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

Given that line $l$ is perpendicular to plane $\alpha$, and line $m$ is contained in plane $\beta$. Consider the following propositions:
(1) If $\alpha \parallel \beta$, then $l \perp m$.
(2) If $\alpha \perp \beta$, then $l \parallel m$.
(3) If $l \parallel m$, then $\alpha \perp \beta$.
(4) If $l \perp m$, then $\alpha \parallel \beta$.
Identify the correct propositions.

Answer: $(3)$


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

The number \( x \) is such that among the four numbers \( x - \sqrt{2} \), \( x - \frac{1}{x} \), \( x + \frac{1}{x} \), \( x^2 + 2 \sqrt{2} \), exactly one is not an integer. Find all such \( x \).

Answer: $\sqrt{2} - 1$


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

Given the sequence $\{a_n\}$, $a_1=1$, $a_{n+1} > a_n$, $(a_n+a_{n+1}-1)^2=4a_na_{n+1}$ $(n\in\mathbb{N}^*)$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Let $b_n=\frac{1}{\sqrt[4]{a_n}}$, $T_n=b_1+b_2+\ldots+b_n$, estimate the integer part of $T_{2017}$.

(Reference data: $1.41 < \sqrt{2} < 1.42$, $44.92 < \sqrt{2018} < 45$)

Answer: $88$


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

Circle inscribed in square  $ABCD$  , is tangent to sides  $AB$  and  $CD$  at points  $M$  and  $K$  respectively. Line  $BK$  intersects this circle at the point  $L, X$  is the midpoint of  $KL$ . Find the angle  $\angle MXK $ .

Answer: $135$


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

Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, point $X$ is selected on the edge $A_1 D_1$, and point $Y$ is selected on the edge $B C$. It is known that $A_1 X = 5$, $B Y = 3$, and $B_1 C_1 = 14$. The plane $C_1 X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.

Answer: $20$


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

In triangle $ABC$, given that $a + c = 2b$, find the value of $\operatorname{tg} \frac{A}{2} \operatorname{tg} \frac{C}{2}$.

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>.

Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point A 10 minutes after the second meeting. Find the ratio of the speeds of the pedestrians.

Answer: $15/8$


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>.

Given two sets $A$ and $B$ that satisfy the conditions $A\cap B \neq \emptyset$ and $A\cup B = \{1, 2, 3, 4, 5\}$. When $A \neq B$, the pair $(A, B)$ and $(B, A)$ are considered as two different pairs of sets. The total number of such set pairs $(A, B)$ that meet the conditions is __________.

Answer: $211$


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

Rachel and Steven play games of chess. If either wins two consecutive games, they are declared the champion. The probability that Rachel will win any given game is 0.6, the probability that Steven will win any given game is 0.3, and the probability that any given game is drawn is 0.1. Find the value of \(1000P\), where \(P\) is the probability that neither is the champion after at most three games.

Answer: $343$


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

The number \( n \) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \( n \)?

Answer: $2891$


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

What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that if any two nodes fail, it still remains possible to transmit information between any two remaining nodes (at least through a chain via other nodes)?

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 the equation of a circle $x^2 + y^2 - 6x - 8y = 0$, if the longest chord AC and the shortest chord BD both pass through the point (-1, 4) on the circle, find the area of the quadrilateral ABCD.

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>.

In triangle \(ABC\), \(AC = 8\) and \(BC = 5\). A line parallel to the bisector of the external angle at \(C\) passes through the midpoint of side \(AB\) and point \(E\) on side \(AC\). Find \(AE\).

Answer: $1.5$


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

Let  $ABCD$  be a quadrilateral with an inscribed circle  $\omega$  and let  $P$  be the intersection of its diagonals  $AC$  and  $BD$ . Let  $R_1$ ,  $R_2$ ,  $R_3$ ,  $R_4$  be the circumradii of triangles  $APB$ ,  $BPC$ ,  $CPD$ ,  $DPA$  respectively. If  $R_1=31$  and  $R_2=24$  and  $R_3=12$ , find  $R_4$ .

Answer: $19$


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

When someone tries to withdraw money from an ATM with a forgotten last digit of the password, calculate the probability that they will press the correct digit in no more than 2 attempts.

Answer: $\dfrac{1}{5}$


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

How many of the natural numbers from 1 to 1000, inclusive, contain the digit 5 at least once?

