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

Consider the polynomial \( p(x) = x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_n \) having all real roots. If \( r_1^{16} + r_2^{16} + \cdots + r_n^{16} = n \), where the \( r_j \) are the roots of \( p(x) \), find all such roots.

Answer: $-1$


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

Find the largest prime factor of $11236$.

Answer: $53$


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

Let  $x_1$ ,  $x_2$ , and  $x_3$  be the roots of the polynomial  $x^3+3x+1$ .  There are relatively prime positive integers  $m$  and  $n$  such that  $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ .  Find  $m+n$ .

Answer: $10$


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

Given that \( a \) is a real number, and for any \( k \in [-1,1] \), when \( x \in (0,6] \), the following inequality is always satisfied:
\[ 6 \ln x + x^2 - 8 x + a \leq k x. \]

Find the maximum value of \( a \).

Answer: $6 - 6 \ln 6$


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>.

A ship sails on a river. After 6 hours, it returns to its starting point, having covered a distance of 36 km according to the map (naturally, the ship had to move in one direction and then in the opposite direction).

What is the speed of the ship if we assume that it did not spend any time turning around and the speed of the river current is $3 \mathrm{~km} / \mathrm{h}$?

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


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

Two dice are differently designed. The first die has faces numbered $1$, $1$, $2$, $2$, $3$, and $5$. The second die has faces numbered $2$, $3$, $4$, $5$, $6$, and $7$. What is the probability that the sum of the numbers on the top faces of the two dice is $3$, $7$, or $8$?
A) $\frac{11}{36}$
B) $\frac{13}{36}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$

Answer: $\frac{13}{36}$


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

Alexey plans to buy one of two car brands: "A" costing 900,000 rubles or "B" costing 600,000 rubles. On average, Alexey drives 15,000 km per year. The cost of gasoline is 40 rubles per liter. The cars use the same type of gasoline. The car is planned to be used for 5 years, after which the car of brand "A" can be sold for 500,000 rubles, and the car of brand "B" for 350,000 rubles.

| Car Brand | Fuel Consumption (L/100km) | Annual Insurance Cost (rubles) | Average Annual Maintenance Cost (rubles) |
| :--- | :--- | :--- | :--- |
| "A" | 9 | 35,000 | 25,000 |
| "B" | 10 | 32,000 | 20,000 |

Using the data from the table, answer the question: How much more expensive will it be for Alexey to buy and own the more expensive car?

Answer: $160000$


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

Given the function $f(x)=x^{2}-3x$. If for any $x_{1}$, $x_{2}$ in the interval $[-3,2]$, we have $|f(x_{1})-f(x_{2})| \leqslant m$, then the minimum value of the real number $m$ is _______.

Answer: $\frac{81}{4}$


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

Given a circle with a diameter of $1$ unit, if a point $P$ on the circle starts from point $A$ representing $3$ on the number line and rolls one round to the left along the number line, find the real number represented by the point $B$ where $P$ arrives on the number line.

Answer: $3 - \pi$


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>.

Two quadratic equations with unequal leading coefficients,
$$
(a-1) x^{2} - \left(a^{2}+2\right) x + \left(a^{2}+2a\right) = 0
$$
and
$$
(b-1) x^{2} - \left(b^{2}+2\right) x + \left(b^{2}+2b\right) = 0
$$
(where \(a\) and \(b\) are positive integers), have a common root. Find the value of \(\frac{a^{b} + b^{a}}{a^{-b} + b^{-a}}\).

Answer: $256$


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

Adding two dots above the decimal 0.142857 makes it a repeating decimal. The 2020th digit after the decimal point is 5. What is the repeating cycle? $\quad$ .

Answer: $142857$


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

Given that the plane containing $\triangle PAD$ is perpendicular to the plane containing rectangle $ABCD$, and $PA = PD = AB = 2$, with $\angle APD = 60^\circ$. If points $P, A, B, C, D$ all lie on the same sphere, find the surface area of this sphere.

Answer: $\frac{28}{3}\pi$


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>.

Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros).

