \mathbb{R}$ and $a \neq 0$, find the values of the six trigonometric functions of $\alpha$.

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


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>.

In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid.

Answer: $3\sqrt{3}$


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

Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{4}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\]

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>.

Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription "MAME" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.

Answer: $1/8$


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

Segment \( BD \) is the median of an isosceles triangle \( ABC \) (\( AB = BC \)). A circle with a radius of 4 passes through points \( B \), \( A \), and \( D \), and intersects side \( BC \) at point \( E \) such that \( BE : BC = 7 : 8 \). Find the perimeter of triangle \( ABC \).

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>.

From the vertex  $ A$  of the equilateral triangle  $ ABC$  a line is drown that intercepts the segment  $ [BC]$  in the point  $ E$ . The point  $ M \in (AE$  is such that  $ M$  external to  $ ABC$ ,  $ \angle AMB \equal{} 20 ^\circ$  and  $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle  $ \angle MAB$ ?

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>.

Let \( f(x) \) be a function with the property that \( f(x) + f\left(\frac{x-1}{3x-2}\right) = x \) for all real numbers \( x \) other than \( \frac{2}{3} \). What is the sum \( f(0) + f(1) + f(2) \)?

Answer: $\frac{87}{40}$


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

A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceiling such that the following conditions are satisfied: every point on the soccer field is illuminated by at least one floodlight; the height of the ceiling must be a multiple of 0.1 meters (e.g., 19.2 meters, 26 meters, 31.9 meters).

Answer: $27.1$


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 magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same height? (The trees grow uniformly for the specified number of hours and then stop growing).

Answer: 15 or 18 hours.

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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $2b\sin \left(C+ \frac {\pi}{6}\right)=a+c$.
(I) Find the magnitude of angle $B$;
(II) If point $M$ is the midpoint of $BC$, and $AM=AC=2$, find the value of $a$.

Answer: $\frac {4 \sqrt {7}}{7}$


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>.

In triangle $DEF$, points $D'$, $E'$, and $F'$ are on the sides $EF$, $FD$, and $DE$, respectively. Given that $DD'$, $EE'$, and $FF'$ are concurrent at the point $P$, and that $\frac{DP}{PD'}+\frac{EP}{PE'}+\frac{FP}{PF'}=94$, find $\frac{DP}{PD'}\cdot \frac{EP}{PE'}\cdot \frac{FP}{PF'}$.

Answer: $92$


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

Neznaika does not know about multiplication and exponentiation operations. However, he is good at addition, subtraction, division, and square root extraction, and he knows how to use parentheses. While practicing, Neznaika chose three numbers 20, 2, and 2, and formed the expression:

$$
\sqrt{(2+20): 2} .
$$

Can he use the same three numbers 20, 2, and 2 to form an expression whose value is greater than 30?

Answer: $20 + 10\sqrt{2}$


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

A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930?

Answer: $39$


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

Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train?

Answer: $233.33$


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>.

Given the function $f(x)=a\ln x-x^{2}+1$.

(I) If the tangent line of the curve $y=f(x)$ at $x=1$ is $4x-y+b=0$, find the values of the real numbers $a$ and $b$;

(II) Discuss the monotonicity of the function $f(x)$.

Answer: $-4$


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>.

On the board, two sums are written:

$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$

and

$9+98+987+9876+98765+987654+9876543+$

$+98765432+987654321$

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

Answer: $1097393685$


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

Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.

$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$

Answer: $2 + \frac{1}{2} \ln \left( \frac{9}{5} \right)$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 plane, the "rectilinear 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 point $C(x, y)$ has an equal "rectilinear distance" to points $A(1, 3)$ and $B(6, 9)$, where the real numbers $x$ and $y$ satisfy $0 \leqslant x \leqslant 10$ and $0 \leqslant y \leqslant 10$, then the total length of the locus of all such points $C$ 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>.

Let \(x, y, z\) be positive real numbers such that \(xyz = 1\). Find the maximum value of
\[
\frac{x^2y}{x+y} + \frac{y^2z}{y+z} + \frac{z^2x}{z+x}.
\]

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>.

