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

Given the function $f(x)=\cos (\sqrt{3}x+\varphi)$, where $\varphi \in (-\pi, 0)$. If the function $g(x)=f(x)+f'(x)$ (where $f'(x)$ is the derivative of $f(x)$) is an even function, determine the value of $\varphi$.

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


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

Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2.

Answer: $312$


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

Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\sin \left(\pi\left(\frac{x}{2}+1\right)\right)}$$

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


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

A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of white tiles, ignoring part tiles.

Answer: $1: 2$


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

Six positive numbers, not exceeding 3, satisfy the equations \(a + b + c + d = 6\) and \(e + f = 2\). What is the minimum value of the expression

$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$

Answer: $72$


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

Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number.

Answer: $1681$


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

The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 years. How old is the mother now if she gave birth to her daughter at the age of 26?

Answer: $33$


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

Given $sinα-\sqrt{3}cosα=1$, then the value of $sin({\frac{{7π}}{6}-2α})$ is ______.

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


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

Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=5$, and $|r-s|=6$. What is the sum of all possible values of $|p-s|$?

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 an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$.

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


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

A certain fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:

| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|---------------------------------------|------------------------|
| First          | $60$                                  | $40$                                  | $1520$                 |
| Second         | $30$                                  | $50$                                  | $1360$                 |

$(1)$ Find the purchase prices of type A and type B fruits.
$(2)$ After selling all the fruits purchased in the first two batches, the fruit store decides to reward customers by launching a promotion. In the third purchase, a total of $200$ $\text{kg}$ of type A and type B fruits are bought, and the capital invested does not exceed $3360$ $\text{元}$. Of these, $m$ $\text{kg}$ of type A fruit and $3m$ $\text{kg}$ of type B fruit are sold at the purchase price, while the remaining type A fruit is sold at $17$ $\text{元}$ per $\text{kg}$ and type B fruit is sold at $30$ $\text{元}$ per $\text{kg}$. If all $200$ $\text{kg}$ of fruits purchased in the third batch are sold, and the maximum profit obtained is not less than $800$ $\text{元}$, find the maximum value of the positive integer $m$.

Answer: $22$


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

Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest for the following month is calculated on the new balance. After how many months will there be enough money in the account to buy the car?

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>.

In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward?

Answer: $129$


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

Find the number ot 6-tuples  $(x_1, x_2,...,x_6)$ , where  $x_i=0,1 or 2$  and  $x_1+x_2+...+x_6$  is even

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>.

Point \( F \) is the midpoint of side \( BC \) of square \( ABCD \). A perpendicular \( AE \) is drawn to segment \( DF \). Find the angle \( CEF \).

Answer: $45$


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

The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\)

Answer: $3999999$


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

How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set?

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>.

Given a parabola $y = ax^2 + bx + c$ ($a \neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \in \{-3, -2, -1, 0, 1, 2, 3\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$.

Answer: $\frac{8}{9}$


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>.

A number \( n \) has a sum of digits equal to 100, while \( 44n \) has a sum of digits equal to 800. Find the sum of the digits of \( 3n \).

Answer: $300$


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

The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45
(c) Between 45 and 55
(d) More than 55
(e) 55

Answer: $55$


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

Among the positive integers less than 1000, there are     how many numbers that are perfect squares but not perfect cubes?

Answer: $28$


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

The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?

Answer: $31254$


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

Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$.

Answer: $4032$


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

Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\).

Answer: $\sqrt{65}$


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 chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\begin{cases} x=2-\frac{1}{2}t \\ y=-1+\frac{1}{2}t \end{cases} (t\text{ is the parameter})$ has a length of $\_\_\_\_\_\_\_.$

Answer: $\sqrt{14}$


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

In a triangle, the lengths of the three sides are integers \( l, m, n \), with \( l > m > n \). It is known that \( \left\{ \frac{3^{l}}{10^{4}} \right\} = \left\{ \frac{3^{m}}{10^{4}} \right\} = \left\{ \frac{3^{n}}{10^{4}} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \) and \( [x] \) denotes the greatest integer less than or equal to \( x \). Determine the smallest possible value of the perimeter of such a triangle.

