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

A set \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?

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>.

Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?

Answer: $402$


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

Let  $S$  be a subset of  $\{1,2,\dots,2017\}$  such that for any two distinct elements in  $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in  $S$ .

Answer: $865$


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

In a version of SHORT BINGO, a $5\times5$ card has specific ranges for numbers placed in each column. In the first column, 5 distinct numbers must be chosen from the set $1-15$, but they must all be prime numbers. As before, the middle square is labeled as WILD and the other columns have specific number ranges as in the original game. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card?

Answer: $720$


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

Consider one positive even integer and one positive odd integer less than $16$, where the even integer is a multiple of $3$. Compute how many different possible values result when the sum of these two numbers is added to their product.

Answer: $16$


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

How many integers $n$ satisfy $(n+2)(n-8) \le 0$?

Answer: $11$


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

In right triangle \\(ABC\\), where \\(\angle C = 90^{\circ}\\) and \\(\angle A = 30^{\circ}\\), the eccentricity of the ellipse, which has \\(A\\) and \\(B\\) as its foci and passes through point \\(C\\), is \_\_\_\_\_.

Answer: $\sqrt{3} - 1$


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>.

Subtract $555.55$ from $888.88.$ Express the result as a decimal to the nearest hundredth.

Answer: $333.33$


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>.

Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to its height $h$ should be _______.

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


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

Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between?

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>.

Blind boxes are a new type of product. Merchants package different styles of products from the same series in boxes with the same appearance, so that consumers do not know which style of product they are buying. A merchant has designed three types of dolls, $A$, $B$, and $C$, in the same series, and sells them in blind boxes. It is known that the production ratio of the three types of dolls $A$, $B$, and $C$ is $6:3:1$. Using frequency to estimate probability, calculate the probability that a consumer randomly buys $4$ blind boxes at once and finds all three types of dolls inside.

Answer: $0.216$


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

Place 6 balls, labeled from 1 to 6, into 3 different boxes. If each box is to contain 2 balls, and the balls labeled 1 and 2 are to be placed in the same box, calculate the total number of different ways to do this.

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>.

A parallelogram has a base of 6 cm and a height of 20 cm. Its area is \_\_\_\_\_\_ square centimeters. If both the base and the height are tripled, its area will increase by \_\_\_\_\_\_ times, resulting in \_\_\_\_\_\_ square centimeters.

Answer: $1080$


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

Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into these groups?

Answer: $18480$


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

Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=1-5+9-13+17-21+......+{(-1)}^{n-1}(4n-3)$, calculate the value of $S_{15}$.

Answer: $29$


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

A trapezoid has side lengths 4, 6, 8, and 10. The trapezoid can be rearranged to form different configurations with sides 4 and 8 as the parallel bases. Calculate the total possible area of the trapezoid with its different configurations.
A) $24\sqrt{2}$
B) $36\sqrt{2}$
C) $42\sqrt{2}$
D) $48\sqrt{2}$
E) $54\sqrt{2}$

Answer: $48\sqrt{2}$


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

The points \((2,3)\) and \((3, 7)\) lie on a circle whose center is on the \(x\)-axis. What is the radius of the circle?

Answer: $\frac{\sqrt{1717}}{2}$


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>.

Simplify the expression: $$\dfrac{\sqrt{450}}{\sqrt{288}} + \dfrac{\sqrt{245}}{\sqrt{96}}.$$ Express your answer as a common fraction.

Answer: $\frac{30 + 7\sqrt{30}}{24}$


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

Given that the left focus of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is $F$, the eccentricity is $\frac{\sqrt{2}}{2}$, and the distance between the left intersection point of the ellipse with the $x$-axis and point $F$ is $\sqrt{2} - 1$.
(I) Find the equation of the ellipse;
(II) The line $l$ passing through point $P(0, 2)$ intersects the ellipse at two distinct points $A$ and $B$. When the area of triangle $OAB$ is $\frac{\sqrt{2}}{2}$, find $|AB|$.

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


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

In triangle $ABC$, point $A$ is at $(1, 1)$, point $B$ is at $(4, 2)$, and point $C$ is at $(-4, 6)$.
(1) Determine the equation of the line where the median to side $BC$ lies;
(2) Determine the length of the altitude to side $BC$ and the area of triangle $ABC$.