Answer: $197$


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

Petya and Vasya were collecting mushrooms. It turned out that Petya collected as many mushrooms as the percentage of mushrooms Vasya collected from the total number of mushrooms they collected together. Additionally, Vasya collected an odd number of mushrooms. How many mushrooms did Petya and Vasya collect together?

Answer: $25$


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

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>.

What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$?

Answer: $97$


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

We label the sides of a dodecagon as \(C_{1}, C_{2}, \ldots, C_{12}\). In how many ways can we color the sides of a dodecagon with four colors such that two adjacent sides are always colored differently? (Two colorings are considered different if any one side \(C_{i}\) is colored differently in the two colorings).

Answer: $531444$


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

A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments of lengths $6 \mathrm{~cm}$ and $7 \mathrm{~cm}$. Calculate the area of the triangle.

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>.

Given the hyperbola $C$: $\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\_1$ and $F\_2$ respectively, find the area of the quadrilateral $P\_1P\_2P\_3P\_4$ for a point $P$ on the hyperbola that satisfies $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$.

Answer: $\frac{8\sqrt{6}}{5}$


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 product of the digits $X$ and $Y$ such that the number $590\,X34\,52Y$ is divisible by $45$. What is the sum of all distinct possible products $XY$, with each product counted only once regardless of the pair ($X$, $Y$) variations that produce it?

Answer: $15$


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

In the Cartesian coordinate system, the parametric equations of the line $C_{1}$ are $\left\{\begin{array}{l}x=1+t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is the parameter). Using the origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of the curve $C_{2}$ is ${\rho}^{2}=\frac{4}{3-\cos2\theta}$.
$(1)$ Find the Cartesian equation of the curve $C_{2}$.
$(2)$ If the line $C_{1}$ intersects the curve $C_{2}$ at points $A$ and $B$, and $P(1,0)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

Answer: $2\sqrt{2}$


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

Transform the following expression into a product: 447. \(\sin 75^{\circ} + \sin 15^{\circ}\).

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


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 club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female?

Answer: $133,938$


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

Person A, Person B, Person C, and Person D share 2013 candies. Person A gets 10 more candies than twice the amount Person B has, 18 more candies than three times the amount Person C has, and 55 less candies than five times the amount Person D has. How many candies does Person A get?

Answer: $990$


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

Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).

Answer: $\frac{1}{9}$


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>.

Given that $a$, $b$, $c$ are three positive real numbers, and $a(a+b+c)=bc$, find the maximum value of $\frac{a}{b+c}$.

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


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>.

Calculate the limit of the function:

$$
\lim _{x \rightarrow 0} \ln \left(\left(e^{x^{2}}-\cos x\right) \cos \left(\frac{1}{x}\right)+\operatorname{tg}\left(x+\frac{\pi}{3}\right)\right)
$$

Answer: $\frac{1}{2} \ln (3)$


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

Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]

Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).

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>.

In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest $k$ such that, in a series of such moves, every chip is shifted one position clockwise relative to its initial position?

Answer: $50$


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

Given point P(-2,0) and the parabola C: y^2=4x, let A and B be the intersection points of the line passing through P and the parabola. If |PA|= 1/2|AB|, find the distance from point A to the focus of parabola C.

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


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

Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______.

Answer: $-7$


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

Given \( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \), and \( f(m) = 2017m \) for \( m = 1, 2, 3, 4 \), find \( f(10) - f(-5) \).

Answer: $75615$


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

Given $x > 0$, $y > 0$, and $2x+8y-xy=0$, find the minimum value of $x+y$.

Answer: $18$


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

Find the smallest natural number $A$ that satisfies the following conditions:

a) Its notation ends with the digit 6;

b) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold.

Answer: $153846$


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

There is a four-digit positive integer whose thousand's place is 2. If the digit 2 is moved to the unit's place, the new number formed is 66 greater than twice the original number. Let x be the original number in the units and tens places. Express the original number as 2000 + 100x + 10y + 2, and the new number formed as 2000 + 100x + 2 + 10y.

Answer: $2508$


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

Given $|a|=1$, $|b|=2$, and $a+b=(1, \sqrt{2})$, the angle between vectors $a$ and $b$ is _______.

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


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>.

