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

There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is the maximum number of cells that the cube can visit without soiling the board with paint?

Answer: $84$


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

When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?

Answer: $100$


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

It is known that there are a total of $n$ students in the first grade of Shuren High School, with $550$ male students. They are divided into layers based on gender, and $\frac{n}{10}$ students are selected to participate in a wetland conservation knowledge competition. It is given that there are $10$ more male students than female students among the participants. Find the value of $n$.

Answer: $1000$


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

Given Joy has 50 thin rods, one each of every integer length from 1 cm through 50 cm, and rods with lengths 8 cm, 12 cm, and 25 cm are already placed on a table, determine the number of the remaining rods that can be chosen as the fourth rod to form a quadrilateral with positive area.

Answer: $36$


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

If the graph of the linear function $y=-3x+m$ (where $m$ is a constant) passes through the second, third, and fourth quadrants, then the possible values of $m$ are _____. (Write down one possible value)

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>.

Three boxes each contain an equal number of hockey pucks. Each puck is either black or gold. All 40 of the black pucks and exactly $\frac{1}{7}$ of the gold pucks are contained in one of the three boxes. Determine the total number of gold hockey pucks.

Answer: $140$


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

We roll five dice, each a different color. In how many ways can the sum of the rolls be 11?

Answer: $205$


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

Calculate the square of 1007 without using a calculator.

Answer: $1014049$


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

The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers?

Answer: $285$


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

Let  $ABC$  be a right-angled triangle with  $\angle ABC=90^\circ$ , and let  $D$  be on  $AB$  such that  $AD=2DB$ . What is the maximum possible value of  $\angle ACD$ ?

Answer: $30$


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

Given that the sequence $\{a_n\}$ forms a geometric sequence, and $a_n > 0$.

(1) If $a_2 - a_1 = 8$, $a_3 = m$. ① When $m = 48$, find the general formula for the sequence $\{a_n\}$. ② If the sequence $\{a_n\}$ is unique, find the value of $m$.

(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, where $k \in \mathbb{N}^*$, find the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$.

Answer: $32$


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

A pedestrian walked 5.5 kilometers in 1 hour but did not reach point \( B \) (short by \(2 \pi - 5.5\) km). Therefore, the third option is longer than the first and can be excluded.

In the first case, they need to cover a distance of 5.5 km along the alley. If they move towards each other, the required time is \(\frac{5.5}{20 + 5.5}\) hours.

In the second case, moving towards each other, after \(\frac{2 \pi - 5.5}{5.5}\) hours the pedestrian will reach point \( B \), while the cyclist will still be riding on the highway (since \(\frac{4}{15} > \frac{2 \pi - 5.5}{5.5}\)). Thus, the cyclist will always be on the highway and the closing speed of the pedestrian and the cyclist will be \(15 + 5.5 = 20.5\) km/h. They will meet in \(\frac{2 \pi - 1.5}{20.5}\) hours.

Compare the numbers obtained in cases 1 and 2:

$$
\frac{5.5}{25.5} = \frac{11}{51} < 0.22 < 0.23 < \frac{4 \cdot 3.14 - 3}{41} < \frac{2 \pi - 1.5}{20.5}
$$

Therefore, the answer is given by the first case.

**Note**: Some participants interpreted the condition differently, considering that the pedestrian and the cyclist met after an hour, and at that time, the pedestrian asked the cyclist to bring him the keys. Since the condition does not specify where exactly the neighbor was, and with a different interpretation of the condition, a similar problem arises (although with slightly more cumbersome calculations), the jury decided to accept both interpretations of the condition. The plan for solving the second interpretation is provided.

Answer: $11/51$


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

There is a card game called "Twelve Months" that is played only during the Chinese New Year. The rules are as follows:

Step 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.

Step 2: Lay out the shuffled cards face down into 12 columns, each column consisting of 4 cards.

Step 3: Start by turning over the first card in the first column. If the card is numbered \(N \ (N=1,2, \cdots, 12\), where J and Q correspond to 11 and 12 respectively, regardless of suit, place the card face up at the end of the \(N\)th column.