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>.

Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.
$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;
$(2)$ Her father said she guessed wrong, the standard answer's simplification does not contain quadratic terms. Please calculate and determine the value of $\square$ in the original question.

Answer: $-2$


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

What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \equiv 30 \pmod{10}$?

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>.

Shift the graph of the function $y=\cos(\frac{π}{2}-2x)$ by an amount corresponding to the difference between the arguments of $y=\sin(2x-\frac{π}{4})$ and $y=\cos(\frac{π}{2}-2x)$.

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


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

At a school fundraiser, $3109 was raised. The money was shared equally among 25 charities. The amount received by each charity from the school was:

Answer: $124.36


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>.

Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units.

Answer: $\frac{83}{3}$


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 $f(x) = \ln x + \ln (ax+1) - \frac {3a}{2}x + 1$ ($a \in \mathbb{R}$).

(1) Discuss the intervals of monotonicity for the function $f(x)$.

(2) When $a = \frac {2}{3}$, if the inequality $xe^{x-\frac {1}{2}} + m \geqslant f(x)$ holds true for all $x$, find the minimum value of $m$, where $e$ is the base of the natural logarithm.

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


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>.

For any positive integer \( n \), define
\( g(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. \)
What is \( \sum_{n = 1}^{1023}{g(n)} \)?
**A** \( \frac{40}{2} \)
**B** \( \frac{42}{2} \)
**C** \( \frac{45}{2} \)
**D** \( \frac{48}{2} \)
**E** \( \frac{50}{2} \)

Answer: $\frac{45}{2}$


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 sets $A=\{2,3,4\}$ and $B=\{a+2,a\}$, if $A \cap B = B$, find $A^cB$ ___.

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 company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \ge 0$) satisfy the equation $x = 3 - \frac{k}{m + 1}$ ($k$ is a constant). Without the technical reform, the annual production volume can only reach 1 ten thousand units. The fixed investment for producing the product in 2013 is 8 million yuan, and an additional investment of 16 million yuan is required for each ten thousand units produced. Due to favorable market conditions, all products produced can be sold. The company sets the selling price of each product at 1.5 times its production cost (including fixed and additional investments).

1. Determine the value of $k$ and express the profit $y$ (in million yuan) of the product in 2013 as a function of the technical reform investment $m$ (profit = sales revenue - production cost - technical reform investment).
2. When does the company's profit reach its maximum with the technical reform investment in 2013? Calculate the maximum profit.

Answer: $21$


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

(1) If $\cos (\frac{\pi}{4}+x) = \frac{3}{5}$, and $\frac{17}{12}\pi < x < \frac{7}{4}\pi$, find the value of $\frac{\sin 2x + 2\sin^2 x}{1 - \tan x}$.

(2) Given the function $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1 (x \in \mathbb{R})$, if $f(x_0) = \frac{6}{5}$, and $x_0 \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find the value of $\cos 2x_0$.

Answer: $\frac{3 - 4\sqrt{3}}{10}$


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>.

According to the latest revision of the "Regulations on the Application and Use of Motor Vehicle Driving Licenses" by the Ministry of Public Security: each driving license applicant must pass the "Subject One" (theoretical subject) and "Comprehensive Subject" (combination of driving skills and some theoretical knowledge of "Subject One") exams. It is known that Mr. Li has passed the "Subject One" exam, and the score of "Subject One" is not affected by the "Comprehensive Subject". The "Comprehensive Subject" offers 5 chances to take the exam within three years. Once an exam is passed, the driving license will be issued, and the applicant will no longer participate in subsequent exams; otherwise, the exams will continue until the 5th attempt. The probabilities of Mr. Li passing the "Comprehensive Subject" in each attempt are 0.5, 0.6, 0.7, 0.8, and 0.9, respectively.
(1) Calculate the distribution and mathematical expectation of the number of times ξ that Mr. Li takes the driving license test within three years.
(2) Calculate the probability of Mr. Li obtaining a driving license within three years.

Answer: $0.9988$


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

Given $\overrightarrow{m}=(2\sqrt{3},1)$, $\overrightarrow{n}=(\cos^2 \frac{A}{2},\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$;
$(1)$ When $A= \frac{\pi}{2}$, find the value of $|\overrightarrow{n}|$;
$(2)$ If $C= \frac{2\pi}{3}$ and $|AB|=3$, when $\overrightarrow{m} \cdot \overrightarrow{n}$ takes the maximum value, find the magnitude of $A$ and the length of side $BC$.

