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

Consider the 800-digit integer
$$
234523452345 \cdots 2345 .
$$

The first \( m \) digits and the last \( n \) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \( m+n \).

Answer: $130$


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

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

Answer: $16$


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

The absolute value of a number \( x \) is equal to the distance from 0 to \( x \) along a number line and is written as \( |x| \). For example, \( |8|=8, |-3|=3 \), and \( |0|=0 \). For how many pairs \( (a, b) \) of integers is \( |a|+|b| \leq 10 \)?

Answer: $221$


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

Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond vertex $D$. Line $BE$ intersects segment $AG$ at point $H$. Find the ratio $AH: HG$.

Answer: $1:1$


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

A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of the sixth pyramid?

Answer: $600$


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

In triangle \(ABC\) with sides \(BC=7\), \(AC=5\), and \(AB=3\), an angle bisector \(AD\) is drawn. A circle is circumscribed around triangle \(ABD\), and a circle is inscribed in triangle \(ACD\). Find the product of their radii.

Answer: $35/32$


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>.

Adam, Bendeguz, Cathy, and Dennis all see a positive integer  $n$ . Adam says, " $n$  leaves a remainder of  $2$  when divided by  $3$ ." Bendeguz says, "For some  $k$ ,  $n$  is the sum of the first  $k$  positive integers." Cathy says, "Let  $s$  be the largest perfect square that is less than  $2n$ . Then  $2n - s = 20$ ." Dennis says, "For some  $m$ , if I have  $m$  marbles, there are  $n$  ways to choose two of them." If exactly one of them is lying, what is  $n$ ?

Answer: $210$


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

Let $a$, $b$, and $c$ be positive real numbers such that $abc = 4$. Find the minimum value of
\[(3a + b)(2b + 3c)(ac + 4).\]

Answer: $384$


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

Given the function \( y = \sqrt{a x^2 + b x + c} \) (where \(a, b, c \in \mathbb{R}\) and \(a < 0\)), the domain is \( D \). If the points \( (s, f(t)) \) (where \( s, t \in D \)) form a square, then the real number \( a \) equals ______.

Answer: $-4$


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

How many numbers, divisible by 4 and less than 1000, do not contain any of the digits 6, 7, 8, 9, or 0?

Answer: $31$


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

The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent.

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>.

Let $p$ and $q$ be constants. Suppose that the equation
\[\frac{(x+p)(x+q)(x-15)}{(x-5)^2} = 0\]
has exactly $3$ distinct roots, while the equation
\[\frac{(x-2p)(x-5)(x+10)}{(x+q)(x-15)} = 0\]
has exactly $2$ distinct roots. Compute $100p + q.$

Answer: $240$


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

After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible area of the spots becomes $S_{1}$. However, if the tablecloth is folded in half along the other diagonal, the total visible area of the spots remains $S$. What is the smallest possible value of the ratio $S_{1}: S$?

Answer: $2/3$


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

Solve the system

$$
\left\{\begin{array}{l}
x^{3}+3 y^{3}=11 \\
x^{2} y+x y^{2}=6
\end{array}\right.
$$

Calculate the values of the expression $\frac{x_{k}}{y_{k}}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the smallest among them. If necessary, round your answer to two decimal places.

Answer: $-1.31$


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

The geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1}=1536$, and common ratio $q=-\frac{1}{2}$. Let $\Pi_{n}$ denote the product of its first $n$ terms $\left(n \in \mathbf{N}^{*}\right)$. Find the value of $n$ that maximizes $\Pi_{n}\$.

Answer: $11$


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

What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time?

Answer: $2419$


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

A gives 24 apples to B and C, and each of the three people has at least two apples. How many different ways are there to distribute the apples?

Answer: $190$


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

Given $O$ as the circumcenter of $\triangle ABC$ and $D$ as the midpoint of $BC$. If $\overrightarrow{AO} \cdot \overrightarrow{AD}=4$ and $BC=2 \sqrt{6}$, then find the length of $AD$.