Answer: $3003$


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

Group the set of positive odd numbers {1, 3, 5, ...} from smallest to largest, where the $n$-th group contains $2n-1$ odd numbers. That is, the first group, the second group, the third group... consist of the sets {1}, {3, 5, 7}, {9, 11, 13, 15, 17}, ..., respectively. In which group does 2007 belong?

Answer: $32$


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

During a break between voyages, a sailor turned 20 years old. All six crew members gathered in the cabin to celebrate. "I am twice the age of the cabin boy and 6 years older than the engineer," said the helmsman. "And I am as much older than the cabin boy as I am younger than the engineer," noted the boatswain. "In addition, I am 4 years older than the sailor." "The average age of the crew is 28 years," reported the captain. How old is the captain?

Answer: $40$


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

We define \( a \star b = a \times a - b \times b \). Find the value of \( 3 \star 2 + 4 \star 3 + 5 \star 4 + \cdots + 20 \star 19 \).

Answer: $396$


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

A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 50 cm to the main gas pipe.

Answer: $1/16$


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>.

Consider a 5x5 grid of squares. How many different squares can be traced using the lines in this grid?

Answer: $55$


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

On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.

The first person said: "Exactly 1 percent of the people present in this room are liars."

The second person said: "Exactly 2 percent of the people present in this room are liars."

and so on.

The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."

How many people could be in the room, given that it is known that at least one of them is a knight?

Answer: $100$


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

Acme Corporation has released a new version of its vowel soup where each vowel (A, E, I, O, U) appears six times, and additionally, each bowl contains one wildcard character that can represent any vowel. How many six-letter "words" can be formed from a bowl of this new Acme Enhanced Vowel Soup?

Answer: $46656$


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

Let  $a,b,c$  be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum  $a + b + c$ . Find this sum.

Answer: $55$


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

For the cubic function $f(x)=ax^3+bx^2+cx+d$ ($a\neq 0$), define: Let $f''(x)$ be the derivative of the derivative of the function $y=f(x)$, that is, the second derivative of $f(x)$. If the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called the "inflection point" of the function $y=f(x)$. Some students found that "every cubic function has an 'inflection point'; every cubic function has a center of symmetry; and the 'inflection point' is the center of symmetry." Based on this discovery, for the function $$f(x)=x^3- \frac{3}{2}x^2+3x- \frac{1}{4},$$ its center of symmetry is ___________; calculate $$f\left( \frac{1}{2013}\right)+f\left( \frac{2}{2013}\right)+f\left( \frac{3}{2013}\right)+\cdots +f\left( \frac{2012}{2013}\right)$$ = ___________.

Answer: $2012$


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

The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.

Answer: $37.5\%$


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>.

Célia wants to trade with Guilherme stickers from an album about Brazilian animals. Célia wants to trade four butterfly stickers, five shark stickers, three snake stickers, six parakeet stickers, and six monkey stickers. All of Guilherme's stickers are of spiders. They know that:

(a) one butterfly sticker is worth three shark stickers;

(b) one snake sticker is worth three parakeet stickers;

(c) one monkey sticker is worth four spider stickers;

(d) one parakeet sticker is worth three spider stickers;

(e) one shark sticker is worth two parakeet stickers.

How many stickers can Célia receive if she trades all she wants?

Answer: $171$


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

A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number?

Answer: $1221$


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

A "fifty percent mirror" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two "fifty percent mirrors" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to the left of the two mirrors?

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


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

Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$

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


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 a trapezoid $ABCD$ with $\angle A = \angle B = 90^{\circ}$, $|AB| = 5 \text{cm}$, $|BC| = 1 \text{cm}$, and $|AD| = 4 \text{cm}$, point $M$ is taken on side $AB$ such that $2 \angle BMC = \angle AMD$. Find the ratio $|AM| : |BM|$.

Answer: $3/2$


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

For any positive integer \( n \), let \( f(n) \) represent the last digit of \( 1 + 2 + 3 + \cdots + n \). For example, \( f(1) = 1 \), \( f(2) = 3 \), \( f(5) = 5 \), and so on. Find the value of \( f(2) + f(4) + f(6) + \cdots + f(2012) \).