Answer: $\sqrt{3}$


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

Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$.

Answer: $\dfrac{8}{27}$


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>.

Select 4 out of 6 sprinters to participate in a 4×100 relay race. If neither A nor B runs the first leg, then there are $\boxed{\text{different}}$ possible team compositions.

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 point O is the circumcenter of triangle ABC, and |BA|=2, |BC|=6, calculate the value of the dot product of the vectors BO and AC.

Answer: $16$


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

Given that $E, U, L, S, R,$ and $T$ represent the digits $1, 2, 3, 4, 5, 6$ (each letter represents a unique digit), and the following conditions are satisfied:
1. $E + U + L = 6$
2. $S + R + U + T = 18$
3. $U \times T = 15$
4. $S \times L = 8$

Determine the six-digit number $\overline{EULSRT}$.

Answer: $132465$


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

Compute: \(93 \times 107\).

Answer: $9951$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 five-digit number whose digits add up to 20?

Answer: $99200$


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

Given two points A (-2, 0), B (0, 2), and point C is any point on the circle $x^2+y^2-2x=0$, determine the minimum area of $\triangle ABC$.

Answer: $3- \sqrt{2}$


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

In triangle $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM} = x\overrightarrow{AB}$ and $\overrightarrow{AN} = y\overrightarrow{AC}$ ($x, y \neq 0$), then the minimum value of $4x+y$ is \_\_\_\_\_\_.

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


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

Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$.

Answer: $\frac{7}{5}$


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 coefficient of $\frac{1}{x}$ in the expansion of $((1-x^{2})^{4}(\frac{x+1}{x})^{5})$.

Answer: $-29$


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

Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{5}}{5}$, and its right latus rectum equation is $x=5$.
(1) Find the equation of the ellipse;
(2) A line $l$ with a slope of $1$ passes through the right focus $F$ of the ellipse, intersecting the ellipse $C$ at points $A$ and $B$. $P$ is a moving point on the ellipse. Find the maximum area of $\triangle PAB$.

Answer: $\frac{16\sqrt{10}}{9}$


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

Given $\vec{a}=(\cos \alpha, \sin \alpha), \vec{b}=(\cos \beta, \sin \beta)$, and $|\vec{a}-\vec{b}|=\frac{2 \sqrt{5}}{5}$. If $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, and $\sin \beta=-\frac{5}{13}$, then $\sin \alpha=$

Answer: $\frac{33}{65}$


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

A student typed out several circles on the computer as follows: ○●○○●○○○●○○○○●○○○○○●… If this pattern continues, forming a series of circles, then the number of ● in the first 120 circles is ______.

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 numbers x and y satisfy the conditions  $\begin{cases} x^2 + y^2 - xy = 2
x^4 + y^4 + x^2y^2 = 8  \end{cases}$ The value of  $P = x^8 + y^8 + x^{2014}y^{2014}$  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>.

Given the function $f(x)=\sin ^{2}x+ \frac{ \sqrt{3}}{2}\sin 2x$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) In $\triangle ABC$, if $f(\frac{A}{2})=1$ and the area of the triangle is $3\sqrt{3}$, find the minimum value of side $a$.

Answer: $2\sqrt{3}$


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

In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than Anne does Bob run for every lap around the block?

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>.

In a triangle $\triangle ABC$, $\angle A = 45^\circ$ and $\angle B = 60^\circ$. A line segment $DE$, with $D$ on $AB$ and $\angle ADE = 30^\circ$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{3}}{4}$
B) $\frac{\sqrt{6} + \sqrt{2}}{4}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$

Answer: $\frac{\sqrt{6} + \sqrt{2}}{4}$


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

Solve the following equations:

a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;

b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;

c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;

d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$.

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>.

A piece of string fits exactly once around the perimeter of a rectangle with a length of 16 and a width of 10. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?

Answer: $215$


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

Four friends have a total of 8 identical pencils, and each friend has at least one pencil. Additionally, the first friend always has at least two pencils. In how many ways can this happen?

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>.

Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.