Answer: $270$


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

There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack?

Answer: $10$


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

Given a triangle \( \triangle ABC \) and a point \( D \) on its side \( BC \) such that \( AB = 2BD \) and \( AC = 3CD \), and an ellipse \( \Gamma \) can be constructed with \( A \) and \( D \) as foci passing through points \( B \) and \( C \), find the eccentricity of \( \Gamma \).

Answer: $\frac{\sqrt{21}}{7}$


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

Let the function $f(x)=\tan \frac {x}{4}\cdot \cos ^{2} \frac {x}{4}-2\cos ^{2}\left( \frac {x}{4}+ \frac {\pi}{12}\right)+1$.

(Ⅰ) Find the domain of $f(x)$ and its smallest positive period;

(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-\pi,0]$.

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


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

On the lateral side \( CD \) of the trapezoid \( ABCD (AD \parallel BC) \), point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped onto segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \) if it is known that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).

Answer: $18$


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

Each of two baskets contains white and black balls such that the total number of balls in both baskets is 25. One ball is randomly drawn from each basket. It is known that the probability that both drawn balls are white is 0.54. Find the probability that both drawn balls are black.

Answer: $0.04$


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

There are 306 different cards with numbers \(3, 19, 3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}\) (each card has exactly one number, and each number appears exactly once). How many ways can you choose 2 cards such that the product of the numbers on the selected cards is a perfect square?

Answer: $17328$


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

A right triangle with legs each measuring 7 cells was drawn on graph paper. Then all the lines of the grid inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing?

Answer: $28$


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

Given real numbers \( x, y, z, w \) such that \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \).

Answer: $3/2$


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>.

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

$$
\left\{
\begin{array}{l}
2 \sin x = \tan y \\
2 \cos y = \cot z \\
\sin z = \tan x
\end{array}
\right.
$$

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

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


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

Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?

Answer: $29112$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 labeled the top with "MAME". Then he unfolded the note, added something else, folded it along the creases in a random manner (not necessarily the same as before), and left it on the table with a random side 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>.

A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-acquaintances not knowing each other. It turned out that in the gathered company, there are neither sociable nor shy people. What is the maximum number of people that can be in this company?

Answer: $40$


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

In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid.

Answer: $4.8$


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>.

Find the area of trapezoid \(ABCD (AD \| BC)\) if its bases are in the ratio \(5:3\), and the area of triangle \(ADM\) is 50, where \(M\) is the point of intersection of lines \(AB\) and \(CD\).

Answer: $32$


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

A cylinder has both its front and left views as rectangles with length 4 and height 3. Calculate the surface area of this cylinder.

Answer: $20\pi$


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>.

Calculate the length of the arc of the astroid given by \(x=\cos^{3} t\), \(y=\sin^{3} t\), where \(0 \leq t \leq 2\pi\).

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>.

The natural number \( a \) is divisible by 21 and has 105 distinct divisors, including 1 and \( a \). Find the smallest such \( a \).

Answer: $254016$


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

A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?

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>.

The segment connecting the centers of two intersecting circles is divided by their common chord into segments of 4 and 1. Find the length of the common chord, given that the radii of the circles are in the ratio $3:2$.

Answer: $2 \sqrt{11}$


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>.

Find the smallest positive integer $m$ such that for all positive integers $n \geq m$, there exists a positive integer $l$ satisfying
$$
n < l^2 < \left(1+\frac{1}{2009}\right)n.
$$

Answer: $16144325$


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

Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors.

Answer: $11250$


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

If the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ .

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>.

The height of an isosceles trapezoid is \( h \). The upper base of the trapezoid is viewed from the midpoint of the lower base at an angle of \( 2\alpha \), and the lower base is viewed from the midpoint of the upper base at an angle of \( 2\beta \). Find the area of the trapezoid in general, and calculate it without tables if \( h = 2 \), \( \alpha = 15^\circ \), and \( \beta = 75^\circ \).

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>.

A workshop produces transformers of types $A$ and $B$. One transformer of type $A$ uses 5 kg of transformer iron and 3 kg of wire, while one transformer of type $B$ uses 3 kg of iron and 2 kg of wire. The profit from selling one transformer of type $A$ is 12 thousand rubles, and for type $B$ it is 10 thousand rubles. The shift's iron inventory is 481 kg, and the wire inventory is 301 kg. How many transformers of types $A$ and $B$ should be produced per shift to obtain the maximum profit from sales, given that the resource usage does not exceed the allocated shift inventories? What will be the maximum profit?