Answer: $3523$


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

When $8000^{50}$ is expanded out, the result is $1$ followed by how many zeros?

Answer: $150$


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

A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse.

Answer: $\frac{48\pi}{5}$


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

The diagram shows a regular dodecagon and a square, whose vertices are also vertices of the dodecagon. What is the value of the ratio of the area of the square to the area of the dodecagon?

Answer: $2:3$


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

A right triangle  $ABC$  is inscribed in the circular base of a cone. If two of the side lengths of  $ABC$  are  $3$  and  $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is  $3$ , then the minimum possible volume of the cone can be written as  $\frac{m\pi\sqrt{n}}{p}$ , where  $m$ ,  $n$ , and  $p$  are positive integers,  $m$  and  $p$  are relatively prime, and  $n$  is squarefree. Find  $m + n + p$ .

Answer: $60$


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

Given that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\frac{\sqrt{6}}{3},-1)$, calculate the area of the ellipse.

Answer: $12\sqrt{3}\pi$


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

We are given a triangle  $ABC$ . Points  $D$  and  $E$  on the line  $AB$  are such that  $AD=AC$  and  $BE=BC$ , with the arrangment of points  $D - A - B - E$ . The circumscribed circles of the triangles  $DBC$  and  $EAC$  meet again at the point  $X\neq C$ , and the circumscribed circles of the triangles  $DEC$  and  $ABC$  meet again at the point  $Y\neq C$ . Find the measure of  $\angle ACB$  given the condition  $DY+EY=2XY$ .

Answer: $60$


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

Calculate the limit of the function:

\[
\lim_{x \rightarrow 0} \frac{\arcsin(2x)}{2^{-3x} - 1} \cdot \ln 2
\]

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


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

Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $60^{\circ}$, and $\overrightarrow{a}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{e_{1}}$.

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


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>.

It is known that
\[
\begin{array}{l}
1 = 1^2 \\
1 + 3 = 2^2 \\
1 + 3 + 5 = 3^2 \\
1 + 3 + 5 + 7 = 4^2
\end{array}
\]
If \(1 + 3 + 5 + \ldots + n = 20^2\), find \(n\).

Answer: $39$


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

If the real numbers \(x\) and \(y\) satisfy the equation \((x-2)^{2}+(y-1)^{2}=1\), then the minimum value of \(x^{2}+y^{2}\) is \_\_\_\_\_.

Answer: $6-2\sqrt{5}$


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

For the subset \( S \) of the set \(\{1,2, \cdots, 15\}\), if a positive integer \( n \) and \( n+|S| \) are both elements of \( S \), then \( n \) is called a "good number" of \( S \). If a subset \( S \) has at least one "good number", then \( S \) is called a "good set". Suppose 7 is a "good number" of a "good set" \( X \). How many such subsets \( X \) are there?

Answer: $4096$


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

Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$

Find all possible values of $a_{2021}$.

Answer: $2021$


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

If the integer part of $\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\_\_\_\_\_\_$.

Answer: $\sqrt{10} - 3$


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 group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300?

Answer: $295$


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

In $\triangle ABC$, it is known that $\cos A= \frac{1}{7}$, $\cos (A-B)= \frac{13}{14}$, and $0 < B < A < \frac{\pi}{2}$. Find the measure of angle $B$.

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


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

The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\).

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


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

If \(100^a = 4\) and \(100^b = 5\), then find \(20^{(1 - a - b)/(2(1 - b))}\).

Answer: $\sqrt{5}$


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

Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c}
&&8&6&7&4_h \\
&+&4&3&2&9_h \\
\cline{2-6}
&1&3&0&0&3_h.
\end{array}$$

Answer: $10$


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

On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.

Answer: $52.5$


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>.

What is the area of the region defined by the equation $x^2 + y^2 - 3 = 6y - 18x + 9$?

Answer: $102\pi$


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

When a granary records the arrival of 30 tons of grain as "+30", determine the meaning of "-30".