Let  $ a_1, a_2,\ldots  ,a_8$  be  $8$  distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every  $4$  of the  $8$  points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the  $8$  points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair  $(i j)$ , with  $ 1 \le i < j \le 8$ , let  $r(i,j)$  be the number of quads, each containing  $ a_i, a_j$  as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer  $n$  such that it is possible to colour  $n$  of the bullets red so that  $r(i,j)$  is a constant for all pairs  $(i,j)$ .

Answer: $14$


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

Suppose \( x \) and \( y \) are integers such that \( xy + 3x + 2y = -4 \). Find the greatest possible value of \( y \).

Answer: $-1$


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

A three-digit number is called a "concave number" if the digit in the tens place is smaller than both the digit in the hundreds place and the digit in the units place. For example, 504 and 746 are concave numbers. How many three-digit concave numbers are there if all the digits are distinct?

Answer: $240$


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

Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store.

Answer: $0.64$


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

Simplify $\dfrac{12}{11}\cdot\dfrac{15}{28}\cdot\dfrac{44}{45}$.

Answer: $\frac{4}{7}$


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

Among all the factors of $11!$ (where $11! = 11 \times 10 \times \cdots \times 1$), the largest factor that can be expressed in the form $6k + 1$ (where $k$ is a natural number) is $\qquad$.

Answer: $385$


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

The sequence $(x_n)$ is defined by $x_1 = 25$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dotsb.\]

Answer: $\frac{1}{25}$


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

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1600$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$.

Answer: $400$


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

A gardener plans to enclose a rectangular garden with 480 feet of fencing. However, one side of the garden will be twice as long as another side. What is the maximum area of this garden?

Answer: $12800$


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

The flea Kuzya can make a jump on the plane in any direction for a distance of exactly 17 mm. Its task is to get from point \( A \) to point \( B \) on the plane, the distance between which is 1947 cm. What is the minimum number of jumps it must make?

Answer: $1146$


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

There are 46 ones written on the board. Each minute, Carlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 46 minutes?

Answer: $1035$


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

Find the minimum value of the function \( f(x) = x^{2} + 2x + \frac{6}{x} + \frac{9}{x^{2}} + 4 \) for \( x > 0 \).

Answer: $10 + 4 \sqrt{3}$


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

Determine the total degrees that exceed 90 for each interior angle of a regular pentagon.

Answer: $90$


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

Given two lines $l_1: ax+2y+6=0$, and $l_2: x+(a-1)y+a^2-1=0$.
(1) If $l_1 \perp l_2$, find the value of $a$;
(2) If $l_1 \parallel l_2$, find the value of $a$.

Answer: $-1$


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

In the triangular prism $P-ABC$, the three edges $PA$, $PB$, and $PC$ are mutually perpendicular, with $PA=1$, $PB=2$, and $PC=2$. If $Q$ is any point on the circumsphere of the triangular prism $P-ABC$, what is the maximum distance from $Q$ to the plane $ABC$?

Answer: $\frac{3}{2} + \frac{\sqrt{6}}{6}$


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 the Cartesian coordinate system, if the terminal side of angle α passes through point P(sin (5π/3), cos (5π/3)), evaluate sin(π+α).

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>.

Positive numbers \(a\), \(b\), and \(c\) satisfy the equations

\[
a^{2} + ab + b^{2} = 1, \quad b^{2} + bc + c^{2} = 3, \quad c^{2} + ca + a^{2} = 4
\]

Find the value of \(a + b + c\).

Answer: $\sqrt{7}$


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

Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$.

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


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

Given the formula $d=\sqrt[3]{\frac{16}{9}V}$, where $d$ is the diameter of a sphere and $V$ is its volume, find the volume of a sphere with a radius of $\frac{1}{3}$.

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


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

In the isosceles trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are related by the equation \(AD = (1 + \sqrt{15}) BC\). A circle is constructed with its center at point \(C\) and radius \(\frac{2}{3} BC\), which cuts a chord \(EF\) on the base \(AD\) of length \(\frac{\sqrt{7}}{3} BC\). In what ratio does the circle divide side \(CD\)?