Answer: $1502$


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

Find the difference between the greatest and least values of  $lcm (a,b,c)$ , where  $a$ ,  $b$ , and  $c$  are distinct positive integers between  $1$  and  $10$ , inclusive.

Answer: $500$


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

In $\Delta ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $b=6$, and the area of $\Delta ABC$ is $15$. The radius of the circumcircle is $5$.

(Ⅰ) Find the value of $\sin 2B$;

(Ⅱ) Find the perimeter of $\Delta ABC$.

Answer: $6+6 \sqrt {6}$


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>.

Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle.

Answer: $\sqrt{1105}$


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

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

Answer: $12$


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

The Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: take one pill daily. The pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect—absent-mindedness disappears, if present.

Another doctor prescribed the Scientist pills for absent-mindedness—also one per day for 10 consecutive days. These pills cure absent-mindedness in $80\%$ of cases, but in $5\%$ of cases, there is a side effect—the knee stops hurting.

The bottles with the pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but didn't pay attention to which one. For ten days, he took one pill per day and returned completely healthy. Both the absent-mindedness and the knee pain were gone. Find the probability that the Scientist took pills for absent-mindedness.

Answer: $0.69$


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

Yesterday, Sasha cooked soup and added too little salt, so he had to add more salt later. Today he doubled the amount of salt, but still had to add more salt, though only half as much as yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any more salt?

(Each day, Sasha cooks the same amount of soup.)

Answer: $1.5$


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

Given three composite numbers \( A, B, C \) that are pairwise coprime and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)?

Answer: $1626$


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

What weights can be measured using a balance scale with weights of $1, 3, 9, 27$ grams? Generalize the problem!

Answer: $40$


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

Given that \( a, b, \) and \( c \) are all positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If both \( A \) and \( B \) are less than 1 unit away from the origin, what is the minimum value of \( a + b + c \)?

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>.

There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers?

Answer: $10890$


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

Let  $G$  be the centroid of triangle  $ABC$  with  $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from  $G$  to the three sides of the triangle.

Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.

*2019 CCA Math Bonanza Individual Round #11*

Answer: $\frac{2348}{195}$


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

Mrs. Kučerová was on a seven-day vacation, and Káta walked her dog and fed her rabbits during this time. Káta received a large cake and 700 CZK as compensation. After another vacation, this time lasting four days, Káta received the same cake and 340 CZK for the same tasks.

What was the cost of the cake?

Answer: $140$


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

How many employees of the enterprise can be insured by the given insurance company if the enterprise is willing to pay an insurance premium of 5,000,000 rubles?

The cost of service for one person in the clinic (without hospitalization), with which the insurance company has a contract, averages 18,000 rubles. The average cost of treatment for one insured person in case of hospitalization is 60,000 rubles (the proportion of hospitalized individuals is 25% of the insured number). Overhead costs of the insurance company for insuring one person are on average 2,800 rubles. The profit of the insurance company is 15% of the cost price.

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>.

A weight of 200 grams rests on a table. It was flipped and placed on the table on its other side, the area of which is 15 sq. cm smaller. As a result, the pressure on the table increased by 1200 Pa. Find the area of the side on which the weight originally rested. Give your answer in sq. cm, rounded to two decimal places if necessary.

Answer: $24.85$


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

In a certain company, there are 100 shareholders, and any 66 of them own no less than 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own?

Answer: $25$


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

How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0
$$

have, where \( f(x) = |x| - 1 \)?

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 \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \).

Answer: $27$


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

In the AU tribe's language, there are two letters - "a" and "u". Certain sequences of these letters form words, where each word contains no fewer than one and no more than 13 letters. It is known that if any two words are written consecutively, the resulting sequence will not be a word. Find the maximum possible number of words in such a language.

Answer: $16256$


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

On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$?

Answer: $1221$


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

There are  $2024$  cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest  $k$  for which the tourist will always be able to visit at least  $k$  cities? During the route, he can return to the cities he has already visited.