The probability of inducing cerebrovascular disease by smoking 5 cigarettes in one hour is 0.02, and the probability of inducing cerebrovascular disease by smoking 10 cigarettes in one hour is 0.16. An employee of a certain company smoked 5 cigarettes in one hour without inducing cerebrovascular disease. Calculate the probability that he can continue to smoke 5 cigarettes without inducing cerebrovascular disease in that hour.

Answer: $\frac{6}{7}$


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 an ellipse $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\frac{\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\frac{4\sqrt{5}}{5}$$.

(Ⅰ) Find the equation of ellipse C;
(Ⅱ) Suppose line $l$ intersects ellipse C at points A and B. If the circle with diameter AB passes through the origin O, investigate whether the distance from point O to line AB is a constant. If so, find this constant; otherwise, explain why;
(Ⅲ) Under the condition of (Ⅱ), try to find the minimum value of the area $S$ of triangle $\triangle AOB$.

Answer: $\frac{4}{5}$


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>.

What is the probability that two trainees were born on the same day (not necessarily the same year)? Note: There are 62 trainees.

Answer: $99.59095749$


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

Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ that passes through the point $(0,1)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.

$(1)$ Find the standard equation of the ellipse $E$;

$(2)$ Suppose a line $l: y = \frac{1}{2}x + m$ intersects the ellipse $E$ at points $A$ and $C$. A square $ABCD$ is constructed with $AC$ as its diagonal. Let the intersection of line $l$ and the $x$-axis be $N$. Is the distance between points $B$ and $N$ a constant value? If yes, find this constant value; if no, explain why.

Answer: $\frac{\sqrt{10}}{2}$


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

Given a sequence of 15 zeros and ones, determine the number of sequences where all the zeros are consecutive.

Answer: $121$


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

In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table?

Answer: $168$


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

Inside a circle, 16 radii of this circle and 10 circles with the same center as the circle are drawn. Into how many regions do the radii and circles divide the circle?

Answer: $176$


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

Divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest?

Answer: $1.9$


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

In the acute-angled triangle \(ABC\), it is known that \(\sin (A+B)=\frac{3}{5}\), \(\sin (A-B)=\frac{1}{5}\), and \(AB=3\). Find the area of \(\triangle ABC\).

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


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>.

Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ?

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>.

The side of the base of a regular quadrilateral pyramid \( \operatorname{ABCDP} \) (with \( P \) as the apex) is \( 4 \sqrt{2} \), and the angle between adjacent lateral faces is \( 120^{\circ} \). Find the area of the cross-section of the pyramid by a plane passing through the diagonal \( BD \) of the base and parallel to the lateral edge \( CP \).

Answer: $4\sqrt{6}$


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>.

Given an ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.

(Ⅰ) Find the equation of the ellipse C;

(Ⅱ) On the ellipse C, is there a point R $(m, n)$ such that the line $l: mx+ny=1$ intersects the circle $O: x^2+y^2=1$ at two distinct points M and N, and the area of $\triangle OMN$ is maximized? If such a point exists, find the coordinates of point R and the corresponding area of $\triangle OMN$; if not, explain why.

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>.

A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.

$(1)$ Individuals Jia, Yi, and Bing each buy $1$ set of blind box set $B$. Let the random variable $\xi$ represent the number of small rabbit toys among them. Find the distribution table and mathematical expectation of $\xi$.

$(2)$ A consumer bought $1$ set of blind box set $A$ and $1$ set of blind box set $B$ on the first and second days of sale, respectively. After mixing all $6$ items together and randomly selecting $1$ to open, a small rabbit toy was found. Find the probability that this small rabbit toy came from blind box set $B$.

Answer: $\frac{1}{3}$


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>.

Find the value of $\frac{\frac{1}{2}-\frac{1}{3}}{\frac{1}{3}-\frac{1}{4}} \times \frac{\frac{1}{4}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{6}} \times \frac{\frac{1}{6}-\frac{1}{7}}{\frac{1}{7}-\frac{1}{8}} \times \ldots \times \frac{\frac{1}{2004}-\frac{1}{2005}}{\frac{1}{2005}-\frac{1}{2006}} \times \frac{\frac{1}{2006}-\frac{1}{2007}}{\frac{1}{2007}-\frac{1}{2008}}$.

Answer: $1004$


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

Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to assess 25 students?

Answer: $110$


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

In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the area of triangle \(BHC\) is \(\frac{1}{8}\), find the area of triangle \(ADG\).