Step 4: Continue by turning over the first face-down card in the \(N\)th column and follow the same process as in step 3.

Step 5: Repeat this process until you cannot continue. If all 12 columns are fully turned over, it signifies that the next 12 months will be smooth and prosperous. Conversely, if some columns still have face-down cards remaining at the end, it indicates that there will be some difficulties in the corresponding months.

Calculate the probability that all columns are fully turned over.

Answer: $1/12$


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>.

Calculate the definite integral:

$$
\int_{\pi / 4}^{\operatorname{arctg} 3} \frac{d x}{(3 \operatorname{tg} x+5) \sin 2 x}
$$

Answer: $\frac{1}{10} \ln \frac{12}{7}$


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 Yule log is shaped like a right cylinder with height  $10$  and diameter  $5$ . Freya cuts it parallel to its bases into  $9$  right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by  $a\pi$ . Compute  $a$ .

Answer: $100$


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

An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$?

Answer: $9/35$


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

A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?

Answer: $77$


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

For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).)

Answer: $1003$


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

For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6?

Answer: $17$


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

Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$ respectively, and $2\sin \frac{7\pi }{6}\sin (\frac{\pi }{6}+C)+ \cos C=-\frac{1}{2}$.

(1) Find $C$;

(2) If $c=2\sqrt{3}$, find the maximum area of $\triangle ABC$.

Answer: $3\sqrt{3}$


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

Given the function $f(x)=\frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))$ where $n \in \mathbb{N}^{*}$, vector $\overrightarrow{a}=(0,1)$, and $\theta_{n}$ is the angle between vector $\overrightarrow{OA}_{n}$ and $\overrightarrow{a}$. Compute the value of $\frac{\cos \theta_{1}}{\sin \theta_{1}} + \frac{\cos \theta_{2}}{\sin \theta_{2}} + ... + \frac{\cos \theta_{2016}}{\sin \theta_{2016}}$.

Answer: $\frac{2016}{2017}$


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>.

Let \( P \) be any point inside a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \). The distances from point \( P \) to the four faces are \( d_1, d_2, d_3, d_4 \) respectively. What is the minimum value of \( d_1^2 + d_2^2 + d_3^2 + d_4^2 \)?

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>.

Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). We compute 98 differences: \(a_{1} = 1 - \frac{1}{3}, a_{2} = \frac{1}{2} - \frac{1}{4}, \ldots, a_{98} = \frac{1}{98} - \frac{1}{100}\). What is the sum of all these differences?

Answer: $\frac{14651}{9900}$


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

Given that $a$ and $b$ are positive real numbers satisfying $a + 2b = 1$, find the minimum value of $a^2 + 4b^2 + \frac{1}{ab}$.

Answer: $\frac{17}{2}$


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

Calculate the definite integral:

$$
\int_{1}^{e} \sqrt{x} \cdot \ln^{2} x \, dx
$$

Answer: $\frac{10e\sqrt{e} - 16}{27}$


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

Simplify $\sin ^{2}\left( \alpha-\frac{\pi}{6} \right)+\sin ^{2}\left( \alpha+\frac{\pi}{6} \right)-\sin ^{2}\alpha$.

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


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

A and B play a guessing game where A first thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b| \leqslant 1$, then A and B are considered to have a "telepathic connection". If two people are randomly chosen to play this game, the probability that they have a "telepathic connection" is ______.

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>.

Given a sequence $\{a_n\}$ with the sum of its first n terms $S_n$, if $S_2 = 4$, and $a_{n+1} = 2S_n + 1$ for $n \in N^*$, find the values of $a_1$ and $S_5$.

Answer: $121$


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

Compute the square of 1085 without using a calculator.

Answer: $1177225$


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

Let the function
$$
f(x) = x^3 + ax^2 + bx + c \quad \text{for } x \in \mathbf{R},
$$
where \( a \), \( b \), and \( c \) are distinct non-zero integers, and
$$
f(a) = a^3 \quad \text{and} \quad f(b) = b^3.
$$
Find \( a + b + c = \quad \).