Answer: $\frac{1}{12}$


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>.

Given an arithmetic sequence $\{a\_n\}$ with first term $a\_1$ and common difference $d$, let $S\_n$ denote the sum of its first $n$ terms. If the two intersection points of the line $y=a\_1x+m$ and the circle $x^2+(y-1)^2=1$ are symmetric about the line $x+y-d=0$, find the sum of the first 100 terms of the sequence $\{(\frac{1}{S\_n})\}$.

Answer: $\frac{200}{101}$


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

Let $M$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, with $F_1$ and $F_2$ as its foci. If $\angle F_1MF_2 = \frac{\pi}{6}$, calculate the area of $\triangle MF_1F_2$.

Answer: $16(2 - \sqrt{3})$


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

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with a common difference of 5. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n$ for all positive integers $n$, find the first term.

Answer: $2.5$


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

Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid  $5\$ $. How much did the ball cost?

Answer: $20$


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

Let
\[ x^6 - 3x^3 - x^2 - x - 2 = q_1(x) q_2(x) \dotsm q_m(x), \]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(3) + q_2(3) + \dots + q_m(3)$.

Answer: $634$


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

Exactly three faces of a \(2 \times 2 \times 2\) cube are partially shaded. Calculate the fraction of the total surface area of the cube that is shaded.

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


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

A hotel has three types of rooms available: a triple room, a double room, and a single room (each type of room can only accommodate the corresponding number of people). There are 4 adult men and 2 little boys looking for accommodation. The little boys should not stay in a room by themselves (they must be accompanied by an adult). If all three rooms are occupied, the number of different arrangements for accommodation is $\boxed{36}$.

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>.

In the subtraction shown, \( K, L, M \), and \( N \) are digits. What is the value of \( K+L+M+N \)?

\[
\begin{array}{llll}
5 & K & 3 & L \\
\end{array}
\]
\[
\begin{array}{r}
M & 4 & N & 1 \\
\hline
4 & 4 & 5 & 1 \\
\end{array}
\]

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>.

Compute $18\left(\frac{200}{3} + \frac{50}{6} + \frac{16}{18} + 2\right)$.

Answer: $1402$


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

Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$

Answer: $1998953$


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

In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are then placed on top of the grid. Five of the circles have a diameter equal to a side of a small square (1 cm), and one large circle has a diameter equal to the length of four sides of a small square (4 cm). The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?

Answer: $41.25$


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>.

Let $O$ be the center of the square $ABCD$. If $3$ points are chosen from $O$, $A$, $B$, $C$, and $D$ at random, calculate the probability that the $3$ points are collinear.

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


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

Given the equation \(\left|x^{2}-2ax+b\right|=8\) has exactly three real roots, and these roots are the side lengths of a right triangle. Find the value of \(a+b\).

Answer: $264$


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

Mia is designing a rectangular flower bed against her house on one side, using 450 feet of fencing to enclose the remaining three sides. If 150 feet of it is planned to be used along the house, what is the maximum area she can achieve for her garden?

Answer: $22500$


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

Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$.

Answer: $64\sqrt{2}$


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>.

Compute and simplify the following expressions:
1. $(1)(1 \frac{1}{2})^{0}-(1-0.5^{-2})÷(\frac{27}{8})^{\frac{2}{3}}$
2. $\sqrt{2 \sqrt{2 \sqrt{2}}}$

Answer: $2^{\frac{7}{8}}$


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

In the arithmetic sequence $\{a_n\}$, $a_4 = 2$ and $a_7 = -4$. Suppose we randomly select numbers from the first 10 terms of $\{a_n\}$, replacing the number after each selection. We perform this selection process three times, assuming each selection is independent. What is the probability that among the three selected numbers, exactly two are positive and one is negative?

Answer: $\frac{6}{25}$


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

A certain car beauty company launched a promotional activity to attract customers: for first-time customers, the charge is $200 per visit, and they can register as members. Members will receive corresponding discounts for each subsequent visit, as shown in the table below:

| Number of Visits | 1st | 2nd | 3rd | 4th | 5th and above |
|------------------|-----|-----|-----|-----|---------------|
| Charge Ratio     | 1   | 0.95| 0.90| 0.85| 0.80          |

The company randomly selected 100 registered members for a survey and obtained the following statistical data:

| Number of Visits | 1st | 2nd | 3rd | 4th | 5th and above |
|------------------|-----|-----|-----|-----|---------------|
| Frequency        | 60  | 20  | 10  | 5   | 5             |

Assuming the cost for the car beauty service is $150 each time, answer the following questions based on the given data:

$(1)$ Estimate the probability that a member of the company will consume at least twice.