Answer: $\sqrt{2}$


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

In the diagram, a rectangular ceiling \( P Q R S \) measures \( 6 \mathrm{~m} \) by \( 4 \mathrm{~m} \) and is to be completely covered using 12 rectangular tiles, each measuring \( 1 \mathrm{~m} \) by \( 2 \mathrm{~m} \). If there is a beam, \( T U \), that is positioned so that \( P T = S U = 2 \mathrm{~m} \) and that cannot be crossed by any tile, then the number of possible arrangements of tiles is:

Answer: $180$


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

Given an isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \), points \( D \) and \( E \) lie on \( BC \) and \( AC \) respectively. Lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), find \( \angle BED \).

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 \( S = \{1, 2, 3, \ldots, 100\} \). Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S \) contains 4 pairwise coprime numbers.

Answer: $75$


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

If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B originally have?

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 the following image, there is a hexagon $ABEFGD$. Quadrilaterals $ABCD$ and $EFGC$ are congruent rectangles, and quadrilateral $BEGD$ is also a rectangle. Determine the ratio of the areas of the white and shaded parts of the hexagon, given that $|AB| = 5 \text{ cm}$ and triangle $BEC$ is equilateral.

Answer: $2:1$


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>.

In a tetrahedron \(ABCD\), \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\). The areas of \(\triangle ADB\), \(\triangle BDC\), and \(\triangle CDA\) are \(\frac{\sqrt{3}}{2}\), \(2\), and \(1\) respectively. What is the volume of the tetrahedron?

Answer: $\frac{2\sqrt{6}}{9}$


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

The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, and the service alternates every two balls. The winner of a game is the first side to reach 11 points with a lead of at least 2 points. If the score is tied at 10-10, the service order remains the same, but the service alternates after each point until one side wins by a margin of 2 points. In a singles table tennis match between players A and B, assuming player A serves first, the probability of player A scoring when serving is $\frac{2}{3}$, and the probability of player A scoring when player B serves is $\frac{1}{2}$. The outcomes of each ball are independent.

$(1)$ Find the probability that player A scores 3 points after the first 4 balls in a game.

$(2)$ If the game is tied at 10-10, and the match ends after X additional balls are played, find the probability of the event "X ≤ 4."

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


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

What is the greatest number of consecutive non-negative integers whose sum is $120$?

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>.

Square $ABCD$ is constructed along diameter $AB$ of a semicircle, where both the square and semicircle are coplanar. Line segment $AB$ has a length of 8 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MD$?

Answer: $4\sqrt{10}$


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>.

Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\).

Answer: $110$


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

Given the set $I=\{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$ such that the smallest number in $B$ is greater than the largest number in $A$. The number of different ways to choose such subsets $A$ and $B$ is ______.

Answer: $49$


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

With four standard six-sided dice in play, Vivian rolls all four and can choose to reroll any subset of them. To win, Vivian needs the sum of the four dice after possibly rerolling some of them to be exactly 12. Vivian plays optimally to maximize her chances of winning. What is the probability that she chooses to reroll exactly three of the dice?
**A)** $\frac{1}{72}$
**B)** $\frac{1}{12}$
**C)** $\frac{1}{10}$
**D)** $\frac{1}{8}$
**E)** $\frac{1}{6}$

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>.

Compute the number of digits is  $2015!$ . Your score will be given by  $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$ , where  $A$  is your answer and  $C$  is the actual answer.

Answer: $5787$


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

A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron.

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>.

Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles?

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 a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.
$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expectation of the random variable $X$;
$(2)$ If one ball is drawn without replacement each time, and the color is recorded before putting it back into the bag, the process continues until two red balls are drawn, with the number of draws denoted as $Y$, find the probability of $Y=4$.

Answer: $\frac{108}{625}$


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

On the sides \(AB\) and \(AD\) of square \(ABCD\), points \(E\) and \(F\) are marked such that \(BE : EA = AF : FD = 2022 : 2023\). Segments \(EC\) and \(FC\) intersect the diagonal \(BD\) of the square at points \(G\) and \(H\), respectively. Find the ratio \(GH : BD\).

Answer: $\frac{12271519}{36814556}$


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

Positive integers $a$, $b$, $c$, and $d$ are such that $a<b<c<d$, and the system of equations
\[ 2x + y = 2007 \quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| \]
has exactly one solution. What is the minimum value of $d$?

