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

Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the 2017 integers that began in the list, how many are now missing?

Answer: $373$


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

In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the table?

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>.

Find all values of \( a \) for which the system

\[
\left\{
\begin{array}{l}
x^{2} + 4y^{2} = 1 \\
x + 2y = a
\end{array}
\right.
\]

has a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0.

Answer: $-1.41$


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>.

Let \( S = \{1, 2, \cdots, 2005\} \). If every subset of \( S \) with \( n \) pairwise coprime numbers always contains at least one prime number, find the minimum value of \( n \).

Answer: $16$


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

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

Answer: $18$


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

In a certain class, with any distribution of 200 candies, there will be at least two students who receive the same number of candies (possibly none). What is the minimum number of students in such a class?

Answer: $21$


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

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

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


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

A quadrilateral \(ABCD\) is inscribed in a circle with a diameter of 1, where \(\angle D\) is a right angle and \(AB = BC\).

Find the area of quadrilateral \(ABCD\) if its perimeter is \(\frac{9\sqrt{2}}{5}\).

Answer: $\frac{8}{25}$


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

Given that points $ D$  and  $ E$  lie on $ \overline{BC}$  and $ \overline{AC}$  respectively, if $ \overline{AD}$  and  $ \overline{BE}$  intersect at $ T$  so that $ AT/DT \equal{} 3$  and $ BT/ET \equal{} 4$, calculate the value of $ CD/BD$.

Answer: $\frac{4}{11}$


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>.

Consider the cards $A, 2, \cdots, J, Q, K$ as the numbers $1, 2, \cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two numbers is a perfect square?

Answer: $2/65$


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

What is the greatest divisor of 360 that is smaller than 60 and also a factor of 90?

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>.

Calculate the arc lengths of the curves given by the equations in the rectangular coordinate system.

$$
y=2-e^{x}, \ln \sqrt{3} \leq x \leq \ln \sqrt{8}
$$

Answer: $1 + \frac{1}{2} \ln \frac{3}{2}$


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

Let \( a \) and \( b \) be real numbers such that \( a + b = 1 \). Then, the minimum value of
\[
f(a, b) = 3 \sqrt{1 + 2a^2} + 2 \sqrt{40 + 9b^2}
\]
is ______.

Answer: $5 \sqrt{11}$


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

Rodney uses the following clues to try to guess a secret number:
- It is a two-digit integer.
- The tens digit is odd.
- The units digit is even.
- The number is greater than 75.

What is the probability that Rodney will guess the correct number if he guesses a number that meets all these criteria? Express your answer as a common fraction.

Answer: $\frac{1}{7}$


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

A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$.

Answer: $62$


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

A five-character license plate is composed of English letters and digits. The first four positions must contain exactly two English letters (letters $I$ and $O$ cannot be used). The last position must be a digit. Xiao Li likes the number 18, so he hopes that his license plate contains adjacent digits 1 and 8, with 1 preceding 8. How many different choices does Xiao Li have for his license plate? (There are 26 English letters in total.)

Answer: $23040$


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

A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircase from the 1st floor to the 2nd floor?

Answer: $1512$


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

Given a $5 \times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \( a_{ij} \) (where \( a_{ij} \in \{0, 1\} \)), with the condition that \( a_{ij} = a_{ji} \) for \( 1 \leq i, j \leq 5 \). Calculate the total number of ways to fill the grid such that there are exactly five 1's in the grid.

Answer: $326$


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

Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done?

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 electronic clock on the International Space Station displayed time in the format HH:MM. Due to an electromagnetic storm, the device malfunctioned, and each digit on the display either increased by 1 or decreased by 1. What was the actual time of the storm if the clock displayed 20:09 immediately after it?

Answer: $11:18$


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

A boy wrote the first twenty natural numbers on a piece of paper. He didn't like how one of them was written, so he crossed it out. Among the remaining 19 numbers, there is one number that equals the arithmetic mean of these 19 numbers. Which number did he cross out? If the problem has more than one solution, write down the sum of these numbers.

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 tetrahedron \(ABCD\), the dihedral angle between face \(ABC\) and face \(BCD\) is \(60^\circ\). The orthogonal projection of vertex \(A\) onto face \(BCD\) is \(H\), which is the orthocenter of \(\triangle BCD\). Point \(G\) is the centroid of \(\triangle ABC\). Given that \(AH = 4\) and \(AB = AC\), find the length of \(GH\).