$(2)$ If a member only consumes twice, calculate the average profit the company will make from these two visits.

$(3)$ The company wants to randomly sample 6 customers who have visited twice and three times from the 100 members using stratified random sampling. Then, 2 customers will be selected from these 6 customers to receive souvenirs. Calculate the probability that exactly 1 of the 2 selected customers has visited twice.

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


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>.

Estimate
99×71≈
25×39≈
124÷3≈
398÷5≈

Answer: $80$


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

One hundred bricks, each measuring $3''\times 8''\times 15''$, are stacked to form a tower. Each brick can contribute $3''$, $8''$, or $15''$ to the height of the tower. How many different tower heights can be achieved using all one hundred bricks?

Answer: $1201$


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

Given that $\cos{\alpha}=-\frac{4}{5}$, where $\alpha$ is an angle in the third quadrant, find the value of $\sin{\left(\alpha-\frac{\pi}{4}\right)$.

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


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $(2a-c)\cos B = b\cos C$;
(1) Find the magnitude of angle $B$;
(2) Let $\overrightarrow{m}=(\sin A, \cos 2A), \overrightarrow{n}=(4k,1)$ ($k>1$), and the maximum value of $\overrightarrow{m} \cdot \overrightarrow{n}$ is 5, find the value of $k$.

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


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

Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 \; (a > b > 0)$, one of its vertices is $M(0,1)$, and its eccentricity $e = \dfrac {\sqrt {6}}{3}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Suppose a line $l$ intersects the ellipse at points $A$ and $B$, and the distance from the origin $O$ to line $l$ is $\dfrac {\sqrt {3}}{2}$, find the maximum area of $\triangle AOB$.

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


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

A computer software retailer has 1200 copies of a new software package to sell. From past experience, she knows that:

- Half of them will sell right away at the original price she sets,
- Two-thirds of the remainder will sell later when their price is reduced by 40%, and
- The remaining copies will sell in a clearance sale at 75% off the original price.

In order to make a reasonable profit, the total sales revenue must be $72000. To the nearest cent, what original price should she set?

Answer: $80.90


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

Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$.

Answer: $-\frac{17}{18}$


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

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths  $3$ ,  $4$ , and  $5$ . What is the area of the triangle?

Answer: $\frac{9}{\pi^2}\left(\sqrt{3}+3\right)$


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

In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\cos(B-C)+a\cos A=2\sqrt{3}b\sin C\cos A$.
$(1)$ Find angle $A$;
$(2)$ If the perimeter of $\triangle ABC$ is $8$ and the radius of the circumcircle is $\sqrt{3}$, find the area of $\triangle ABC$.

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


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

The function \( f(x) = x^{2} + a x + 3a \) has integer roots. What is the sum of all possible values of \( a \)?

Answer: $24$


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

$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$

Answer: $365$


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

Given the function $f(x)=|2x+1|-|x|-2$.
(1) Solve the inequality $f(x)\geqslant 0$;
(2) If there exists a real number $x$ such that $f(x)-a\leqslant |x|$, find the minimum value of the real number $a$.

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>.

In a variant of SHORT BINGO, a $5\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:
- First column contains 5 distinct numbers from the set $1-15$.
- Second column contains 5 distinct numbers from the set $16-30$.
- Third column (excluding the FREE middle square) contains 4 distinct numbers from $31-45$.
- Fourth column contains 5 distinct numbers from the set $46-60$.
- Fifth column contains 5 distinct numbers from the set $61-75$.

How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card?

Answer: $360360$


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

Let $S_n$ and $T_n$ respectively be the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$. Given that $\frac{S_n}{T_n} = \frac{n}{2n+1}$ for $n \in \mathbb{N}^*$, find the value of $\frac{a_5}{b_5}$.

Answer: $\frac{9}{19}$


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>.