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>.

For which values of \( x \) and \( y \) the number \(\overline{x x y y}\) is a square of a natural number?

Answer: $7744$


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

A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+2k-99\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.

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>.

In a class that includes Petya and Vanya, there are 31 students. In how many ways can a football team be selected from the class?

Answer: $2 \binom{29}{10} + \binom{29}{9}$


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

If \(a\) copies of a right-angled isosceles triangle with hypotenuse \(\sqrt{2} \, \mathrm{cm}\) can be assembled to form a trapezium with perimeter equal to \(b \, \mathrm{cm}\), find the least possible value of \(b\). (Give the answer in surd form.)

Answer: $4 + 2\sqrt{2}$


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>.

Let $A$ and $B$ be two subsets of the set $\{1,2, \cdots, 20\}$ such that $A \cap B = \varnothing$, and if $n \in A$, then $2n + 2 \in B$. Let $M(A)$ be the sum of the elements in $A$. Find the maximum value of $M(A)$.

Answer: $39$


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

Let $A$ be a point on the circle $x^2 + y^2 + 4x - 4y + 4 = 0$, and let $B$ be a point on the parabola $y^2 = 8x$. Find the smallest possible distance $AB$.

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


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

Let \( a \) and \( b \) be positive integers. The quotient of \( a^{2} + b^{2} \) divided by \( a + b \) is \( q \), and the remainder is \( r \), such that \( q^{2} + r = 2010 \). Find the value of \( ab \).

Answer: $1643$


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

What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)?

Answer: $241$


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

Two regular tetrahedrons  $A$  and  $B$  are made with the 8 vertices of a unit cube. (this way is unique)

What's the volume of  $A\cup B$ ?

Answer: $1/2$


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

Let $A(2,0)$ be a fixed point in the plane, and let $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ be a moving point. Find the area swept by the line segment $AP$ as $t$ changes from $15^{\circ}$ to $45^{\circ}$.

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>.

In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?

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>.

According to national regulations, only adults aged between 18 and 70 are eligible to apply for a motor vehicle driver's license. A sixth-grade student, Li Ming, says, "My dad has a driver's license. His age equals the product of the month and day of his birth, and that product is 2975." How old is Li Ming's father?

Answer: $35$


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

On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links?

Answer: $41$


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

Given the set
$$
T=\left\{n \mid n=5^{a}+5^{b}, 0 \leqslant a \leqslant b \leqslant 30, a, b \in \mathbf{Z}\right\},
$$
if a number is randomly selected from set $T$, what is the probability that the number is a multiple of 9?

Answer: $5/31$


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

\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there?

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>.

What is the maximum number of cells in an $8 \times 8$ square that can be colored such that the centers of any four colored cells do not form the vertices of a rectangle with sides parallel to the edges of the square?

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>.

The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \leqslant x \leqslant 4, 1 \leqslant y \leqslant 4$, and where $x$ and $y$ are integers is $\qquad$ .

Answer: $516$


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

Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.)

Answer: $9608$


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

From the 200 natural numbers from 1 to 200, how many numbers must be selected to ensure that there are at least 2 numbers whose sum is a multiple of 5?

Answer: $82$


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

The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits?

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 \(ABCD\) be a square of side length 1. \(P\) and \(Q\) are two points on the plane such that \(Q\) is the circumcentre of \(\triangle BPC\) and \(D\) is the circumcentre of \(\triangle PQA\). Find the largest possible value of \(PQ^2\). Express the answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\) and \(b\) are rational numbers.

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>.

At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site exceeded that at the second site by 6 units, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment.

Answer: $17$


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

In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probability that the distance between them after an hour will be more than 6 km.

Answer: $0.375$


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

In a football championship, 16 teams participated. A team receives 2 points for a win; in case of a draw in regular time, both teams shoot penalty kicks, and the team that scores more goals receives one point. After 16 rounds, all teams have accumulated a total of 222 points. How many matches ended in a draw in regular time?