Answer: $\frac{7}{92}$


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

120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each have been stockpiled at the quarry. A railroad platform can hold up to 40 tons. What is the minimum number of platforms required to transport all the slabs?

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>.

Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+P(d)$, where $P(X) = X^{6}-X^{5}-X^{4}-X^{3}-X$.

Answer: $-2$


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 school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule.

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 the four-digit number $\overline{abcd}$ (where $1 \leq a \leq 9$ and $0 \leq b, c, d \leq 9$):
- If $a > b$, $b < c$, and $c > d$, then $\overline{abcd}$ is called a $P$-type number;
- If $a < b$, $b > c$, and $c < d$, then $\overline{abcd}$ is called a $Q$-type number.

Let $N(P)$ and $N(Q)$ denote the number of $P$-type numbers and $Q$-type numbers, respectively. Find the value of $N(P) - N(Q)$.

Answer: $285$


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

Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7?

Answer: $2857$


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

In a store, there are 21 white and 21 purple shirts hanging in a row. Find the smallest $k$ such that, regardless of the initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts, so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively.

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>.

It takes person A 1 minute and 20 seconds to complete a lap, and person B meets person A every 30 seconds. Determine the time it takes for person B to complete a lap.

Answer: $48$


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

Today is 17.02.2008. Natasha noticed that in this date, the sum of the first four digits is equal to the sum of the last four digits. When will this coincidence happen for the last time this year?

Answer: $25.12.2008$


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

Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \).

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>.

Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \).

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>.

Calculate the definite integral:

$$
\int_{0}^{\pi / 4} \frac{7+3 \operatorname{tg} x}{(\sin x+2 \cos x)^{2}} d x
$$

Answer: $3 \ln \left(\frac{3}{2}\right) + \frac{1}{6}$


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>.

Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.

At what time did the bicyclist meet the scooter?

Answer: $15:20$


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

Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, the sum of the first $n$ terms is $S_n$, and $S_3 = 7$. Also, $a_1, a_2, a_3 - 1$ form an arithmetic sequence. For the sequence $\{b_n\}$, the sum of the first $n$ terms is $T_n$, and $6T_n = (3n+1)b_n + 2$, where $n \in \mathbb{N}^+$.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$.
(Ⅱ) Find the general formula for the sequence $\{b_n\}$.
(Ⅲ) Let set $A = \{a_1, a_2, \ldots, a_{10}\}$, $B = \{b_1, b_2, \ldots, b_{40}\}$, and $C = A \cup B$. Calculate the sum of all elements in set $C$.

Answer: $3318$


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

A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse.

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>.

Using each of the digits 1-9 exactly once, form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them?

Answer: $1369$


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

Add the following two numbers: $800,000,000,000 + 299,999,999,999$.
A) $1,000,000,000,000$
B) $1,099,999,999,999$
C) $1,100,000,000,000$
D) $900,000,000,000$
E) $2,099,999,999,999$

Answer: $1,099,999,999,999$


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

What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\circ$ central angle?

Answer: $0.2$


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>.

Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division?

Answer: $\frac{1}{3}$


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>.

Given the sets:
$$
U = \{1, 2, 3, 4, 5\}, \quad I = \{X \mid X \subseteq U\}
$$

If two different elements \( A \) and \( B \) are randomly selected from the set \( I \), what is the probability that \( A \cap B \) contains exactly three elements?

Answer: $5/62$


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

In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar?

Answer: $\frac{1}{3}$


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>.

In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)?

Answer: $17$


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

A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?

Answer: $16/3$


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 minimum number  $n$  such that for any coloring of the integers from  $1$  to  $n$  into two colors, one can find monochromatic  $a$ ,  $b$ ,  $c$ , and  $d$  (not necessarily distinct) such that  $a+b+c=d$ .

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>.

Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed?

Answer: $174$


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

Given that $P$ is a moving point on the line $l: x-2y+4=0$, two tangents are drawn from point $P$ to the circle $C: x^{2}+y^{2}-2x=0$, with tangents intersecting at points $A$ and $B$. Find the minimum area of the circumcircle of quadrilateral $PACB$.

Answer: $\frac{5\pi}{4}$


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

In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Find the ratio of the radius of the circle to leg \(BC\).