Given the function $f\left(x\right)=\frac{1}{3}a{x}^{3}-\frac{1}{2}b{x}^{2}+x$, where $a$ and $b$ are the outcomes of rolling two dice consecutively, find the probability that the function $f'\left( x \right)$ takes its extreme value at $x=1$.

Answer: $\frac{1}{12}$


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>.

Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______.

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>.

In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.

[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W);
[/asy]

Answer: $7.5$


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

Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$

Answer: $80$


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

Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that was spent attending meetings.

Answer: $33.33\%$


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>.

For $\{1, 2, 3, ..., n\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$.

Answer: $5120$


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

Derek fills a square  $10$  by  $10$  grid with  $50$   $1$ s and  $50$   $2$ s. He takes the product of the numbers in each of the  $10$  rows. He takes the product of the numbers in each of the  $10$  columns. He then sums these  $20$  products up to get an integer  $N.$  Find the minimum possible value of  $N.$

Answer: $640$


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

A real number  $ to $  is randomly and uniformly chosen from the  $ [- 3,4] $  interval. What is the probability that all roots of the polynomial  $ x ^ 3 + ax ^ 2 + ax + 1 $  are real?

Answer: $3/7$


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>.

Find all values of $x$ with $0 \le x < \pi$ that satisfy $\sin x - \cos x = 1$. Enter all the solutions, separated by commas.

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


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

In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b+2$ ($k \neq 0$) intersects the positive half of the x-axis at point A and the positive half of the y-axis at point B.
(1) Express the area $S_{\triangle AOB}$ of triangle $AOB$ in terms of $b$ and $k$.
(2) If the area $S_{\triangle AOB} = |OA| + |OB| + 3$,
   ① Express $k$ in terms of $b$ and determine the range of values for $b$.
   ② Find the minimum value of the area of triangle $AOB$.

Answer: $7 + 2\sqrt{10}$


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

The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on the segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$?

Answer: $1/8$


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

There are 7 people standing in a row. How many different arrangements are there according to the following requirements?

(1) Among them, A cannot stand at the head, and B cannot stand at the tail;
(2) Among them, A, B, and C must stand next to each other;
(3) Among them, A, B, and C must not stand next to each other;
(4) There is exactly one person between A and B;
(5) A, B, and C must stand in order from left to right.

Answer: $840$


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

When written out in full, the number \((10^{2020}+2020)^{2}\) has 4041 digits. What is the sum of the digits of this 4041-digit number?
A) 9
B) 17
C) 25
D) 2048
E) 4041

Answer: $25$


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

In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?

Answer: $50$


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

In an arithmetic sequence $\{a_n\}$, $S_n$ represents the sum of the first $n$ terms. Given that $a_4 + a_8 = 4$, find the value of $S_{11} + a_6$.

Answer: $24$


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

In $\Delta ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $c=\frac{7}{2}$, the area of $\Delta ABC$ is $\frac{3\sqrt{3}}{2}$, and $\tan A+\tan B=\sqrt{3}(\tan A\tan B-1)$,

(1) Find the measure of angle $C$;

(2) Find the value of $a+b$.

Answer: $\frac{11}{2}$


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 quadratic function $f(x)=3x^{2}-2x$, the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n \in \mathbb{N}^{*}$) is on the graph of the function $y=f(x)$.

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

$(2)$ Let $b_{n}=\frac{3}{a_{n}a_{n+1}}$, and $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find the smallest positive integer $m$ such that $T_{n} < \frac{m}{20}$ holds for all $n \in \mathbb{N}^{*}$.

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 school organizes a table tennis competition. Class A has 5 male students and 3 female students registered; Class B has 6 male students and 2 female students registered. If 2 students are selected from each of Class A and Class B, there are a total of $\boxed{345}$ different ways to select 4 students such that exactly 1 female student is among them.

Answer: $345$


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

The integer $n > 9$ is a root of the quadratic equation $x^2 - ax + b=0$. In this equation, the representation of $a$ in the base-$n$ system is $19$. Determine the base-$n$ representation of $b$.

Answer: $90_n$


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

The sequence \(\left\{a_{n}\right\}\) satisfies \(a_{1}=1, \sqrt{\frac{1}{a_{n}^{2}}+4}=\frac{1}{a_{n+1}}\). Let \(S_{n}=\sum_{i=1}^{n} a_{i}^{2}\). If \(S_{2 n+1}-S_{n} \leqslant \frac{t}{30}\) holds for any \(n \in \mathbf{N}^{*}\), what is the smallest positive integer \(t\)?