Answer: $\frac{4 \sqrt{21}}{9}$


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 $a\in \mathbb{R}$, $i$ is the imaginary unit. If $z=1+ai$ and $z\cdot \overline{z}=4$, solve for the value of $a$.

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>.

In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\) and \(BC : AD = 5 : 7\). Point \(F\) lies on \(AD\) and point \(E\) lies on \(DC\) such that \(AF : FD = 4 : 3\) and \(CE : ED = 2 : 3\). If the area of quadrilateral \(ABEF\) is 123, determine the area of trapezoid \(ABCD\).

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>.

Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{|6a - 4b| + |3(a + b\sqrt{3}) + 2(a\sqrt{3} - b)|}{\sqrt{a^2 + b^2}}
$$

Answer: $\sqrt{39}$


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>.

A regular octagon has a side length of 8 cm. What is the number of square centimeters in the area of the shaded region formed by diagonals connecting alternate vertices (forming a square in the center)?

Answer: $192 + 128\sqrt{2}$


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>.

Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.

- At 12:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "82."
- At 13:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "71."
- At 15:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "46."

It is known that Erema rounded the distance to the nearest whole number each time, and if there were two nearest whole numbers, he could choose either.

At 16:00, Foma asked again: "How many versts to Moscow?" This time Erema gave an exact answer, without rounding. What was Erema's answer?

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>.

A pedestrian and a cyclist set off from point $A$ to point $B$ simultaneously. At point $B$, the cyclist turns back and meets the pedestrian 20 minutes after starting. Without stopping, the cyclist continues to point $A$, turns around, and catches up with the pedestrian 10 minutes after their first meeting. How long will it take for the pedestrian to travel from $A$ to $B$?

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>.

The three-digit even numbers \( A \, , B \, , C \, , D \, , E \) satisfy \( A < B < C < D < E \). Given that \( A + B + C + D + E = 4306 \), find the smallest value of \( A \).

Answer: $326$


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

A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.

Answer: $132$


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

For each positive integer \(1 \leqq k \leqq 100\), let \(a_{k}\) denote the sum \(\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}\). Calculate the value of

\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \ldots + a_{100}^{2}. \]

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>.

The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1 is

Answer: $4436$


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

The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$.

Answer: $3\sqrt{2} + \sqrt{10}$


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>.

Three clients are at the hairdresser, each paying their bill at the cash register.

- The first client pays the same amount that is in the register and takes 10 reais as change.
- The second client performs the same operation as the first.
- The third client performs the same operation as the first two.

Find the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty.

Answer: $8.75$


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>.

When the number "POTOP" was added together 99,999 times, the resulting number had the last three digits of 285. What number is represented by the word "POTOP"? (Identical letters represent identical digits.)

Answer: $51715$


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

In an $11 \times 11$ table, integers from 0 to 10 are placed (naturally, numbers can repeat, and not necessarily all listed numbers occur). It is known that in every $3 \times 2$ or $2 \times 3$ rectangle, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers in the entire table.

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>.

Let $\overline{AB}$ be a diameter in a circle with radius $6$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at point $E$ such that $BE = 3$ and $\angle AEC = 60^{\circ}$. Find the value of $CE^{2} + DE^{2}$.

Answer: $108$


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

In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$.

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>.

A sphere is inscribed in a cone, and the surface area of the sphere is equal to the area of the base of the cone. Find the cosine of the angle at the vertex in the axial section of the cone.

Answer: $\frac{7}{25}$


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

The numbers \( x \) and \( y \) are such that the equalities \( \operatorname{ctg} x - \operatorname{ctg} y = 2 \) and \( 5 \sin (2x - 2y) = \sin 2x \sin 2y \) hold. Find \( \operatorname{tg} x \operatorname{tg} y \).

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


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 diameters of two pulleys with parallel axes are 80 mm and 200 mm, respectively, and they are connected by a belt that is 1500 mm long. What is the distance between the axes of the pulleys if the belt is tight (with millimeter precision)?

Answer: $527$


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

In the trapezoid \(MPQF\), the bases are \(MF = 24\) and \(PQ = 4\). The height of the trapezoid is 5. Point \(N\) divides the side into segments \(MN\) and \(NP\) such that \(MN = 3NP\).

