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

Given the function \( f(x) = \lg \frac{1 + x}{1 - x} \), if \( f\left(\frac{y + z}{1 + y z}\right) = 1 \) and \( f\left(\frac{y - z}{1 - y z}\right) = 2 \), where \( -1 < y, z < 1 \), find the value of \( f(y) \cdot f(z) \).

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>.

The shortest distance from a point on the curve $f(x) = \ln(2x-1)$ to the line $2x - y + 3 = 0$ is what?

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 $f(\sin \alpha + \cos \alpha) = \sin \alpha \cdot \cos \alpha$, determine the domain of $f(x)$ and the value of $f\left(\sin \frac{\pi}{6}\right)$.

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


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>.

Find the maximum value of the product \(x^{2} y^{2} z^{2} u\) given the condition that \(x, y, z, u \geq 0\) and:

\[ 2x + xy + z + yz u = 1 \]

Answer: $1/512$


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

Point \( M \) is the midpoint of side \( BC \) of the triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible length of the segment \( MX \)?

Answer: $6.5$


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

What is the remainder when $5x^8 - 3x^7 + 4x^6 - 9x^4 + 3x^3 - 5x^2 + 8$ is divided by $3x - 6$?

Answer: $1020$


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

Given the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) with three points \(P\), \(Q\), and \(R\) on it, where \(P\) and \(Q\) are symmetric with respect to the origin. Find the maximum value of \(|RP| + |RQ|\).

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 \( a \) and \( b \) are positive integers, and \( b - a = 2013 \). If the equation \( x^{2} - a x + b = 0 \) has a positive integer solution, what is the smallest value of \( a \)?

Answer: $93$


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

Let $ABCD$ be a parallelogram with $\angle ABC=135^\circ$, $AB=14$ and $BC=8$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of segment $FD$.
A) $\frac{6}{17}$
B) $\frac{18}{17}$
C) $\frac{24}{17}$
D) $\frac{30}{17}$

Answer: $\frac{24}{17}$


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

Let \( a \star b = ab + a + b \) for all integers \( a \) and \( b \). Evaluate \( 1 \star (2 \star (3 \star (4 \star \ldots (99 \star 100) \ldots))) \).

Answer: $101! - 1$


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 basketball team consists of 12 players, including two pairs of twins, Alex and Brian, and Chloe and Diana. In how many ways can we choose a team of 5 players if no pair of twins can both be in the team lineup simultaneously?

Answer: $560$


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

How many different positive three-digit integers can be formed using only the digits in the set $\{1, 2, 2, 3, 4, 4, 4\}$ if no digit may be used more times than it appears in the given set of available digits?

Answer: $43$


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>.

Given the parabola $y=ax^{2}+bx+c$ ($a\neq 0$) with its axis of symmetry to the left of the $y$-axis, where $a$, $b$, $c \in \{-3,-2,-1,0,1,2,3\}$, let the random variable $X$ be the value of "$|a-b|$". Then, the expected value $EX$ is \_\_\_\_\_\_.

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


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

Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \).

Answer: $-1$


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

The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid.

Answer: $224/3$


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

Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0)=2$, then $f(4)=$ ?

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>.

For a positive number $x$, define $f(x)=\frac{2x}{x+1}$. Calculate: $f(\frac{1}{101})+f(\frac{1}{100})+f(\frac{1}{99})+\ldots +f(\frac{1}{3})+f(\frac{1}{2})+f(1)+f(2)+f(3)+\ldots +f(99)+f(100)+f(101)$.

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>.

Given that $α$ is an angle in the third quadrant and $\cos(α+π)=\frac{4}{5}$, find the value of $\tan 2α$.

Answer: $\frac{24}{7}$


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>.

Let \( x \neq y \), and the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. Then \(\frac{b_{4}-b_{3}}{a_{2}-a_{1}}\) equals $\qquad$.

Answer: $2.6666666666666665$


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

Given the complex number $z=(2m^{2}+3m-2)+(m^{2}+m-2)i$ where $(m\in\mathbb{R})$, find the value of $m$ under the following conditions:
$(1) z$ is a real number;
$(2) z$ is an imaginary number;
$(3) z$ is a pure imaginary number;
$(4) z=0$.

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>.

Given a sequence \( x_{n} \), satisfying \( (n+1) x_{n+1}=x_{n}+n \), and \( x_{1}=2 \), find \( x_{2009} \).

Answer: $\frac{2009! + 1}{2009!}$


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

Given a four-digit number \(\overline{abcd}\), when divided by 2, 3, 4, 5, 6, and 7, the remainders are all different and none of them are 0. Find the minimum value of \(\overline{abcd}\).