Answer: $2:1$


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

Given real numbers \( x \) and \( y \) satisfy
\[
\left\{
\begin{array}{l}
x - y \leq 0, \\
x + y - 5 \geq 0, \\
y - 3 \leq 0
\end{array}
\right.
\]

If the inequality \( a(x^2 + y^2) \leq (x + y)^2 \) always holds, then the maximum value of the real number \( a \) is $\qquad$.

Answer: $25/13$


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>.

Let $f(x) = |x+1| - |x-4|$.
(1) Find the range of real number $m$ such that $f(x) \leq -m^2 + 6m$ always holds;
(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$.

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 \( n \geq 2 \) be a fixed integer. Find the smallest constant \( C \) such that for all non-negative reals \( x_1, x_2, \ldots, x_n \):
\[ \sum_{i < j} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. \]
Determine when equality occurs.

Answer: $\frac{1}{8}$


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

The endpoints of a line segment $MN$ with a fixed length of $4$ move on the parabola $y^{2}=x$. Let $P$ be the midpoint of the line segment $MN$. The minimum distance from point $P$ to the $y$-axis is ______.

Answer: $\dfrac{7}{4}$


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

A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their non-negative difference. When just one number remains, it is compared to \(x\). If it has the same parity A wins, otherwise B wins. Find the probability that A wins.

Answer: $3/4$


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

Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$              .

Answer: $-7$


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

In the Cartesian coordinate plane, the area of the region formed by the points \((x, y)\) that satisfy \( |x| + |y| + |x - 2| \leqslant 4 \) is ______.

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>.

Three squares with a perimeter of 28cm each are combined to form a rectangle. What is the area of this rectangle? Additionally, if four such squares are combined to form a larger square, what is the perimeter of this larger square?

Answer: $56$


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

Find the integer \( m \) so that
\[ 10^{m} < \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \ldots \times \frac{99}{100} < 10^{m+1} \]

Answer: $-2$


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

Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.

Answer: $67$


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

An arbitrary point \( E \) inside the square \( ABCD \) with side length 1 is connected by line segments to its vertices. Points \( P, Q, F, \) and \( T \) are the points of intersection of the medians of triangles \( BCE, CDE, DAE, \) and \( ABE \) respectively. Find the area of the quadrilateral \( PQFT \).

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


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>.

In triangle $\triangle ABC$, $AC=2AB=4$ and $\cos A=\frac{1}{8}$. Calculate the length of side $BC$.

Answer: $3\sqrt{2}$


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

Arrange the numbers from 1 to 25 in a random order. Then subtract them sequentially from 1, 2, 3, ..., 25, always subtracting the smaller number from the larger number. What is the maximum number of even differences that can appear among these 25 differences?

Answer: $25$


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

In the coordinate plane, points whose x-coordinates and y-coordinates are both integers are called lattice points. For any natural number \( n \), connect the origin \( O \) with the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( OA_n \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).

Answer: $1326$


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

There are red and blue socks in a drawer, with a total number not exceeding 2017. If two socks are randomly drawn, the probability that they are of the same color is $\frac{1}{2}$. What is the maximum number of red socks in the drawer?

Answer: $990$


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

Let  $x_{1}$  be a positive real number and for every integer  $n \geq 1$  let  $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ .
If  $x_{5} = 43$ , what is the sum of digits of the largest prime factors of  $x_{6}$ ?

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>.

Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than they disassemble one. When they arrived at the workshop in the morning, several grinders were already assembled.

At 7:00 AM, Hubert started assembling and Robert started disassembling. Exactly at 12:00 PM, Hubert finished assembling a grinder and Robert finished disassembling another one. In total, 70 grinders were added during this shift.

At 1:00 PM, Robert started assembling and Hubert started disassembling. Exactly at 10:00 PM, Robert finished assembling the last grinder and Hubert finished disassembling another one. In total, 36 grinders were added during this shift.

How long would it take for Robert and Hubert to assemble 360 grinders if both of them worked together assembling?

Answer: $15$


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

The equations of the asymptotes of the hyperbola $\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________.

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


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>.

They paid 100 rubles for a book and still need to pay as much as they would need to pay if they had paid as much as they still need to pay. How much does the book cost?

Answer: $200$


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

Given that \( 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} = 252000 \), what is the probability that a three-digit number formed by any 3 of the natural numbers \( a, b, c, d \) is divisible by 3 and less than 250?