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 number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \( a_{22} = 2 \), find the sum of all 9 numbers in the matrix.

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>.

The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. Ahead of Peter, there were 50 people; ahead of Nicholas, there were 100; ahead of Denis, there were 170. When the command "About-face!" was given, everyone turned to face the opposite direction. It turned out that in front of one of the brothers, there are now four times as many people as in front of another. How many recruits, including the brothers, could there be? List all possible options.

Answer: $211$


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

In the geometric sequence $\{a_n\}$, if $a_2a_5 = -\frac{3}{4}$ and $a_2 + a_3 + a_4 + a_5 = \frac{5}{4}$, calculate the value of $\frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5}$.

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


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>.

Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote?
A) 997
B) 999
C) 1111
D) 1113
E) 1115

Answer: $1113$


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

How many positive integers, not exceeding 200, are multiples of 2 or 5 but not 6?

Answer: $87$


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

Lei Lei bought some goats and sheep. If she had bought 2 more goats, the average price of each sheep would increase by 60 yuan. If she had bought 2 fewer goats, the average price of each sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total.

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>.

Find the smallest constant  $ C$  such that for all real  $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds.

Answer: $4/3$


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

In how many ways can 9 identical items be distributed into four boxes?

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 that the sum of the coefficients of the expansion of $(\frac{3}{x}-\sqrt{x})^n$ is $512$. Find:

$(1)$ The coefficient of the term containing $x^{3}$ in the expansion;

$(2)$ The constant term in the expansion of $(1+\frac{1}{x})(2x-1)^n$.

Answer: $17$


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

What is the sum of all the integers from 100 to 2000?

Answer: $1996050$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?

Answer: $2016$


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

Certain integers, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$, result in a mixed number where the fractional part is $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$, respectively. Find the smallest integer greater than 1 that satisfies these conditions.

Answer: $316$


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

In a certain country, the airline system is arranged in such a way that any city is connected by airlines with no more than three other cities, and from any city to any other, one can travel with no more than one stopover. What is the maximum number of cities that can be in this country?

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>.

$A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$.

Answer: $48$


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

In \(\triangle ABC\), \(AB = 13\), \(BC = 14\), and \(CA = 15\). \(P\) is a point inside \(\triangle ABC\) such that \(\angle PAB = \angle PBC = \angle PCA\). Find \(\tan \angle PAB\).

Answer: $\frac{168}{295}$


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

Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of the first one. What is the total surface area of the first bar?

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_n)_{n \equal{} 1}^\infty$  is defined on real numbers with  $ a_n \not \equal{} 0$ ,  $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$  and  $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So   $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$

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>.

Let $f(x)$ have a derivative, and satisfy $\lim_{\Delta x \to 0} \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$. Find the slope of the tangent line at point $(1,f(1))$ on the curve $y=f(x)$.

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>.

The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take, if it is known that the arc measure of $A B$ is $24^\circ$?

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>.

Solve the fractional equation application problem.

On the eve of Children's Day, a certain shopping mall purchased a certain electric toy for $7200. Due to good sales, after a period of time, the mall purchased the same toy again for $14800. The quantity purchased the second time was twice the quantity purchased the first time, but the price per item was $2 more expensive.

(1) Find out how many items of this toy the shopping mall purchased the first time.

(2) Assuming that the toys purchased by the shopping mall twice are sold at the same retail price, and the remaining 80 toys are sold at a 40% discount on the retail price, if all the toys purchased twice are sold out and the profit is not less than $4800, what is the minimum retail price per item?

Answer: $100$


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

Mr. Wang drives from his home to location $A$. On the way there, he drives the first $\frac{1}{2}$ of the distance at a speed of 50 km/h and increases his speed by $20\%$ for the remaining distance. On the way back, he drives the first $\frac{1}{3}$ of the distance at a speed of 50 km/h and increases his speed by $32\%$ for the remaining distance. The return trip takes 31 minutes less than the trip to $A$. What is the distance in kilometers between Mr. Wang's home and location $A$?