Answer: $13/20$


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

Given the user satisfaction ratings: 1 person with 10 points, 1 person with 9 points, 2 people with 8 points, 4 people with 7 points, 1 person with 5 points, and 1 person with 4 points, calculate the average, mode, median, and 85% percentile of the user satisfaction rating for this product.

Answer: $8.5$


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>.

There are \( k \) people and \( n \) chairs in a row, where \( 2 \leq k < n \). There is a couple among the \( k \) people. The number of ways in which all \( k \) people can be seated such that the couple is seated together is equal to the number of ways in which the \( k-2 \) people, without the couple present, can be seated. Find the smallest value of \( n \).

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>.

Given non-zero complex numbers \( x \) and \( y \) satisfying
\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \]
find the value of
\[ \sum_{m=0}^{29} \sum_{n=0}^{29} x^{18mn} y^{-18mn}. \]

Answer: $180$


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

Let \( f(x) \) be the polynomial \( (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) \) where \( a_1, a_2, a_3, a_4, \) and \( a_5 \) are distinct integers. Given that \( f(104) = 2012 \), evaluate \( a_1 + a_2 + a_3 + a_4 + a_5 \).

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>.

Let \(ABCD\) be a convex quadrilateral such that \(AB + BC = 2021\) and \(AD = CD\). We are also given that \(\angle ABC = \angle CDA = 90^\circ\). Determine the length of the diagonal \(BD\).

Answer: $\frac{2021}{\sqrt{2}}$


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

A math competition problem: The probabilities that A, B, and C solve the problem independently are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$ respectively, where $a$, $b$, and $c$ are all single-digit numbers. If A, B, and C attempt the problem independently and the probability that exactly one of them solves the problem is $\frac{7}{15}$, then the probability that none of them solves the problem is $\qquad$.

Answer: $\frac{4}{15}$


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

Given \( x, y, z \in \mathbb{R}^{+} \) and \( x + 2y + 3z = 1 \), find the minimum value of \( \frac{16}{x^{3}}+\frac{81}{8y^{3}}+\frac{1}{27z^{3}} \).

Answer: $1296$


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

The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees.

Answer: $48$


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

In a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \), it is known that \( \overrightarrow{AP} = \frac{1}{2} \overrightarrow{AB} \), \( \overrightarrow{AQ} = \frac{1}{3} \overrightarrow{AC} \), and \( \overrightarrow{AR} = \frac{1}{4} \overrightarrow{AD} \). If point \( K \) is the centroid of \( \triangle BCD \), then what is the volume of the tetrahedron \( KPQR \)?

Answer: $\frac{1}{36}$


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 two distinct geometric progressions with first terms both equal to 1, and the sum of their common ratios equal to 3, find the sum of the fifth terms of these progressions if the sum of the sixth terms is 573. If the answer is ambiguous, provide the sum of all possible values of the required quantity.

Answer: $161$


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

Calculate the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$

Answer: $\frac{5\pi}{8}$


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>.

Ten elves are sitting around a circular table, each with a basket of nuts. Each elf is asked, "How many nuts do your two neighbors have together?" and the answers, going around the circle, are 110, 120, 130, 140, 150, 160, 170, 180, 190, and 200. How many nuts does the elf who answered 160 have?

Answer: $55$


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

There are \( R \) zeros at the end of \(\underbrace{99\ldots9}_{2009 \text{ of }} \times \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}} + 1 \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}}\). Find the value of \( R \).

Answer: $4018$


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

Let an "operation" be randomly changing a given positive integer $n$ to a smaller nonnegative integer (each number has the same probability). What is the probability that after performing the operation on 2019 several times to obtain 0, the numbers 10, 100, and 1000 all appear during the process?

Answer: $1/2019000000$


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

Sasha and Misha are playing a game: they take turns naming a number from 1 to 213 (Misha goes first, and the numbers must be different). Then each counts the number of different rectangles with integer sides whose perimeter equals the named number. The one with the greater number of rectangles wins. What number should Misha name to win? Rectangles that differ by rotation are considered the same. For example, rectangles $2 \times 3$ and $3 \times 2$ are the same.