Answer: $-30$


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

Given the function $f(x)=\frac{1}{3}x^3-ax^2+(a^2-1)x+b$, where $(a,b \in \mathbb{R})$
(I) If $x=1$ is an extreme point of $f(x)$, find the value of $a$;
(II) If the equation of the tangent line to the graph of $y=f(x)$ at the point $(1, f(1))$ is $x+y-3=0$, find the maximum and minimum values of $f(x)$ on the interval $[-2, 4]$.

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>.

The distance from the point of intersection of a circle's diameter with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.

$$
AB = 18, EO = 7, AE = 2BE, R = ?
$$

Answer: $11$


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

If digits  $A$ ,  $B$ , and  $C$  (between  $0$  and  $9$  inclusive) satisfy
    \begin{tabular}{c@{\,}c@{\,}c@{\,}c}
      &  $C$   &  $C$  &  $A$
    + &  $B$  &  $2$  &  $B$  \hline
      &  $A$  &  $8$  &  $8$
    \end{tabular}
    what is  $A \cdot B \cdot C$ ?

*2021 CCA Math Bonanza Individual Round #5*

Answer: $42$


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

For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$

Answer: $\frac{22}{5}$


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

Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit.

Answer: $8281$


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

Count all the distinct anagrams of the word "YOANN".

Answer: $60$


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

Given a finite sequence $\{a\_1\}, \{a\_2\}, \ldots \{a\_m\} (m \in \mathbb{Z}^+)$ that satisfies the conditions: $\{a\_1\} = \{a\_m\}, \{a\_2\} = \{a\_{m-1}\}, \ldots \{a\_m\} = \{a\_1\}$, it is called a "symmetric sequence" with the additional property that in a $21$-term "symmetric sequence" $\{c\_n\}$, the terms $\{c\_{11}\}, \{c\_{12}\}, \ldots, \{c\_{21}\}$ form an arithmetic sequence with first term $1$ and common difference $2$. Find the value of $\{c\_2\}$.

Answer: $19$


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

Given that the terminal side of the angle $α+ \frac {π}{6}$ passes through point P($-1$, $-2\sqrt {2}$), find the value of $\sinα$.

Answer: $\frac{1-2\sqrt {6}}{6}$


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 which numeral system is 792 divisible by 297?

Answer: $19$


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

Can you use the four basic arithmetic operations (addition, subtraction, multiplication, division) and parentheses to write the number 2016 using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence?

Answer: $2016$


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

Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover?

Answer: $4.75$


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>.

Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value.

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


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>.

Consider a "Modulo $m$ graph paper," with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \leq x, y < 17$. We graph the congruence $$5x \equiv 3y + 2 \pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < 17$. Determine $x_0 + y_0$.

Answer: $19$


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

In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?

Answer: $176$


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

For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?

Answer: $500$


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

A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two, Xiao Hua actually paid ____ yuan at checkout.

Answer: $486$


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

Let \(P\) and \(P+2\) be both prime numbers satisfying \(P(P+2) \leq 2007\). If \(S\) represents the sum of such possible values of \(P\), find the value of \(S\).

Answer: $106$


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

Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence.

Answer: $1039$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, with the origin O as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. The polar coordinate of point P is $(1, \pi)$. Given the curve $C: \rho=2\sqrt{2}a\sin(\theta+ \frac{\pi}{4}) (a>0)$, and a line $l$ passes through point P, whose parametric equation is:
$$
\begin{cases}
x=m+ \frac{1}{2}t \\
y= \frac{\sqrt{3}}{2}t
\end{cases}
$$
($t$ is the parameter), and the line $l$ intersects the curve $C$ at points M and N.
(1) Write the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If $|PM|+|PN|=5$, find the value of $a$.

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


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>.

Find the smallest integer \( n \) such that the expanded form of \( (xy - 7x - 3y + 21)^n \) has 2012 terms.

Answer: $44$


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

How many integers from 1 to 1997 have a sum of digits that is divisible by 5?

Answer: $399$


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

Given a positive geometric sequence $\{a_{n}\}$, if ${a_m}{a_n}=a_3^2$, find the minimum value of $\frac{2}{m}+\frac{1}{{2n}}$.