Find the area of triangle \(NQF\).

Answer: $22.5$


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

The numbers \( 62, 63, 64, 65, 66, 67, 68, 69, \) and \( 70 \) are divided by, in some order, the numbers \( 1, 2, 3, 4, 5, 6, 7, 8, \) and \( 9 \), resulting in nine integers. The sum of these nine integers is \( S \). What are the possible values of \( S \)?

Answer: $187$


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

Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \).

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


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

Arrange all positive integers whose digits sum to 8 in ascending order to form a sequence $\{a_n\}$, called the $P$ sequence. Then identify the position of 2015 within this sequence.

Answer: $83$


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

Given an equilateral triangle with side length \( n \), divided into unit triangles as illustrated, let \( f(n) \) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a common edge, never revisiting any triangle, and never moving upwards (from a lower row to an upper row). An example path is illustrated for \( n = 5 \). Determine the value of \( f(2012) \).

Answer: $2011!$


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

Given that (1+ex)2019=a0+a1x+a2x2+……+a2019x2019, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$

Answer: $-1$


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

Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get?

Answer: $8654232$


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

From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20.

Answer: $25$


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

Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number?

Answer: $70$


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

The diagonals of a trapezoid are mutually perpendicular, and one of them is 13. Find the area of the trapezoid if its height is 12.

Answer: $1014/5$


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>.

For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a "five-divisible number". What is the smallest "five-divisible number" greater than 2000?

Answer: $2004$


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

How many days have passed from March 19, 1990, to March 23, 1996, inclusive?

Answer: $2197$


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

In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).

A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\).

Answer: $3\sqrt{2}$


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

The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.

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>.

On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)?

Answer: $\frac{1}{6}$


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

There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples remain. How many apples are in the basket, at minimum?

Answer: $23$


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

There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the chips in reverse order?

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 equation \(x^{2}+5x+1=0\) has roots \(x_{1}\) and \(x_{2}\). Find the value of the expression

\[
\left(\frac{x_{1} \sqrt{6}}{1+x_{2}}\right)^{2}+\left(\frac{x_{2} \sqrt{6}}{1+x_{1}}\right)^{2}
\]

Answer: $220$


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

Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers.

Answer: $2525$


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

Find all the roots of the equation

\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]

(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))

In the answer, specify the sum of the found roots.

Answer: $21$


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

The function \( f \) satisfies the equation \((x-1) f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x-1}\) for each value of \( x \) not equal to 0 and 1. Find \( f\left(\frac{2016}{2017}\right) \).

Answer: $2017$


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

Find all positive real numbers \(c\) such that the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x) = x^3 - cx\) has the property that the circle of curvature at any local extremum is centered at a point on the \(x\)-axis.

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


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

Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake weighing 900 grams. How many grams of the cake did Winnie the Pooh eat on the way?

Answer: $112.5$


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

A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called "good" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs?

Answer: $22$


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

A large batch of tires contains $1.5\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$

Answer: $168$


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

If Greg rolls five fair eight-sided dice, what is the probability that he rolls more 1's than 8's?

Answer: $\frac{10246}{32768}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 the parameter \(a\) does the equation \(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\) have exactly three distinct solutions?

Answer: $48$


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

Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time?

Answer: $8.4$


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

On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the round table? Indicate all possible options.

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>.

Two vertices of a square with an area of \( 256 \, \text{cm}^2 \) lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the 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>.

$ABCD$  forms a rhombus.  $E$  is the intersection of  $AC$  and  $BD$ .  $F$  lie on  $AD$  such that  $EF$  is perpendicular to  $FD$ . Given $EF=2$  and  $FD=1$ . Find the area of the rhombus  $ABCD$

Answer: $20$


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

At the end of the school year, teachers of the third grade met with the parents of some of their students; exactly 31 people were present at this meeting. The Latin teacher was asked questions by 16 parents, the French teacher by 17 parents, the English teacher by 18 parents, and so on up to the Math teacher, who was asked questions by all the parents present at the meeting. How many parents were present at the meeting?

Answer: $23$


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

Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain?

Answer: $2118$


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

Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)?

Answer: $1/3$


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

In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\).

Answer: $20/3$


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>.