Answer: $212$


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

There were cards with numbers from 1 to 20 in a bag. Vlad pulled out 6 cards and asserted that all these cards can be divided into pairs such that the sums of the numbers in each pair were the same. Lena managed to sneak a peek at 5 of Vlad's cards: they had the numbers $2, 4, 9, 17, 19$ written on them. Which number on the sixth card did Lena not manage to see? (It is sufficient to provide one suitable answer.)

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>.

Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \, \mathrm{m}^{2}$, $720 \, \mathrm{m}^{2}$, and $900 \, \mathrm{m}^{2}$ respectively. What is the area of the plot received by the fourth sibling?

Answer: $540$


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

Every Sunday, a married couple has breakfast with their mothers. Unfortunately, the relationships each spouse has with the other's mother are quite strained: both know that there is a two-thirds chance of getting into an argument with the mother-in-law. In the event of a conflict, the other spouse sides with their own mother (and thus argues with their partner) about half of the time; just as often, they defend their partner and argue with their own mother.

Assuming that each spouse's arguments with the mother-in-law are independent of each other, what is the proportion of Sundays where there are no arguments between the spouses?

Answer: $4/9$


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

Each of two sides of a triangle is divided into six equal parts; the corresponding division points are connected by line segments. Find the sum of the lengths of these segments if the third side of the triangle is equal to 12.

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>.

Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the areas of \(\odot O_1\) and \(\odot O_2\) be denoted as \( S_1 \) and \( S_2 \) respectively. Solve for \( S_1 + S_2 \).

Answer: $41 \pi$


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>.

What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?

Answer: $15$


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

Given that there is a gathering attended by 1982 people, and among any group of 4 people, at least 1 person knows the other 3. How many people, at minimum, must know all the attendees at this gathering?

Answer: $1979$


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

Given a rhombus \(ABCD\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \(B\), draw the altitudes \(BE\) and \(BF\). Calculate the area of the quadrilateral \(BFDE\).

Answer: $4.32$


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>.

Previously, on an old truck, I traveled from village $A$ through $B$ to village $C$. After five minutes, I asked the driver how far we were from $A$. "Half as far as from $B," was the answer. Expressing my concerns about the slow speed of the truck, the driver assured me that while the truck cannot go faster, it maintains its current speed throughout the entire journey.

$13$ km after $B$, I inquired again how far we were from $C$. I received exactly the same response as my initial inquiry. A quarter of an hour later, we arrived at our destination. How many kilometers is the journey from $A$ to $C$?

Answer: $26$


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

To qualify for the competition, wrestler Vladimir must have three matches and win at least two of them consecutively. His opponents are Andrei (A) and Boris (B). Vladimir can choose to schedule the matches in the order ABA or BAB. The probability of Vladimir losing one match to Boris is 0.3, and to Andrei is 0.4. These probabilities are constant. Under which schedule does Vladimir have a higher probability of qualifying, and what is this probability?

Answer: $0.742$


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

For a triangular piece of land with sides measuring 198 meters, 180 meters, and 210 meters, plant trees equidistantly along the three sides with 6 meters between each pair of consecutive trees, also requiring one tree planted at each of the three vertices. Determine the total number of trees planted.

Answer: $98$


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

The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).

Find the angle \( \angle AMB \).

Answer: $72$


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

A circle constructed on the side $AD$ of the parallelogram $ABCD$ as its diameter passes through the midpoint of diagonal $AC$ and intersects side $AB$ at point $M$. Find the ratio $AM: AB$ if $AC = 3BD$.

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 a bag, there are 70 balls which differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.

What is the minimum number of balls that must be drawn from the bag, without seeing them, to ensure that there are at least 10 balls of one color among them?

Answer: $38$


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

In triangle \( \triangle ABC \), point \( D \) is on side \( BC \) such that \( AD \perp BC \) and \( AD = BC = a \). Find the maximum value of \( \frac{b}{c} + \frac{c}{b} \).

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


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

Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?

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>.

Hooligan Vasya loves to run on the escalator in the metro, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is running downwards, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is running upwards? (The escalator always moves at a constant speed.)

Answer: $324$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 palindrome that is larger than 2015?

Answer: $2112$


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

The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$.

Answer: $\frac{2\sqrt{\pi^2 - 1}}{\pi}$


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

Given a regular tetrahedron with four vertices \(A, B, C, D\) and each edge of length 1 meter, a bug starts at point \(A\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability and crawls to the other end of the chosen edge. What is the probability that the bug will be exactly at vertex \(A\) after crawling a total of 7 meters?