Answer: $1/4$


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

A fair 10-sided die is rolled repeatedly until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
- **A** $\frac{1}{300}$
- **B** $\frac{1}{252}$
- **C** $\frac{1}{500}$
- **D** $\frac{1}{100}$

Answer: $\frac{1}{252}$


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

A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(0)=0$, $f(x)+f(1-x)=1$, $f(\frac{x}{5})=\frac{1}{2}f(x)$, and $f(x_1)\leqslant f(x_2)$ when $0\leqslant x_1 < x_2\leqslant 1$. Find the value of $f(\frac{1}{2007})$.

A) $\frac{1}{2}$
B) $\frac{1}{16}$
C) $\frac{1}{32}$
D) $\frac{1}{64}$

Answer: $\frac{1}{32}$


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>.

Let \( M = \{1, 2, \ldots, 20\} \) and \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). When \( i \neq j \), the intersection of \( A_i \) and \( A_j \) has at most two elements. Find the maximum value of \( n \).

Answer: $1350$


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

Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inches, and Dan's cookies are circles with a radius of 3 inches. If Art makes 18 cookies in one batch, how many cookies will be in one batch of Dan's cookies?

Answer: $15$


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

Suppose $a$ and $b$ are the points obtained by throwing a dice in order, and the function is $f(x)=\frac{1}{2}ax^{2}+bx+1$.

(1) Find the probability that $f(x)$ is a decreasing function in the interval $(-\infty,-1]$;

(2) Find the probability that the function $f(x)$ has zero points.

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>.

A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table?

Answer: $15$


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

Given a triangle \(ABC\) with an area of 1. The first player chooses a point \(X\) on side \(AB\), the second player chooses a point \(Y\) on side \(BC\), and then the first player chooses a point \(Z\) on side \(AC\). The first player's goal is to maximize the area of triangle \(XYZ\), and the second player's goal is to minimize it. What is the largest area that the first player can guarantee for themselves?

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


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

Given the parabola $C$: $x^2 = 2py (p > 0)$ and the line $2x-y+2=0$, they intersect at points $A$ and $B$. A vertical line is drawn from the midpoint of the line segment $AB$ to the $x$-axis, intersecting the parabola $C$ at point $Q$. If $\overrightarrow{QA} \cdot \overrightarrow{QB}=0$, calculate the value of $p$.

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


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

Distribute 5 volunteers from the Shanghai World Expo to work in the pavilions of China, the United States, and the United Kingdom. Each pavilion must have at least one volunteer, with the requirement that two specific volunteers, A and B, do not work in the same pavilion. How many different distribution schemes are possible?

Answer: $114$


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

Find the sum of all four-digit numbers, in which only the digits $1, 2, 3, 4, 5$ are used, and each digit occurs no more than once.

Answer: $399960$


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

Andrey drove a car to the airport in a neighboring city. After one hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. He then sharply increased his speed, and as a result, covered the remaining distance at an average speed of 90 km/h and arrived at the airport 20 minutes earlier than originally planned. What is the distance from Andrey's home to the airport?

Answer: $180$


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

Given $f(x) = 4\cos x\sin \left(x+ \frac{\pi}{6}\right)-1$.
(Ⅰ) Determine the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac{\pi}{6}, \frac{\pi}{4}\right]$.

Answer: $-1$


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

Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.

| $x$     | $\ldots $ | $0.064$ | $0.64$ | $64$  | $6400$ | $64000$ | $\ldots $ |
|---------|-----------|---------|--------|-------|--------|---------|-----------|
| $\sqrt{x}$ | $\ldots $ | $0.25298$ | $0.8$ | $8$   | $m$    | $252.98$ | $\ldots $ |
| $\sqrt[3]{x}$ | $\ldots $ | $n$     | $0.8618$ | $4$   | $18.566$ | $40$    | $\ldots $ |

$(1)$ $m=$______, $n= \_\_\_\_\_\_.$
$(2)$ From the numbers in the table, it can be observed that when finding the square root, if the decimal point of the radicand moves two places to the left (or right), the decimal point of its square root moves one place to the left (or right). Please describe in words the pattern of cube root: ______.
$(3)$ If $\sqrt{a}≈14.142$, $\sqrt[3]{700}≈b$, find the value of $a+b$. (Reference data: $\sqrt{2}≈1.4142$, $\sqrt{20}≈4.4721$, $\sqrt[3]{7}≈1.9129$, $\sqrt[3]{0.7}≈0.8879$)

Answer: $208.879$


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

What is the total number of digits used when the first 1500 positive odd integers are written?