Answer: $330$


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

I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not necessarily different)?

Answer: $26460$


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

To meet the shopping needs of customers during the "May Day" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is $6 yuan per kilogram.
$(1)$ Find out how many kilograms of cherries and cantaloupes the fruit store purchased.
$(2)$ After selling the first batch of purchased fruits, the fruit supermarket decided to reward customers by launching a promotional activity. They purchased a total of $300$ kilograms of cherries and cantaloupes for no more than $5280$ yuan. Among them, $a$ kilograms of cherries and $2a$ kilograms of cantaloupes were sold at the purchase price, while the remaining cherries were sold at $55$ yuan per kilogram and cantaloupes were sold at $10$ yuan per kilogram. If the supermarket owner plans to make a profit of at least $2120$ yuan in this transaction, find the maximum value of the positive integer $a$.

Answer: $35$


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

Calculate $7.45 + 2.56$ as a decimal.

Answer: $10.01$


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

Find the largest integer \( k \) such that for at least one natural number \( n > 1000 \), the number \( n! = 1 \cdot 2 \cdot \ldots \cdot n \) is divisible by \( 2^{n+k+2} \).

Answer: $-3$


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

A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \mapsto 15 + 2 \times 7 = 29$, $5 \mapsto 0 + 2 \times 5 = 10$. A natural number is called ‘good’ if after repeatedly applying this operation, the resulting number stops changing. Find the smallest such good number.

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>.

Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.

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


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

**p1.** The Evergreen School booked buses for a field trip. Altogether,  $138$  people went to West Lake, while  $115$  people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?**p2.** In New Scotland there are three kinds of coins:  $1$  cent,  $6$  cent, and  $36$  cent coins. Josh has  $99$  of the  $36$ -cent coins (and no other coins). He is allowed to exchange a  $36$  cent coin for  $6$  coins of  $6$  cents, and to exchange a  $6$  cent coin for  $6$  coins of  $1$  cent. Is it possible that after several exchanges Josh will have  $500$  coins?**p3.** Find all solutions  $a, b, c, d, e, f, g, h$  if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc}
 &  & a & b & c
+ &  &  & d & e
\hline
 & f & a & g & c
x & b & b & h &
\hline
f & f & e & g & c
\end{tabular}$ **p4.** Is it possible to find a rectangle of perimeter  $10$  m and cut it in rectangles (as many as you want) so that the sum of the perimeters is  $500$  m?**p5.** The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber  $C$  tries to catch a mouse that was originally in the chamber  $M$ . The cat makes the first move, moving from chamber  $C$  to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse?
![Image](https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png)

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: $23$


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

In a bucket, there are $34$ red balls, $25$ green balls, $23$ yellow balls, $18$ blue balls, $14$ white balls, and $10$ black balls. Find the minimum number of balls that must be drawn from the bucket without replacement to guarantee that at least $20$ balls of a single color are drawn.

Answer: $100$


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

Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$        .

Answer: $-\dfrac{5}{2}$


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

Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum.

Answer: $105$


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

In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling station, the ratio was $3:7$ in favor of Mr. Schopný. The election ended in a tie, with both candidates receiving the same number of votes. In what ratio were the valid votes cast in each polling station if we know that the same number of people cast valid votes in the first and second polling stations?

Answer: $24 : 24 : 25$


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>.

Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet.

Answer: $375$


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

The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).

Answer: $1705$


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

A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____.

Answer: $11.0$


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

Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$.

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>.

Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 17x\]for all integers $x.$  Find the smallest possible value of $a.$

Answer: $17$


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

Given the function $f(x)=\cos^2x+\cos^2\left(x-\frac{\pi}{3}\right)-1$, where $x\in \mathbb{R}$,

$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;

$(2)$ The function $f(x)$ is translated to the right by $\frac{\pi}{3}$ units to obtain the function $g(x)$. Find the expression for $g(x)$;

$(3)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4},\frac{\pi}{3}\right]$;

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


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

At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Dishes} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Burger & Soda & Apple Pie \\ \hline
Fish and Chips & Juice & Chocolate Cake \\ \hline
Pasta &  &  \\ \hline
Vegetable Salad &  &  \\ \hline
\end{tabular}

Answer: $14$


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

Let the function $f(x) = (\sin x + \cos x)^2 - \sqrt{3}\cos 2x$.

(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding value of $x$ when the maximum value is attained.

Answer: $\frac{5\pi}{12}$


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>.

Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$.

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 that 3 people are to be selected from 5 girls and 2 boys, if girl A is selected, determine the probability that at least one boy is selected.

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


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

Find the distance \( B_{1} H \) from point \( B_{1} \) to the line \( D_{1} B \), given \( B_{1}(5, 8, -3) \), \( D_{1}(-3, 10, -5) \), and \( B(3, 4, 1) \).

Answer: $2\sqrt{6}$


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

Given \( f(x)=\frac{2x+3}{x-1} \), the graph of the function \( y=g(x) \) is symmetric with the graph of the function \( y=f^{-1}(x+1) \) with respect to the line \( y=x \). Find \( g(3) \).

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


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

$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is:

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>.

Let the function $f(x)$ be defined on $\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$.

Answer: $802$


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

Ali Baba and the 40 thieves decided to divide a treasure of 1987 gold coins in the following manner: the first thief divides the entire treasure into two parts, then the second thief divides one of the parts into two parts, and so on. After the 40th division, the first thief picks the largest part, the second thief picks the largest of the remaining parts, and so on. The last, 41st part goes to Ali Baba. For each of the 40 thieves, determine the maximum number of coins he can secure for himself in such a division irrespective of the actions of other thieves.

Answer: $49$


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

Find all three-digit numbers that are equal to the sum of all their digits plus twice the square of the sum of their digits. List all possible numbers in ascending order without spaces and enter the resulting concatenated multi-digit number.

Answer: $171465666$


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

Given the sets $A=\{x|x^{2}-px-2=0\}$ and $B=\{x|x^{2}+qx+r=0\}$, if $A\cup B=\{-2,1,5\}$ and $A\cap B=\{-2\}$, find the value of $p+q+r$.

Answer: $-14$


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

There are 85 beads in total on a string, arranged in the pattern "three green, four red, one yellow, three green, four red, one yellow, ...". How many red beads are there?

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>.

A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific heat of fusion of snow is $\lambda = 330$ kJ/kg. Assume that all the energy released upon impact is used for melting.

Answer: $0.005$


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

The polynomial
\[px^4 + qx^3 + rx^2 + sx + t = 0\]
has coefficients that are all integers, and roots $-3$, $4$, $6$, and $\frac{1}{2}$. If $t$ is a positive integer, find its smallest possible value.

Answer: $72$


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

A certain type of beverage with a prize promotion has bottle caps printed with either "reward one bottle" or "thank you for your purchase". If a bottle is purchased and the cap is printed with "reward one bottle", it is considered a winning bottle, with a winning probability of $\frac{1}{6}$. Three students, A, B, and C, each bought a bottle of this beverage.

1. Find the probability that none of the three students win a prize.
2. Find the probability that at least two of the three students do not win a prize.

Answer: $\frac{25}{27}$


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

Given that the length of the major axis of the ellipse is 4, the left vertex is on the parabola \( y^2 = x - 1 \), and the left directrix is the y-axis, find the maximum value of the eccentricity of such an ellipse.

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


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

Find the smallest prime \( p > 100 \) for which there exists an integer \( a > 1 \) such that \( p \) divides \( \frac{a^{89} - 1}{a - 1} \).

Answer: $179$


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

Using only the digits  $2,3$  and  $9$ , how many six-digit numbers can be formed which are divisible by  $6$ ?