Answer: $\frac{182}{729}$


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

What two digits should be added to the right of the number 2013, so that the resulting six-digit number is divisible by 101? Find all possible solutions.

Answer: $94$


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

Find the ratio of the volume of a regular hexagonal pyramid to the volume of a regular triangular pyramid, given that the sides of their bases are equal and their slant heights are twice the length of the sides of the base.

Answer: $\frac{6 \sqrt{1833}}{47}$


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

A natural number's proper divisors are defined as positive integer divisors other than 1 and the number itself. A natural number greater than 1 is called "good" if it is equal to the product of all its distinct proper divisors. What is the sum of the first 10 "good" natural numbers?

Answer: $182$


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

Find the minimum of the expression \(6 t^{2} + 3 s^{2} - 4 s t - 8 t + 6 s + 5\).

Answer: $\frac{8}{7}$


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>.

A coin is tossed. If heads appear, point \( P \) moves +1 on the number line; if tails appear, point \( P \) does not move. The coin is tossed no more than 12 times, and if point \( P \) reaches coordinate +10, the coin is no longer tossed. In how many different ways can point \( P \) reach coordinate +10?

Answer: $66$


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

In triangle \(ABC\), point \(P\) is located one-third of the way along segment \(AB\) closer to point \(A\). Point \(R\) is one-third of the way along segment \(PB\) closer to point \(P\), and point \(Q\) lies on segment \(BC\) such that angles \(PCB\) and \(RQB\) are congruent.

Determine the ratio of the areas of triangles \(ABC\) and \(PQC\).

Answer: $9:2$


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

A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of stationery. How many sheets of stationery did they each buy?

Answer: $120$


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

A line parallel to side \( AB \) of triangle \( ABC \) intersects side \( BC \) at point \( M \) and side \( AC \) at point \( N \). The area of triangle \( MCN \) is twice the area of trapezoid \( ABMN \). Find the ratio \( CM:MB \).

Answer: $2 + \sqrt{6}$


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>.

Let \( x, y, z \) be positive numbers that satisfy the system of equations:

\[
\begin{cases}
x^{2}+xy+y^{2}=27 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=43
\end{cases}
\]

Find the value of the expression \( xy+yz+xz \).

Answer: $24$


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

In a kindergarten, there are two small Christmas trees and five children. The teachers want to divide the children into two circles around each of the Christmas trees, with at least one child in each circle. The teachers can distinguish between the children but not between the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the Christmas trees (along with the corresponding circles) and rotating each circle around its respective tree. How many ways can the children be divided into circles?

Answer: $50$


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

In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles?

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>.

Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers.

Answer: $26$


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

Given a triangle \( ABC \) where \( |AB| = |AC| \) and \( \angle BAC = 80^\circ \). Inside the triangle, there is a point \( M \) such that \( \angle MBC = 30^\circ \) and \( \angle MCB = 10^\circ \). Find \( \angle AMC \).

Answer: $70$


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

A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be scored during the match.

Answer: $0.502$


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

Vitya has five math lessons a week, one on each day from Monday to Friday. Vitya knows that with a probability of \( \frac{1}{2} \) the teacher will not check his homework at all during the week, and with a probability of \( \frac{1}{2} \) the teacher will check it exactly once during one of the math lessons, but it is impossible to predict on which day - each day has an equal chance.

At the end of the math lesson on Thursday, Vitya realized that so far the teacher has not checked his homework this week. What is the probability that the homework will be checked on Friday?

Answer: $1/6$


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

In an $8 \times 8$ grid filled with different natural numbers, where each cell contains only one number, if a cell's number is greater than the numbers in at least 6 other cells in its row and greater than the numbers in at least 6 other cells in its column, then this cell is called a "good cell". What is the maximum number of "good cells"?

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 the sides \(BC\) and \(CD\) of the square \(ABCD\), points \(M\) and \(K\) are marked respectively such that \(\angle BAM = 30^{\circ}\) and \(\angle CKM = 30^{\circ}\). Find the angle \(AKD\).

Answer: $75$


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

Calculate the limit of the function:

$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$

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


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

A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye scores 315 points. How many points does hitting the blue (second to last) zone score?

Answer: $35$