Answer: $1259$


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

Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form.

Answer: $92$


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

There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?

Answer: $2601$


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

Given four points \( O, A, B, C \) on a plane, with \( OA=4 \), \( OB=3 \), \( OC=2 \), and \( \overrightarrow{OB} \cdot \overrightarrow{OC}=3 \), find the maximum area of triangle \( ABC \).

Answer: $2 \sqrt{7} + \frac{3\sqrt{3}}{2}$


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

Suppose $p$ and $q$ are both real numbers, and $\sin \alpha$ and $\cos \alpha$ are the two real roots of the equation $x^{2}+px+q=0$ with respect to $x$. Find the minimum value of $p+q$.

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 area of triangle ABC, whose vertices have coordinates A(0,0), B(1424233,2848467), C(1424234,2848469). Round the answer to two decimal places.

Answer: $0.50$


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

The value of $\sin 210^\circ$ is equal to $\frac{-\sqrt{3}}{2}$.

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>.

Find the smallest natural number ending in the digit 6, which increases fourfold when its last digit is moved to the beginning of the number.

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>.

Does there exist an integer \( n \) such that \( 21n \equiv 1 \mod 74 \)?

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>.

One of the angles in a triangle is $120^{\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle.

Answer: $3 : 5 : 7$


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>.

Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis?

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


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

A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which all 8 lights are green?

Answer: $20$


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

The sum of the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}- \frac{\pi}{3}\right)$ where $(0\leqslant x\leqslant 9)$ is to be determined.

Answer: $2-\sqrt{3}$


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

By how many zeros does the number 2012! end?

Answer: $501$


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

Given the complex numbers \( z_1 \) and \( z_2 \) such that \( \left| z_1 + z_2 \right| = 20 \) and \( \left| z_1^2 + z_2^2 \right| = 16 \), find the minimum value of \( \left| z_1^3 + z_2^3 \right| \).

Answer: $3520$


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

Find the limits:
1) \(\lim_{x \to 3}\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)\)
2) \(\lim_{x \to \infty}\left(\sqrt{x^2 + 1}-x\right)\)
3) \(\lim_{n \to \infty} 2^n \sin \frac{x}{2^n}\)
4) \(\lim_{x \to 1}(1-x) \tan \frac{\pi x}{2}\).

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


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>.

To encourage residents to conserve water, a city charges residents for domestic water use in a tiered pricing system. The table below shows partial information on the tiered pricing for domestic water use for residents in the city, each with their own water meter:

| Water Sales Price | Sewage Treatment Price |
|-------------------|------------------------|
| Monthly Water Usage per Household | Unit Price: yuan/ton | Unit Price: yuan/ton |
| 17 tons or less | $a$ | $0.80$ |
| More than 17 tons but not more than 30 tons | $b$ | $0.80$ |
| More than 30 tons | $6.00$ | $0.80$ |

(Notes: 1. The amount of sewage generated by each household is equal to the amount of tap water used by that household; 2. Water bill = tap water cost + sewage treatment fee)

It is known that in April 2020, the Wang family used 15 tons of water and paid 45 yuan, and in May, they used 25 tons of water and paid 91 yuan.

(1) Find the values of $a$ and $b$;
(2) If the Wang family paid 150 yuan for water in June, how many tons of water did they use that month?

Answer: $35$


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

From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron.

Answer: $1/2$


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

The product of all integers whose absolute value is greater than 3 but not greater than 6 is ____.

Answer: $-14400$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $453_6$, $436_6$, and $42_6$ in base 6.

Answer: $1415_6$


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>.

The cheetah takes strides of 2 meters each and the fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. Given that the distance between the cheetah and the fox is 30 meters, find the distance the cheetah must run to catch up with the fox.

Answer: $120$


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

Given that the odd function $f(x)$ is a monotonically increasing function defined on $\mathbb{R}$ and the sequence $\{x_n\}$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, find the value of $x_{2012}$.

Answer: $4005$


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

In the diagram, $D$ and $E$ are the midpoints of $\overline{AC}$ and $\overline{BC}$ respectively, where $A(0,8)$, $B(0,0)$, and $C(10,0)$. Find the sum of the slope and $y$-intercept of the line passing through the points $C$ and $D$.

Answer: $\frac{36}{5}$


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

The integers \( m \) and \( n \) satisfy the equation \( 3^{m} \times n = 7! + 8! + 9! \). What is the smallest possible value for \( n \)?

Answer: $560$


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

A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers.

Answer: $1782$


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

In the six-digit number $1 A B C D E$, each letter represents a digit. Given that $1 A B C D E \times 3 = A B C D E 1$, calculate the value of $A+B+C+D+E$.