Answer: $81$


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

Given that $a_1, a_2, b_1, b_2, b_3$ are real numbers, and $-1, a_1, a_2, -4$ form an arithmetic sequence, $-4, b_1, b_2, b_3, -1$ form a geometric sequence, calculate the value of $\left(\frac{a_2 - a_1}{b_2}\right)$.

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


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

When $x=1$, the value of the expression $px^3+qx-10$ is 2006; when $x=-1$, find the value of the expression $px^3+qx-10$.

Answer: $-2026$


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

In the rectangular coordinate system xOy, the parametric equation of curve C is $$\begin{cases} x=3\cos\theta \\ y=3\sin\theta \end{cases}$$ (θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of line l is ρ(cosθ - sinθ) = 1.

1. Find the rectangular coordinate equations of C and l.
2. Given that line l intersects the y-axis at point M and intersects curve C at points A and B, find the value of $|$$\frac {1}{|MA|}- \frac {1}{|MB|}$$|$.

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


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

Given $A=a^{2}-2ab+b^{2}$, $B=a^{2}+2ab+b^{2}$, where $a\neq b$.
$(1)$ Determine the sign of $A+B$ and explain the reason;
$(2)$ If $ab$ are reciprocals of each other, find the value of $A-B$.

Answer: $-4$


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

Calculate: $$\frac {\cos 2^\circ}{\sin 47^\circ} + \frac {\cos 88^\circ}{\sin 133^\circ}$$.

Answer: $\sqrt{2}$


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

What is the largest possible area of a quadrilateral with sidelengths  $1, 4, 7$  and  $8$  ?

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>.

Given 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder, and the recorder is seated between the leader and vice leader, determine the number of different seating arrangements possible, considering that arrangements that can be obtained by rotation are identical.

Answer: $240$


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

Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017).

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


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

Given the sets
$$
\begin{array}{c}
M=\{x, xy, \lg (xy)\} \\
N=\{0, |x|, y\},
\end{array}
$$
and that \( M = N \), determine the value of
$$
\left(x+\frac{1}{y}\right)+\left(x^2+\frac{1}{y^2}\right)+\left(x^3+\frac{1}{y^3}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).
$$

Answer: $-2$


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

Let the function \( f(x) \) satisfy the following conditions:
(i) If \( x > y \), and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \), such that \( f(z) = w - z \);
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there exists one solution that is not greater than all other solutions;
(iii) \( f(0) = 1 \);
(iv) \( f(-2003) \leq 2004 \);
(v) \( f(x) \cdot f(y) = f[x \cdot f(y) + y \cdot f(x) + xy] \).

Find the value of \( f(-2003) \).

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>.

A cyclist is riding on a track at a constant speed. It is known that at 11:22, he covered a distance that is 1.4 times greater than the distance he covered at 11:08. When did he start?

Answer: $10:33$


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

In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.
$(1)$ If one ball is randomly drawn from the bag, the probability that the character on the ball is exactly "康" is ________;
$(2)$ If two balls are drawn from the bag by person A, using a list or a tree diagram, find the probability $P$ that one ball has the character "阳" and the other ball has the character "康".

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


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

Given the function $f(x)=( \frac {1}{3})^{x}$, the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $f(n)-c$, and the first term of the sequence $\{b\_n\}_{b\_n > 0}$ is $c$. The sum of the first $n$ terms, $S\_n$, satisfies $S\_n-S_{n-1}= \sqrt {S\_n}+ \sqrt {S_{n-1}}(n\geqslant 2)$.
(I) Find the general term formula for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(II) If the sum of the first $n$ terms of the sequence $\{ \frac {1}{b\_nb_{n+1}}\}$ is $T\_n$, what is the smallest positive integer $n$ such that $T\_n > \frac {1005}{2014}$?