Answer: $34$


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

Given the line $l: \sqrt{3}x-y-4=0$, calculate the slope angle of line $l$.

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


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

Find the sum of all divisors \(d=2^a \cdot 3^b\) (where \(a, b > 0\)) of \(N=19^{88}-1\).

Answer: $744$


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

How many kings can be placed on an $8 \times 8$ chessboard without putting each other in check?

Answer: $16$


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

Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$

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 sum of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ is denoted as $a$. If the sum of the digits of $a$ is 50, then what is the smallest value of $x$?

Answer: $99950$


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

Dylan has a \( 100 \times 100 \) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?

Answer: $9999$


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

During the manufacture of a steel cable, it was found that the cable has the same length as the curve given by the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=14
\end{array}\right.
$$

Find the length of the cable.

Answer: $4\pi \sqrt{\frac{11}{3}}$


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 angle between the slant height of a cone and the base plane is $30^{\circ}$. The lateral surface area of the cone is $3 \pi \sqrt{3}$ square units. Determine the volume of a regular hexagonal pyramid inscribed in the cone.

Answer: $\frac{27 \sqrt{2}}{8}$


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

Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A?

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>.

Let \( a \in \mathbf{R} \). A complex number is given by \(\omega = 1 + a\mathrm{i}\). A complex number \( z \) satisfies \( \overline{\omega} z - \omega = 0 \). Determine the value of \( a \) such that \(|z^2 - z + 2|\) is minimized, and find this minimum value.

Answer: $\frac{\sqrt{14}}{4}$


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

Let \( f(x) \) be a function defined on \( \mathbf{R} \) such that for any real number \( x \), \( f(x+3) f(x-4) = -1 \). When \( 0 \leq x < 7 \), \( f(x) = \log_{2}(9-x) \). Find the value of \( f(-100) \).

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


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

The integer \( n \) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \( \frac{n}{3} \). Find the maximum value of the integer \( n \).

Answer: $162$


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

There are 16 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?

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>.

Consider a rectangle \( ABCD \) where the side lengths are \( \overline{AB}=4 \) and \( \overline{BC}=8 \). Points \( M \) and \( N \) are fixed on sides \( BC \) and \( AD \), respectively, such that the quadrilateral \( BMDN \) is a rhombus. Calculate the area of this rhombus.

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 that the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x-1)=f(x+1)$ for all $x \in \mathbb{R}$. When $x \in (0,1]$ and $x_1 \neq x_2$, we have $\frac{f(x_2) - f(x_1)}{x_2 - x_1} < 0$. Determine the correct statement(s) among the following:

(1) $f(1)=0$

(2) $f(x)$ has 5 zeros in $[-2,2]$

(3) The point $(2014,0)$ is a symmetric center of the function $y=f(x)$

(4) The line $x=2014$ is a symmetry axis of the function $y=f(x)$

Answer: $(1) (2) (3)$


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>.

8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the second smallest number?

Answer: $26$


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

A force of $60 \mathrm{H}$ stretches a spring by 2 cm. The initial length of the spring is $14 \mathrm{~cm}$. How much work is required to stretch it to 20 cm?

Answer: $5.4$


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 positive integer \( n \) is said to be 'good' if \( n^2 - 1 \) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers.

Answer: $104$


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

There are 700 cards in a box, in six colors: red, orange, yellow, green, blue, and white. The ratio of the number of red, orange, and yellow cards is $1: 3: 4$, and the ratio of the number of green, blue, and white cards is $3:1:6$. Given that there are 50 more yellow cards than blue cards, determine the minimum number of cards that must be drawn to ensure that there are at least 60 cards of the same color among the drawn cards.

Answer: $312$


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

Given the function $f(x) = x^3 - 3x^2 - 9x + 1$, find the intervals of monotonicity and the extrema of $f(x)$.

Answer: $-26$


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

The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02.

Answer: $0.7698$


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

\( 427 \div 2.68 \times 16 \times 26.8 \div 42.7 \times 16 \)

Answer: $25600$


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

A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain.

Answer: $50.5$


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

A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result?

Answer: $14$


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

In an isosceles triangle \( ABC \), the bisectors \( AD, BE, CF \) are drawn.