How many positive rational numbers less than \(\pi\) have denominator at most 7 when written in lowest terms? (Integers have denominator 1.)

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>.

Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \in \mathbb{R}$, the range of $f(x)$ is $[0, +\infty)$.
(1) Find the expression for $f(x)$.
(2) Let $g(x) = f(x) - 2kx$, where $k \in \mathbb{R}$.
   (i) If $g(x)$ is monotonic on $x \in [-2, 2]$, find the range of the real number $k$.
   (ii) If the minimum value of $g(x)$ on $x \in [-2, 2]$ is $g(x)_{\text{min}} = -15$, find the value of $k$.

Answer: $k = 6$


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>.

How many points can be placed inside a circle of radius 2 such that one of the points coincides with the center of the circle and the distance between any two points is not less than 1?

Answer: $19$


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

Determine the largest natural number \( n \) so that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.

Answer: $2004$


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

Find all three-digit integers \( abc = n \) such that \( \frac{2n}{3} = a! \cdot b! \cdot c! \).

Answer: $432$


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

Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels?

Answer: $46$


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

\( n \) is a positive integer that is not greater than 100 and not less than 10, and \( n \) is a multiple of the sum of its digits. How many such \( n \) 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>.

Given that the probability mass function of the random variable $X$ is $P(X=k)= \frac{k}{25}$ for $k=1, 2, 3, 4, 5$, find the value of $P(\frac{1}{2} < X < \frac{5}{2})$.

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>.

A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how these students schedule their visit, we can always find \( k \) students such that either all \( k \) students are present in the school at the same time, or at any time, no two of them are present in the school simultaneously. Find the maximum value of \( k \).

Answer: $32$


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

Given that \( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \), find the value of \( x \).

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>.

There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?

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>.

If \( f(x) = x^{6} - 2 \sqrt{2006} x^{5} - x^{4} + x^{3} - 2 \sqrt{2007} x^{2} + 2 x - \sqrt{2006} \), then find \( f(\sqrt{2006} + \sqrt{2007}) \).

Answer: $\sqrt{2007}$


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

Let's call a natural number "remarkable" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit "remarkable" numbers exist?

Answer: $1800$


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

The strengths of the two players are equal, meaning they have equal chances of winning each game. They agreed that the prize would go to the first player to win 6 games. They had to stop the game after the first player won 5 games and the second won 3. In what proportion should the prize be fairly divided?

Answer: $7:1$


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>.

The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of the pyramid.

Answer: $192$


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

Let $u_0 = \frac{1}{3}$, and for $k \ge 0$, let $u_{k+1} = \frac{3}{2}u_k - \frac{3}{2}u_k^2$. This sequence tends to a limit; call it $M$. Determine the least value of $k$ such that $|u_k - M| \le \frac{1}{2^{1000}}$.

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>.

From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood?

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 two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), then the real number \(9x+81y+729z\) equals \(\qquad\)

Answer: $439$


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

The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the lesson. What fraction of the way to school had he covered when he remembered about the pen?

Answer: $\frac{1}{4}$


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

Given points $A\left(\begin{matrix} \cos \alpha , & \sin \alpha \end{matrix}\right)$ and $B\left(\begin{matrix} \cos \beta , & \sin \beta \end{matrix}\right)$, where $\alpha$ and $\beta$ are acute angles, and the distance between $A$ and $B$ is $\frac{\sqrt{10}}{5}$.

$(1)$ Find the value of $\cos (\alpha -\beta)$; $(2)$ If $\tan \frac{\alpha}{2}=\frac{1}{2}$, find the values of $\cos \alpha$ and $\cos \beta$.

Answer: $\frac{24}{25}$


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

A positive integer n is called *primary divisor* if for every positive divisor  $d$  of  $n$  at least one of the numbers  $d - 1$  and  $d + 1$  is prime. For example,  $8$  is divisor primary, because its positive divisors  $1$ ,  $2$ ,  $4$ , and  $8$  each differ by  $1$  from a prime number ( $2$ ,  $3$ ,  $5$ , and  $7$ , respectively), while  $9$  is not divisor primary, because the divisor  $9$  does not differ by  $1$  from a prime number (both  $8$  and  $10$  are composite). Determine the largest primary divisor number.