Answer: $252$


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

Find the maximum number of Permutation of set { $1,2,3,...,2014$ } such that for every 2 different number  $a$  and  $b$  in this set  at last in one of the permutation  $b$  comes exactly after  $a$

Answer: $1007$


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

Given that $\tan \alpha = -\frac{1}{3}$ and $\cos \beta = \frac{\sqrt{5}}{5}$, with $\alpha, \beta \in (0, \pi)$, find:
1. The value of $\tan(\alpha + \beta)$;
2. The maximum value of the function $f(x) = \sqrt{2} \sin(x - \alpha) + \cos(x + \beta)$.

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>.

In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number)

Answer: $-21600$


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

Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. If $\frac{S_3}{T_3}$ is a positive integer, then find the possible value of $q$.

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


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

The height \(CH\), dropped from the vertex of the right angle of the triangle \(ABC\), bisects the angle bisector \(BL\) of this triangle. Find the angle \(BAC\).

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>.

A bus with programmers departed from Novosibirsk to Pavlodar. After traveling 70 km, another car with Pavel Viktorovich left Novosibirsk on the same route and caught up with the bus in Karasuk. After that, Pavel traveled another 40 km, while the bus traveled only 20 km in the same time. Find the distance from Novosibirsk to Karasuk, given that both the car and the bus traveled at constant speeds. (Provide a complete solution, not just the answer.)

Answer: $140$


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

On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( M, N, \) and \( K \) are taken respectively so that \( AM:MB = 2:3 \), \( AK:KC = 2:1 \), and \( BN:NC = 1:2 \). In what ratio does the line \( MK \) divide the segment \( AN \)?

Answer: $6:7$


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

The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excellent comprehensive quality or outstanding basic subject knowledge. The school exam of the Strong Foundation Plan is independently formulated by pilot universities. In the school exam process of a certain pilot university, students can only enter the interview stage after passing the written test. The written test score $X$ of students applying to this pilot university in 2022 approximately follows a normal distribution $N(\mu, \sigma^2)$. Here, $\mu$ is approximated by the sample mean, and $\sigma^2$ is approximated by the sample variance $s^2$. It is known that the approximate value of $\mu$ is $76.5$ and the approximate value of $s$ is $5.5$. Consider the sample to estimate the population.

$(1)$ Assuming that $84.135\%$ of students scored higher than the expected average score of the university, what is the approximate expected average score of the university?

$(2)$ If the written test score is above $76.5$ points to enter the interview, and $10$ students are randomly selected from those who applied to this pilot university, with the number of students entering the interview denoted as $\xi$, find the expected value of the random variable $\xi$.

$(3)$ Four students, named A, B, C, and D, have entered the interview, and their probabilities of passing the interview are $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{1}{2}$, respectively. Let $X$ be the number of students among these four who pass the interview. Find the probability distribution and the mathematical expectation of the random variable $X$.

Reference data: If $X \sim N(\mu, \sigma^2)$, then: $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$.

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


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>.

There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts and snowy areas. In a group of camels that has 23 humps and 60 feet, how many camels are there?

Answer: $15$


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

For a natural number $n \geq 1$, it satisfies: $2002 \times n$ is a perfect cube, and $n \div 2002$ is a perfect square. The smallest such $n$ is

Answer: $2002^5$


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

In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b^{2}=ac$ and $a^{2}-c^{2}=ac-bc$.
1. Find the measure of $\angle A$.
2. Let $f(x)=\cos (wx-\frac{A}{2})+\sin (wx) (w > 0)$ and the smallest positive period of $f(x)$ is $\pi$. Find the maximum value of $f(x)$ on $[0, \frac{\pi}{2}]$.

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>.

What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$?

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>.

Olga Ivanovna, the homeroom teacher of class 5B, is staging a "Mathematical Ballet". She wants to arrange the boys and girls so that every girl has exactly 2 boys at a distance of 5 meters from her. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating?

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>.

\(a_n\) is the last digit of \(1 + 2 + \ldots + n\). Find \(a_1 + a_2 + \ldots + a_{1992}\).

Answer: $6984$


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

A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not both?

Answer: $15/52$


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

Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$.

Answer: $-206$


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

We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$.

What is the length of the broken line?

Answer: $\sqrt{2}$


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

Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits?

Answer: $163$