Find \( BC \), given that \( AB = AC = 1 \), and the vertex \( A \) lies on the circle passing through the points \( D, E, \) and \( F \).

Answer: $\frac{\sqrt{17} - 1}{2}$


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

The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class.

Answer: $33$


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

In triangle \( \triangle ABC \), the angles are \( \angle B = 30^\circ \) and \( \angle A = 90^\circ \). Point \( K \) is marked on side \( AC \), and points \( L \) and \( M \) are marked on side \( BC \) such that \( KL = KM \) (point \( L \) lies on segment \( BM \)).

Find the length of segment \( LM \), given that \( AK = 4 \), \( BL = 31 \), and \( MC = 3 \).

Answer: $14$


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

In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\triangle CDE$ is $22,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$

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>.

Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter?

Answer: $120$


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

Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ common elements.

Answer: $200$


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

In trapezoid \(ABCD\), the angles \(A\) and \(D\) at the base \(AD\) are \(60^{\circ}\) and \(30^{\circ}\) respectively. Point \(N\) lies on the base \(BC\) such that \(BN : NC = 2\). Point \(M\) lies on the base \(AD\), the line \(MN\) is perpendicular to the bases of the trapezoid and divides its area in half. Find the ratio \(AM : MD\).

Answer: $3:4$


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

Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options.

Answer: $2020$


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

In each square of an  $11\times 11$  board, we are to write one of the numbers  $-1$ ,  $0$ , or  $1$  in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

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>.

In Chinese ancient music, the pentatonic scale is the basic scale, hence the idiom "五音不全" (meaning "lacking in musical talent"). The five notes of the pentatonic scale in Chinese ancient music are: 宫 (gong), 商 (shang), 角 (jue), 徵 (zhi), 羽 (yu). If these five notes are all used and arranged into a sequence of five notes, with the condition that 宫 (gong), 角 (jue), and 羽 (yu) cannot be adjacent to each other, the number of different possible sequences is ______.

Answer: $84$


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

Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from the 11th list. How many draws were there?

Answer: $54$


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

Suppose that  $ABCD$  is a rectangle with sides of length  $12$  and  $18$ . Let  $S$  be the region of points contained in  $ABCD$  which are closer to the center of the rectangle than to any of its vertices. Find the area of  $S$ .

Answer: $54$


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

The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)).

Answer: $106$


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

Find the maximum value of the function

$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$

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


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>.

What is the probability that two people, A and B, randomly choosing their rooms among 6 different rooms in a family hotel, which has two rooms on each of the three floors, will stay in two rooms on the same floor?

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


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>.

The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coincides with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 square km more than the sum of the areas of the fields. Find the total area of the forest and fields in square km.

Answer: $135$


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

Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([2, 4]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{5-\sqrt{11}}{2}\right)) \ldots) \). Round your answer to the nearest hundredth if necessary.

Answer: $4.16$


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

In the following diagram, \(ABCD\) is a square, and \(E\) is the center of the square \(ABCD\). \(P\) is a point on a semi-circle with diameter \(AB\). \(Q\) is a point on a semi-circle with diameter \(AD\). Moreover, \(Q, A,\) and \(P\) are collinear (that is, they are on the same line). Suppose \(QA = 14 \text{ cm}\), \(AP = 46 \text{ cm}\), and \(AE = x \text{ cm}\). Find the value of \(x\).

Answer: $34$


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

Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{7}.\]
Find $\mathbf{a} \cdot \mathbf{d}$.

Answer: $-\frac{19}{21}$


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

Given the function $f(x) = e^x \cos x - x$.

(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;

(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$.

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


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

Find a four-digit number that is a perfect square, in which its digits can be grouped into two pairs of equal digits.

Answer: $7744$


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

$ABCDEFGH$ is a cube having a side length of 2. $P$ is the midpoint of $EF$, as shown. The area of $\triangle APB$ is:

Answer: $\sqrt{8}$


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

Given that $|\vec{a}|=1$, $|\vec{b}|=\sqrt{2}$, $\vec{c}=\vec{a}+\vec{b}$, and $\vec{c} \perp \vec{a}$, the angle between vectors $\vec{a}$ and $\vec{b}$ is $\_\_\_\_\_\_\_\_\_.$

Answer: $\frac{3\pi}{4}$


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

How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7?