*Proposed by Bogdan Rublov*

Answer: $1012$


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

In \(\triangle ABC\), \(BC = a\), \(CA = b\), \(AB = c\). If \(2a^{2} + b^{2} + c^{2} = 4\), then the maximum area of \(\triangle ABC\) is ______.

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


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

Given that there is 1 path from point A to the first red arrow, 2 paths from the first red arrow to the second red arrow, 3 paths from the first red arrow to each of the first two blue arrows, 4 paths from the second red arrow to each of the first two blue arrows, 5 paths from each of the first two blue arrows to each of the first two green arrows, 6 paths from each of the third and fourth blue arrows to each of the first two green arrows, 3 paths from each green arrow to point B, and 4 paths from each green arrow to point C, calculate the total number of distinct paths to reach points B and C.

Answer: $4312$


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

Given the function $y=\sin x$ and $y=\sin (2x+ \frac {\pi}{3})$, determine the horizontal shift required to transform the graph of the function $y=\sin x$ into the graph of the function $y=\sin (2x+ \frac {\pi}{3})$.

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


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

A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amount of money she spent.

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>.

The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle.

Answer: $6 \sqrt[4]{3}$


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 second year of junior high school this semester, a basketball game was held. In order to better train the participating athletes, the sports department plans to purchase two brands of basketballs, brand A and brand B. It is known that the unit price of brand A basketball is $40$ yuan lower than the unit price of brand B basketball. The quantity of brand A basketball purchased with $4800$ yuan is $\frac{3}{2}$ times the quantity of brand B basketball purchased with $4000$ yuan.
$(1)$ Find the unit prices of brand A and brand B basketballs.
$(2)$ If the school plans to purchase a total of $90$ brand A and brand B basketballs, and the quantity of brand B basketballs is at least twice the quantity of brand A basketballs, and the total cost of purchasing both brands of basketballs does not exceed $17200$ yuan. How many purchasing plans does the school have?
$(3)$ Under the conditions of $(2)$, the store is preparing to offer a discount on brand B basketballs, with a discount of $a$ yuan $\left(30 \lt a \lt 50\right)$ per brand B basketball. The price of brand A basketball remains unchanged. Which purchasing plan should the school adopt to minimize the total cost?

Answer: $m = 20$


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 circle is divided into 30 equal parts by 30 points on the circle. Randomly selecting 3 different points, what is the probability that these 3 points form an equilateral triangle?

Answer: $1/406$


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

The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other.

Answer: $2 \sqrt{3}$


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

Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \).

Answer: $1475$


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

Given that the two trisection points on the minor axis of an ellipse and its two foci form a square, find the eccentricity of the ellipse.

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


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>.

To walk 2 km, ride 3 km by bicycle, and drive 20 km by car, Uncle Vanya needs 1 hour and 6 minutes. If he needs to walk 5 km, ride 8 km by bicycle, and drive 30 km by car, it takes him 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 4 km, ride 5 km by bicycle, and drive 80 km by car?

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 two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$

If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).

Answer: $2 + \sqrt{2}$


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

Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 being rational and the remaining 50 being irrational. Then, in each cell of the table, he recorded the sum of the numbers written next to its row and column ("addition table"). What is the maximum number of sums in this table that could be rational numbers?

Answer: $1250$


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

Suppose $b$ is an integer such that $1 \le b \le 30$, and $524123_{81}-b$ is a multiple of $17$. What is $b$?

Answer: $11$


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

The positions of cyclists in the race are determined by the total time across all stages: the first place goes to the cyclist with the shortest total time, and the last place goes to the cyclist with the longest total time. There were 500 cyclists, the race consisted of 15 stages, and no cyclists had the same times either on individual stages or in total across all stages. Vasya finished in seventh place every time. What is the lowest position (i.e., position with the highest number) he could have taken?

Answer: $91$


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

Write the four numbers \(1, 9, 8, 8\) in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, then write the difference between these two numbers in between them. This completes one operation. Then, perform the same operation on the new series of 7 numbers. Continue doing this a total of 100 times. Find the sum of the numbers in the final row.

Answer: $726$


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

There are 1000 toy bears in a row. The bears come in three colors: white, brown, and black. Among any three consecutive bears, there is at least one bear of each color. Iskander tries to guess the colors of the bears. He made five guesses:

- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
- The 400th bear from the left is black;
- The 600th bear from the left is brown;
- The 800th bear from the left is white.