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


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

Find the value of $x$ if $\log_8 x = 1.75$.

Answer: $32\sqrt[4]{2}$


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>.

Niall's four children have different integer ages under 18. The product of their ages is 882. What is the sum of their ages?

Answer: $31$


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

Find all three-digit numbers $\overline{\Pi B \Gamma}$, consisting of distinct digits $\Pi, B$, and $\Gamma$, for which the following equality holds: $\overline{\Pi B \Gamma} = (\Pi + B + \Gamma) \times (\Pi + B + \Gamma + 1)$.

Answer: $156$


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

Using $1 \times 2$ tiles to cover a $2 \times 10$ grid, how many different ways are there to cover the grid?

Answer: $89$


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

Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a} - 2\overrightarrow{b}| \leqslant 2$, find the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$.

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


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

Suppose that \(a, b, c,\) and \(d\) are positive integers which are not necessarily distinct. If \(a^{2}+b^{2}+c^{2}+d^{2}=70\), what is the largest possible value of \(a+b+c+d?\)

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>.

We write the equation on the board:

$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$

We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?

Answer: $2016$


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

Let the sequence $x, 3x+3, 5x+5, \dots$ be in geometric progression. What is the fourth term of this sequence?

Answer: $-\frac{125}{12}$


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>.

Evaluate the greatest integer less than or equal to \[\frac{5^{150} + 3^{150}}{5^{147} + 3^{147}}.\]

Answer: $124$


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

How many different right-angled triangles exist, one of the legs of which is \(\sqrt{2016}\), and the other leg and hypotenuse are expressed in natural numbers?

Answer: $12$


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

Given the operation defined as \(a \odot b \odot c = a \times b \times c + (a \times b + b \times c + c \times a) - (a + b + c)\), calculate \(1 \odot 43 \odot 47\).

Answer: $4041$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 length of the side opposite to angle $A$ is $2$, and the vectors $\overrightarrow{m} = (2, 2\cos^2\frac{B+C}{2} - 1)$ and $\overrightarrow{n} = (\sin\frac{A}{2}, -1)$.

1. Find the value of angle $A$ when the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.
2. Under the conditions of part (1), find the maximum area of $\triangle ABC$.

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 an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.

Answer: $33$


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

Given that point \\(A\\) on the terminal side of angle \\(\alpha\\) has coordinates \\(\left( \sqrt{3}, -1\right)\\),
\\((1)\\) Find the set of angle \\(\alpha\\)
\\((2)\\) Simplify the following expression and find its value: \\( \dfrac{\sin (2\pi-\alpha)\tan (\pi+\alpha)\cot (-\alpha-\pi)}{\csc (-\alpha)\cos (\pi-\alpha)\tan (3\pi-\alpha)} \\)

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


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

Given that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,
(1) Find the range of $z=2x+y$;
(2) If $x+y+a\geqslant 0$ always holds, find the range of real numbers $a$;
(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$.

Answer: $6-2\sqrt{73}$


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

A student is given a budget of $10,000 to produce a rectangular banner for a school function. The length and width (in meters) of the banner must be integers. If each meter in length costs $330 while each meter in width costs $450, what is the maximum area (in square meters) of the banner that can be produced?

Answer: $165$


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

Let \(\triangle ABC\) be equilateral, and let \(D, E, F\) be points on sides \(BC, CA, AB\) respectively, with \(FA = 9\), \(AE = EC = 6\), and \(CD = 4\). Determine the measure (in degrees) of \(\angle DEF\).

Answer: $60$


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

A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms?

Answer: $10000$


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

Given an ellipse $\frac{x^2}{25}+\frac{y^2}{m^2}=1\left(m \gt 0\right)$ with one focus at $F\left(0,4\right)$, find $m$.

Answer: $\sqrt{41}$


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

Define the operation "□" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\lg|x+2|$ ($x\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$.

Answer: $-8$


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

Given that the point $P$ on the ellipse $\frac{x^{2}}{64} + \frac{y^{2}}{28} = 1$ is 4 units away from the left focus, find the distance from point $P$ to the right directrix.