Answer: $48$


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

Given vectors $\overrightarrow{a} = (x, -3)$, $\overrightarrow{b} = (-2, 1)$, $\overrightarrow{c} = (1, y)$ on a plane. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} - \overrightarrow{c}$, and $\overrightarrow{b}$ is parallel to $\overrightarrow{a} + \overrightarrow{c}$, find the projection of $\overrightarrow{a}$ onto the direction of $\overrightarrow{b}$.

Answer: $-\sqrt{5}$


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

A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle.

Answer: $\frac{7}{18}$


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

Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$?

Answer: $2 - \sqrt{2}$


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

The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When is it advisable to swap the tires so that they wear out equally? (Assume that the tires are swapped only once, although in practice drivers do this more frequently.)

Answer: $9375$


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

Find the least prime factor of the number represented by \(1 \underbrace{0000 \cdots 00}_{2010 \text{-many}} 1\).

Answer: $11$


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

How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set?

Answer: $42$


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

Add together all natural numbers less than 1980 for which the sum of their digits is even!

Answer: $979605$


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

In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places.

Answer: $4.24$


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 simple (i.e., non-intersecting) quadrilateral \(ABCD\) has sides \(AB\), \(BC\), and \(CD\) with lengths 4, 5, and 20, respectively. If the angles \(B\) and \(C\) are obtuse, and \(\sin C = -\cos B = \frac{3}{5}\), then what is the length of the side \(AD\)?

(Note: Taken from the 30th annual American High School Mathematics Examination, 1979.)

Answer: $25$


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

On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?

Answer: $288$


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

The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs.

Answer: $12\pi$


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

Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12?

Answer: $27720$


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

For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).

Answer: $312$


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

The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider "$a^{2}+2a$" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore "$x$" to "$a^{2}+2a$". The solution process is as follows:
Given: Let $a^{2}+2a=x$, then the original expression $=x(x+2)+1$ (Step 1),
$=x^{2}+2x+1$ (Step 2),
$=(x+1)^{2}$ (Step 3),
$=(a^{2}+2a+1)^{2}$ (Step 4).
Questions:
$(1)$ ① Did the student complete the factoring? If not, please write down the final result directly;
② Please try to factorize the polynomial $(a^{2}-4a)(a^{2}-4a+8)+16$ following the above method;
$(2)$ Please try to calculate using the above method: $\left(1-2-3-\ldots -2023\right)\times \left(2+3+\ldots +2024\right)-\left(1-2-3-\ldots -2024\right)\times \left(2+3+\ldots +2023\right)$.

Answer: $2024$


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

Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:

First method:

a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:

\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.48 \text{ (billion rubles)} \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = \frac{478.48}{1388.01} \approx 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]

Second method:

a) Determine the share of the euro in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{E U R} = \frac{41.89}{1388.01} \approx 3.02\% \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = 100 - 3.02 - 0.2 - 22.31 - 39.98 - 0.02 = 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]

Answer: $-12.6$


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>.

$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$ .  $ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$

Answer: $1005$


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

In a race, four cars each independently run timed laps around a circuit. Each car's lap time is discretely measured in seconds and can be any integer value between 150 and 155 seconds, uniformly distributed. The winner is the car with the shortest lap time. In case of a tie, the involved cars re-run the lap until a single winner is determined. What is the probability that Maria's first lap time was 152 seconds, given that she won the race?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$

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


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

If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is \_\_\_\_\_\_\_\_.

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>.

Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also find the length of the segment $K F$.

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


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

Given a sequence of 0s and 1s of length 23 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consecutive 1s, determine the number of such sequences.

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>.

How many ordered pairs of real numbers  $(x, y)$  are there such that  $x^2+y^2 = 200$  and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer?

Answer: $12$


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

In a plane, there are 10 lines, among which 4 lines are parallel to each other. Then, these 10 lines can divide the plane into at most how many parts?

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>.

Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets, which are 70 yuan per person. What is the minimum amount they need to spend to buy the tickets?

Answer: $800$


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

A mother gives pocket money to her children sequentially: 1 ruble to Anya, 2 rubles to Borya, 3 rubles to Vitya, then 4 rubles to Anya, 5 rubles to Borya, and so on until Anya receives 202 rubles, and Borya receives 203 rubles. How many more rubles will Anya receive compared to Vitya?

Answer: $68$


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

A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink?

Answer: $29$