It turns out that exactly one of his guesses is wrong. What could be the number of the bear whose color Iskander did NOT guess correctly? Select all possible answers.

Answer: $20$


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

Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right triangle with sides of length $5$, $12$, and $13$ so that one side of the square lies on the hypotenuse of the triangle, find the value of $\frac{x}{y}$.

Answer: $\frac{39}{51}$


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

In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done?

Answer: $5184$


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

In which acute-angled triangle is the value of the product \(\operatorname{tg} \alpha \cdot \operatorname{tg} \beta \cdot \operatorname{tg} \gamma\) minimized?

Answer: $\sqrt{27}$


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

ABCDEF is a six-digit number. All of its digits are different and arranged in ascending order from left to right. This number is a perfect square.

Determine what this number is.

Answer: $134689$


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

Write in ascending order the multiples of 3 which, when 1 is added, are perfect squares, i.e., $3, 15, 24, 48, \ldots$ What is the multiple of 3 in the $2006^{\mathrm{th}}$ position?

Answer: $9060099$


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

The function \( f(x) = \begin{cases}
x + 2, & x \leq -1, \\
x^2, & -1 < x < 2, \\
2x, & x \geq 2
\end{cases} \), if \( f(x) = 3 \), then \( x = \)

Answer: $\sqrt{3}$


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

Given point $P(0,-2)$, and an ellipse $E$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$ with eccentricity $\dfrac{\sqrt{2}}{2}$, let $F$ be the right focus of ellipse $E$, and the slope of line $PF$ is $2$, with $O$ being the origin.
$(1)$ Find the equation of ellipse $E$;
$(2)$ A line $l$ intersects the circle $O$: $x^{2}+y^{2}=3$ and forms a chord of length $3$, and intersects ellipse $E$ at points $A$ and $B$. Find the maximum area of $\triangle AOB$.

Answer: $\dfrac{\sqrt{2}}{2}$


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

Given a natural number \( a \), let \( S(a) \) represent the sum of its digits (for example, \( S(123) = 1 + 2 + 3 = 6 \) ). If a natural number \( n \) has all distinct digits, and \( S(3n) = 3S(n) \), what is the maximum value of \( n \)?

Answer: $3210$


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

Let  $x, y$  and  $z$  be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of  $x + y + z$ .

Answer: $402$


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

Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ.

Answer: $32$


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

As usual, let \( n! \) denote the product of the integers from 1 to \( n \) inclusive. Determine the largest integer \( m \) such that \( m! \) divides \( 100! + 99! + 98! \).

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>.

A convex hexagon \( A_{1} A_{2} \ldots A_{6} \) is circumscribed around a circle \( \omega \) with a radius of 1. Consider three segments that connect the midpoints of the opposite sides of the hexagon. What is the greatest \( r \) for which it can be stated that at least one of these segments is not shorter than \( r \)?

Answer: $\sqrt{3}$


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

Suppose \( N \) is a 6-digit number having base-10 representation \( \underline{a} \underline{b} \underline{c} \underline{d} \underline{e} \underline{f} \). If \( N \) is \( \frac{6}{7} \) of the number having base-10 representation \( \underline{d} \underline{e} \underline{f} \underline{a} \underline{b} \underline{c} \), find \( N \).

Answer: $461538$


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

In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can be repeated) so that the resulting 12-digit number is divisible by 75. How many ways can this be done?

Answer: $2592$


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

Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron?

Answer: $\frac{\sqrt{2}}{12}$


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>.

John rode his bicycle for 45 minutes at a speed of 20 mph, then walked for 120 minutes at a speed of 3 mph. Calculate the overall average speed in miles per hour.

Answer: $7.64$


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

Let \(Q\) be a point chosen uniformly at random inside the unit square with vertices at \((0,0), (1,0), (1,1)\), and \((0,1)\). Calculate the probability that the slope of the line determined by \(Q\) and the point \(\left(\frac{1}{4}, \frac{3}{4}\right)\) is greater than or equal to 1.

Answer: $\frac{1}{8}$


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>.

In how many ways can 8 identical rooks be placed on an $8 \times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square?

Answer: $139448$


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

Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \).

Answer: $999$


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>.

The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=$ $\qquad$ .