Answer: $5445$


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

Find all \( a_{0} \in \mathbb{R} \) such that the sequence defined by
\[ a_{n+1} = 2^{n} - 3a_{n}, \quad n = 0, 1, 2, \cdots \]
is increasing.

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


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

Evaluate the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty}\left(\frac{n^{2}-6 n+5}{n^{2}-5 n+5}\right)^{3 n+2}
\]

Answer: $e^{-3}$


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

If the system of inequalities $\left\{\begin{array}{l}9x - a \geqslant 0, \\ 8x - b < 0\end{array}\right.$ has integer solutions only for $1, 2, 3$, how many ordered pairs of integers $(a, b)$ satisfy the system?

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>.

Given the chord length intercepted by the circle $x^{2}+y^{2}+2x-4y+1=0$ on the line $ax-by+2=0$ $(a > 0, b > 0)$ is 4, find the minimum value of $\frac{1}{a} + \frac{1}{b}$.

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


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>.

Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed?

Answer: $102$


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

Given the function $y = x^2 - 8x + 12$, find the vertex of the parabola and calculate the value of $y$ when $x = 3$.

Answer: $-3$


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

Let \( n \) be a positive integer, \( [x] \) be the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \).
(1) Find all positive integers \( n \) satisfying \( \sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n \);
(2) Find all positive integers \( n \) that make \( \sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right\} \) attain its maximum value, and determine this maximum value.

Answer: $1006$


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

Set \( A = \{1, 2, \cdots, n\} \). If there exist nonempty sets \( B \) and \( C \) such that \( B \cap C = \emptyset \), \( B \cup C = A \), and the sum of the squares of the elements in \( B \) is \( M \), and the sum of the squares of the elements in \( C \) is \( N \), and \( M - N = 2016 \), find the smallest value of \( n \).

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>.

Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of these two prime numbers?

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>.

Given a random variable $\xi \sim N(1,4)$, and $P(\xi < 3)=0.84$, then $P(-1 < \xi < 1)=$ \_\_\_\_\_\_.

Answer: $0.34$


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

Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called "significantly dissatisfied" if they were dissatisfied with at least two out of the three aspects of the olympiad. What is the maximum number of "significantly dissatisfied" participants that could have been at this olympiad?

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>.

In the geometric sequence $\{a\_n\}$, it is known that $a\_1+a\_2+a\_3=1$, $a\_4+a\_5+a\_6=-2$, find the sum of the first 15 terms of the sequence, denoted as $S\_{15}$.

Answer: $11$


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

In response to the call of the commander, 55 soldiers came: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.

- To the question "Are you wearing golden armor?" 44 people responded affirmatively.
- To the question "Are you an archer?" 33 people responded affirmatively.
- To the question "Is today Monday?" 22 people responded affirmatively.

How many archers in golden armor came in response to the commander's call?

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>.

Determine the value of the following expressions:
$(1)(2 \frac{7}{9})^{0.5}+0.1^{-2}+(2 \frac{10}{27})\,^{- \frac{2}{3}}-3π^{0}+ \frac{37}{48}$;
$(2)(-3 \frac{3}{8})\,^{- \frac{2}{3}}+(0.002)\,^{- \frac{1}{2}}-10( \sqrt{5}-2)^{-1}+( \sqrt{2}- \sqrt{3})^{0}$.

Answer: $- \frac{167}{9}$


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>.

In a chess tournament, students from the 9th and 10th grades participated. There were 10 times more 10th graders than 9th graders.

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>.

The ratio of the sum of the first $n$ terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ is $\frac{S_n}{T_n} = \frac{7n+1}{4n+2}$. Calculate the value of $\frac{a_{11}}{b_{11}}$.