Answer: $4571$


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

Points \( M \) and \( N \) are the midpoints of sides \( BC \) and \( AD \) of quadrilateral \( ABCD \). It is known that \(\angle B = 150^\circ\), \(\angle C = 90^\circ\), and \(AB = CD\). Find the angle between the lines \(MN\) and \(BC\).

Answer: $60$


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

Find all four-digit numbers that are 9 times larger than their reversed counterparts.

Answer: $9801$


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

A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the wall to the ground in meters, or $\qquad$ meters.

Answer: $16 - 4\sqrt{2}$


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>.

Two circular tracks $\alpha$ and $\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\alpha$ and a car $B$ travels counterclockwise on track $\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\alpha$, and this line is tangent to track $\beta$. After the start, the cars begin to approach the point of tangency of the tracks. Each car completes one full lap on its track in one hour (and never switches to the other track). For how much time during this hour will the distance between the cars be at least the diameter of each track?

Answer: $1/2$


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

Find the minimum value of the constant \(\lambda\) such that for all positive integers \(n\) and any positive real numbers \(x_{k}\) \((1 \leq k \leq n)\) summing to 1, the following inequality holds:
$$
\lambda \prod_{k=1}^{n}\left(1-x_{k}\right) \geq 1-\sum_{k=1}^{n} x_{k}^{2}.
$$

Answer: $e$


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

Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum?

Answer: $\frac{500}{1001}$


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

In triangle \(ABC\), it is known that \(AB = 3\), \(AC = 3\sqrt{7}\), and \(\angle ABC = 60^\circ\). The bisector of angle \(ABC\) is extended to intersect at point \(D\) with the circle circumscribed around the triangle. Find \(BD\).

Answer: $4\sqrt{3}$


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

In the Cartesian coordinate system, define $d(P, Q) = |x_1 - x_2| + |y_1 - y_2|$ as the "polyline distance" between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Then, the minimum "polyline distance" between a point on the circle $x^2 + y^2 = 1$ and a point on the line $2x + y - 2 \sqrt{5} = 0$ is __________.

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


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>.

Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$.

Answer: $8\sqrt{2}-10$


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>.

In triangle \(ABC\), the side lengths are 4, 5, and \(\sqrt{17}\). Find the area of the region consisting of those and only those points \(X\) inside triangle \(ABC\) for which the condition \(XA^{2} + XB^{2} + XC^{2} \leq 21\) is satisfied.

Answer: $\frac{5 \pi}{9}$


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

Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \).

Answer: $137$


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

On a number line, there are three points A, B, and C which represent the numbers -24, -10, and 10, respectively. Two electronic ants, named Alpha and Beta, start moving towards each other from points A and C, respectively. Alpha moves at a speed of 4 units per second, while Beta moves at a speed of 6 units per second.

(1) At which point on the number line do Alpha and Beta meet?
(2) After how many seconds will the sum of Alpha's distances to points A, B, and C be 40 units? If at that moment Alpha turns back, will Alpha and Beta meet again on the number line? If they can meet, find the meeting point; if they cannot, explain why.

Answer: $-44$


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

Let \( f(n) \) be the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Find the value of
\[ f(1) + f(2) + f(3) + \ldots + f(99999) \]

Answer: $38889$


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

In the center of a circular field, there is a geologists' house. Six straight roads radiate from it, dividing the field into six equal sectors. Two geologists set out on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that the distance between them will be more than 8 km after one hour.

Answer: $0.5$


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

Two ferries leave simultaneously from opposite shores of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest shore. Upon reaching the shore, they immediately depart back. On the return trip, they meet 400 meters from the other shore. What is the width of the river?

Answer: $1280$


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

A chest of gold coins is divided among 10 pirates where the kth pirate takes k/10 of the remaining coins. Determine the smallest number of coins initially in the chest such that each pirate gets a positive whole number of coins and find the number of coins the 10th pirate receives.

Answer: $362880$