Answer: $46$


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

Xiao Ming arrives at the departure station between 7:50 and 8:30 to catch the high-speed train departing at 7:00, 8:00, or 8:30. Calculate the probability that his waiting time does not exceed 10 minutes.

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>.

Start with an angle of \(60^\circ\) and bisect it, then bisect the lower \(30^\circ\) angle, then the upper \(15^\circ\) angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original \(60^\circ\) angle into two angles. Find the measure (degrees) of the smaller angle.

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>.

Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and heads towards point B. Half an hour later, it meets Vehicle B again. How many hours does it take for Vehicle B to travel from A to B?

Answer: $7.2$


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>.

An alarm clock gains 9 minutes each day. When going to bed at 22:00, the precise current time is set on the clock. At what time should the alarm be set so that it rings exactly at 6:00? Explain your answer.

Answer: $6:03$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = 1$ and $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the projection of the vector $2\overrightarrow{b} - \overrightarrow{a}$ in the direction of vector $\overrightarrow{a}$.

Answer: $-1$


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

Point \( M \) lies on side \( BC \) of parallelogram \( ABCD \) with a \(45^{\circ}\) angle at vertex \( A \), such that \(\angle AMD = 90^{\circ}\) and the ratio \( BM : MC = 2 : 3 \). Find the ratio of the adjacent sides of the parallelogram.

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


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

Three students  $A, B$  and  $C$  are traveling from a location on the National Highway No. $5$  on direction to Hanoi for participating the HOMC  $2018$ . At beginning,  $A$  takes  $B$  on the motocycle, and at the same time  $C$  rides the bicycle. After one hour and a half,  $B$  switches to a bicycle and immediately continues the trip to Hanoi, while  $A$  returns to pick up  $C$ . Upon meeting,  $C$  continues the travel on the motocycle to Hanoi with  $A$ . Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is  $50$  km per hour and of the both bicycles are  $10$  km per hour. Find the distance from the starting point to Hanoi.

Answer: $100$


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

In a certain region of the planet, seismic activity was studied. 80 percent of all days were quiet. The predictions of the devices promised a calm environment in 64 out of 100 cases, and in 70 percent of all cases where the day was calm, the predictions of the devices came true. What percentage of days with increased seismic activity are those in which the predictions did not match reality?

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 $AC$ and $CE$ be two diagonals of a regular hexagon $ABCDEF$. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $AM:AC = CN:CE = r$. If $B$, $M$, and $N$ are collinear, find $r$.

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


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

Each face of a rectangular prism is painted with a single narrow stripe from one vertex to the diagonally opposite vertex. The choice of the vertex pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the rectangular prism?
A) $\frac{1}{8}$
B) $\frac{3}{16}$
C) $\frac{1}{4}$
D) $\frac{3}{8}$
E) $\frac{1}{2}$

Answer: $\frac{3}{16}$


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>.

Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.

Answer: $60718293$


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

Given that points $A$ and $B$ lie on the curves $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively, what is the minimum value of the distance $|AB|$?

Answer: $\frac{3\sqrt{2}}{4}$


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

A bus arrives randomly between 3:30 pm and 4:30 pm, waits for 40 minutes, and then departs. If Sara also arrives randomly between 3:30 pm and 4:30 pm, what is the probability that the bus will still be there when she arrives?

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>.

Let $M$ be the number of positive integers that are less than or equal to $2048$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.

Answer: $24$


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

A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period?

Answer: $43$


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

The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the platform? The speed of sound in the air is \( c = 340 \) m/s.

Answer: $4.12$


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>.

How many positive perfect cubes are divisors of the product \(1! \cdot 2! \cdot 3! \cdots 10!\)?

Answer: $468$


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

Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position.

Answer: $504$


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

Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function.

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>.

What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits?

Answer: $1/203$


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

Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$?
A) $62$
B) $98 - 4\sqrt{94}$
C) $94 - 4\sqrt{94}$
D) $98$
E) $100$

Answer: $98 - 4\sqrt{94}$


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>.

There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them?

Answer: $41/56$


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

The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that can be included in the olympiad?

Answer: $27$


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

Find the largest positive integer \( n \) for which we can find a set of distinct positive integers such that each integer is at most 2002 and if \( a \) and \( b \) are in the set, then \( a^2 \) and \( ab \) are not in the set.

Answer: $1958$