Answer: $\frac{74}{43}$


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

In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visit the sauna. Visiting only the swimming pool costs 200 rubles more than visiting the sauna. How many times must Katya visit the swimming pool without the sauna to save enough to buy the headphones? (Give the answer as a whole number, without spaces and units of measurement.)

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>.

Determine the number of ordered pairs of positive integers \((a, b)\) satisfying the equation
\[ 100(a + b) = ab - 100. \]

Answer: $18$


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

Hooligan Vasya likes to run on the escalator in the subway. He runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving downward, it takes him 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)

Answer: $324$


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

Given a hyperbola with the equation $\frac{y^2}{4} - \frac{x^2}{a} = 1$ and asymptote equations $y = \pm\frac{2\sqrt{3}}{3}x$, find the eccentricity of this hyperbola.

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


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

Find the minimum value of the function $f(x)=\log_{2}^{\sqrt{x}} \cdot \log_{\sqrt{2}}^{(2x)}$.

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


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

Two circles with centers $A$ and $B$ and radii 2 and 1, respectively, are tangent to each other. Point $C$ lies on a line that is tangent to each of the circles and is at a distance of $\frac{3 \sqrt{3}}{2 \sqrt{2}}$ from the midpoint of segment $AB$. Find the area $S$ of triangle $ABC$, given that $S > 2$.

Answer: $\frac{15 \sqrt{2}}{8}$


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

Given a power function $y = f(x)$ whose graph passes through the point $(4, 2)$, find $f\left( \frac{1}{2} \right)$.

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


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

Find the sum of the $2010$ roots of $(x-1)^{2010} + 2(x-2)^{2009} + 3(x-3)^{2008} + \cdots + 2009(x-2009)^2 + 2010(x-2010)^1$.

Answer: $2008$


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

Approximate the number $0.00356$ to the nearest ten-thousandth: $0.00356 \approx$____.

Answer: $0.0036$


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

The base of isosceles $\triangle XYZ$ is 30 units and its area is 60 square units.

Answer: $\sqrt{241}$


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

2005^{2} + 2 \times 2005 \times 1995 + 1995^{2} divided by 800.

Answer: $20000$


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

For the line $l_1: ax - y - a + 2 = 0$ (where $a \in \mathbb{R}$), the line $l_2$ passing through the origin $O$ is perpendicular to $l_1$, and the foot of the perpendicular from $O$ is $M$. Then, the maximum value of $|OM|$ is ______.

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>.

Given the sequence
$$
\begin{array}{l}
a_{0}=134, a_{1}=150, \\
a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \quad (k=1,2, \cdots, n-1).
\end{array}
$$

If \( a_n = 0 \), then \( n \) is:

Answer: $201$


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

Find $\frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}$, if $\frac{a}{2}-\frac{2}{a}=3$.

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>.

Simplify completely: $$\sqrt[3]{40^3 + 70^3 + 100^3}.$$

Answer: $10 \sqrt[3]{1407}$


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

In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\), point \(P\) is a moving point on side \(DC\) (including points \(D\) and \(C\)), and point \(Q\) is a moving point on the extension line of \(CB\) (including point \(B\)), such that \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). Determine the minimum value of the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PQ}\).

Answer: $3/4$


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

Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$.

Answer: $18\sqrt{2}$


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

In an arithmetic sequence \(\{a_{n}\}\), where \(a_{1} > 0\) and \(5a_{8} = 8a_{13}\), what is the value of \(n\) when the sum of the first \(n\) terms \(S_{n}\) is maximized?

Answer: $21$


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

If \(0^{\circ} < \alpha < 30^{\circ}\), and \(\sin^6 \alpha + \cos^6 \alpha = \frac{7}{12}\), then \(1998 \cos \alpha = ?\)

Answer: $333 \sqrt{30}$


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>.

(13 points) A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that remains unrented. The maintenance cost for each rented car is 150 yuan per month, and for each unrented car, it is 50 yuan per month.
(1) How many cars can be rented out when the monthly rent per car is set at 3600 yuan?
(2) At what monthly rent per car will the rental company's monthly revenue be maximized? What is the maximum monthly revenue?

Answer: $307050$