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 starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels parallel to these lines. Astronaut Gavrila recorded that every 7 seconds the ship encounters meteors coming towards it, and every 13 seconds it overtakes meteors traveling in the same direction. He wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavrila correct? If so, write down this arithmetic mean. If not, indicate the correct time in seconds, rounded to the nearest tenth.

Answer: $9.1$


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>.

Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. Calculate the area of square $ABCD$.
A) 100
B) 144
C) 196
D) 256
E) 324

Answer: $144$


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

Let  $\mathcal{P}$  be the set of all polynomials  $p(x)=x^4+2x^2+mx+n$ , where  $m$  and  $n$  range over the positive reals. There exists a unique  $p(x) \in \mathcal{P}$  such that  $p(x)$  has a real root,  $m$  is minimized, and  $p(1)=99$ . Find  $n$ .

*Proposed by **AOPS12142015***

Answer: $56$


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

Given that $\binom{21}{13}=20349$, $\binom{21}{14}=11628$, and $\binom{23}{15}=490314$, find $\binom{22}{15}$.

Answer: $458337$


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

Given the sequence $\{a\_n\}$, where $a\_1=a\_2=1$ and $a_{n+2}-a_{n}=1$, find the sum of the first 100 terms of the sequence.

Answer: $2550$


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

Given \(w\) and \(z\) are complex numbers such that \(|w+z|=2\) and \(|w^2+z^2|=8,\) find the smallest possible value of \(|w^3+z^3|.\)

Answer: $20$


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

Let \( x \) and \( y \) be positive integers such that

\[ x^2 + y^2 - 2017xy > 0 \]

and it is not a perfect square. Find the minimum value of \( x^2 + y^2 - 2017xy \).

Answer: $2019$


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

Suppose the function \( g(x) \) is defined for \( x \in \{a, b, c\} \) such that the points \( (a, g(a)), (b, g(b)), (c, g(c)) \) form a triangle with an area of \( 24 \). The graph of \( y = 3g(3x) \) likewise contains just three points. Determine the area of this triangle.

Answer: $24$


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

Given the function $f(x)=A\sin^2(\omega x+\frac{\pi}{8})$ ($A>0, \omega>0$), the graph of which is symmetric with respect to the point $({\frac{\pi}{2},1})$, and its minimum positive period is $T$, where $\frac{\pi}{2}<T<\frac{3\pi}{2}$. Find the value of $\omega$.

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


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

How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?

Answer: $616$


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

How many times does the digit 9 appear in the list of all integers from 1 to 1000?

Answer: $301$


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

Given that positive real numbers $x$ and $y$ satisfy $e^{x}=y\ln x+y\ln y$, then the minimum value of $\frac{{e}^{x}}{x}-\ln y$ is ______.

Answer: $e-1$


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>.

What is the sum of all the four-digit positive integers?

Answer: $49495500$


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

The graph of the function $y= \sqrt {3}\cos x+\sin x, (x\in \mathbb{R})$ is translated to the right by $\theta$ ($\theta>0$) units. Determine the minimum value of $\theta$ such that the resulting graph is symmetric about the y-axis.

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


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

Suppose there exist constants  $A$ ,  $B$ ,  $C$ , and  $D$  such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers  $n\geq 4$ .  What is  $A+B+C+D$ ?

*Proposed by David Altizio*

Answer: $75$


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

In $\triangle ABC$, $2\sin 2C\cdot\cos C-\sin 3C= \sqrt {3}(1-\cos C)$.
(1) Find the measure of angle $C$;
(2) If $AB=2$, and $\sin C+\sin (B-A)=2\sin 2A$, find the area of $\triangle ABC$.

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


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

Given that $|$$\overrightarrow {a}$$ $|=1$, $\overrightarrow {b}$ $=$ ($    $\frac { \sqrt {3}}{3} $, $    \frac { \sqrt {3}}{3}$), and $|$ $\overrightarrow {a}$ $+3 \overrightarrow {b}$ $|=2$, find the projection of $\overrightarrow {b}$ in the direction of $\overrightarrow {a}$.

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