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>.

15. If \( a = 1.69 \), \( b = 1.73 \), and \( c = 0.48 \), find the value of
$$
\frac{1}{a^{2} - a c - a b + b c} + \frac{2}{b^{2} - a b - b c + a c} + \frac{1}{c^{2} - a c - b c + a b}.
$$

Answer: $20$


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

Given the polar equation of curve C is $\rho - 6\cos\theta + 2\sin\theta + \frac{1}{\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through point P(3, 3) with an inclination angle $\alpha = \frac{\pi}{3}$.
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;
(2) Suppose $l$ intersects curve C at points A and B, find the value of $|AB|$.

Answer: $2\sqrt{5}$


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

Given an equilateral triangle \( ABC \). Point \( K \) is the midpoint of side \( AB \), and point \( M \) lies on side \( BC \) such that \( BM : MC = 1 : 3 \). A point \( P \) is chosen on side \( AC \) such that the perimeter of triangle \( PKM \) is minimized. In what ratio does point \( P \) divide side \( AC \)?

Answer: $2/3$


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

If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam?

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>.

Find the sum of the roots of $\tan^2 x - 8\tan x + \sqrt{2} = 0$ that are between $x=0$ and $x=2\pi$ radians.

Answer: $4\pi$


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

Given that \(\log_8 2 = 0.2525\) in base 8 (to 4 decimal places), find \(\log_8 4\) in base 8 (to 4 decimal places).

Answer: $0.5050$


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

In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.

Answer: $6/5$


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

Determine the number of palindromes between 1000 and 10000 that are multiples of 6.

Answer: $13$


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

Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older brothers paid 800 rubles, and the younger brothers shared this money among themselves, so that everyone ended up with an equal share. What is the value of one house?

Answer: $2000$


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

An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.

Answer: $1/9$


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>.

A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.

Answer: $4/15$


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>.

Find the largest four-digit number in which all digits are different and which is divisible by 2, 5, 9, and 11.

Answer: $8910$


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

Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers.

Answer: $9.625$


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

Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]

Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\).

Answer: $2016$


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

A rectangular metal plate measuring \(10\) cm by \(8\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process.

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>.

Compute the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx
$$

Answer: $\frac{3\pi}{8}$


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>.

Let $a, b$ be real numbers. If the complex number $\frac{1+2i}{a+bi} \= 1+i$, then $a=\_\_\_\_$ and $b=\_\_\_\_$.

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


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

A certain TV station randomly selected $100$ viewers to evaluate a TV program in order to understand the evaluation of the same TV program by viewers of different genders. It is known that the ratio of the number of "male" to "female" viewers selected is $9:11$. The evaluation results are divided into "like" and "dislike", and some evaluation results are organized in the table below.

| Gender | Like | Dislike | Total |
|--------|------|---------|-------|
| Male   | $15$ |         |       |
| Female |      |         |       |
| Total  | $50$ |         | $100$  |

$(1)$ Based on the given data, complete the $2\times 2$ contingency table above. According to the independence test with $\alpha = 0.005$, can it be concluded that gender is related to the evaluation results?
$(2)$ The TV station plans to expand the male audience market. Now, using a proportional stratified sampling method, $3$ viewers are selected from the male participants for a program "suggestions" solicitation reward activity. The probability that a viewer who evaluated "dislike" has their "suggestion" adopted is $\frac{1}{4}$, and the probability that a viewer who evaluated "like" has their "suggestion" adopted is $\frac{3}{4}$. The reward for an adopted "suggestion" is $100$ dollars, and for a non-adopted "suggestion" is $50$ dollars. Let $X$ be the total prize money obtained by the $3$ viewers. Find the distribution table and the expected value of $X$.
Given: ${\chi}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$
| $\alpha$ | $0.010$ | $0.005$ | $0.001$ |
|----------|---------|---------|---------|
| $x_{\alpha}$ | $6.635$ | $7.879$ | $10.828$ |

Answer: $212.5$


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 maximum value of the positive real number \( A \) such that for any real numbers \( x, y, z \), the inequality
$$
x^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \geq 0
$$
holds.

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