Answer: $26$


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

In the interior of a triangle \( ABC \) with area 1, points \( D \), \( E \), and \( F \) are chosen such that \( D \) is the midpoint of \( AE \), \( E \) is the midpoint of \( BF \), and \( F \) is the midpoint of \( CD \). Find the area of triangle \( DEF \).

Answer: $1/7$


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

Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do not have to return at the same time. The cars can lend gasoline to each other. To maximize the distance one car can travel away from the starting point, the other car should turn back at a distance of $\boxed{360}$ km from the starting point.

Answer: $360$


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

In the tetrahedron $P-ABC$, edges $PA$, $AB$, and $AC$ are mutually perpendicular, and $PA = AB = AC$. Points $E$ and $F$ are the midpoints of segments $AB$ and $PC$, respectively. Find the sine of the angle between line $EF$ and plane $PBC$.

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


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

What is the result of $120 \div (6 \div 2 \times 3)$?

Answer: $\frac{120}{9}$


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>.

How many pairs of two-digit positive integers have a difference of 50?

Answer: $40$


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

The diagonals \(AC\) and \(BD\) of the symmetric trapezoid \(ABCD\) intersect at point \(O\). The area of triangle \(AOB\) is \(52 \, \text{m}^2\) and the area of triangle \(COD\) is \(117 \, \text{m}^2\). Calculate the area of the trapezoid.

Answer: $325$


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

Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power.

Answer: $243$


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

In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe?

Answer: $30$


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

Given distinct natural numbers \( k, l, m, n \), it is known that there exist three natural numbers \( a, b, c \) such that each of the numbers \( k, l, m, n \) is a root of either the equation \( a x^{2} - b x + c = 0 \) or the equation \( c x^{2} - 16 b x + 256 a = 0 \). Find \( k^{2} + l^{2} + m^{2} + n^{2} \).

Answer: $325$


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

Alexio now has 100 cards numbered from 1 to 100. He again randomly selects one card from the box. What is the probability that the number on the chosen card is a multiple of 3, 5, or 7? Express your answer as a common fraction.

Answer: $\frac{11}{20}$


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

In rectangle $ABCD$, side $AB$ measures $8$ units and side $BC$ measures $4$ units. Points $F$ and $G$ are on side $CD$ such that segment $DF$ measures $2$ units and segment $GC$ measures $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$?

Answer: $32$


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

\[
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
\]
is a union of disjoint intervals the sum of whose lengths is 1988.

Answer: $1988$


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

When two fair dice are thrown, the numbers obtained are $a$ and $b$, respectively. Express the probability that the slope $k$ of the line $bx+ay=1$ is greater than or equal to $-\dfrac{2}{5}$.

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


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>.

You are in a place where 99% of the inhabitants are vampires and 1% are regular humans. On average, 90% of the vampires are correctly identified as vampires, and 90% of humans are correctly identified as humans. What is the probability that someone identified as a human is actually a human?

Answer: $1/12$


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

Given the parabola C: x² = 2py (p > 0), draw a line l: y = 6x + 8, which intersects the parabola C at points A and B. Point O is the origin, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. A moving circle P has its center on the parabola C and passes through a fixed point D(0, 4). If the moving circle P intersects the x-axis at points E and F, and |DE| < |DF|, find the minimum value of $\frac{|DE|}{|DF|}$.

Answer: $\sqrt{2} - 1$


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 an $8 \times 6$ grid, consider a triangle with vertices at $D=(2,1)$, $E=(7,1)$, and $F=(5,5)$. Determine the fraction of the grid covered by this triangle.

Answer: $\frac{5}{24}$


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

The real numbers \( x, y, z \) satisfy \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \).

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


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

The quadratic \( x^2 + 1800x + 1800 \) can be written in the form \( (x+b)^2 + c \), where \( b \) and \( c \) are constants. What is \( \frac{c}{b} \)?

Answer: $-898$


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

A deck consists of 32 cards divided into 4 suits, each containing 8 cards. In how many ways can we choose 6 cards such that all four suits are represented among them?

Answer: $415744$


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

A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?

Answer: $16$


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

Observing the equations:
$1\times 3+1=4=2^{2}$;
$2\times 4+1=9=3^{2}$;
$3\times 5+1=16=4^{2}$;
$4\times 6+1=25=5^{2}$;
$\ldots $
$(1)7\times 9+1=( $______)$^{2}$;
$(2)$ Using the pattern you discovered, calculate: $(1+\frac{1}{1×3})×(1+\frac{1}{2×4})×(1+\frac{1}{3×5})×⋅⋅⋅×(1+\frac{1}{198×200})$.

Answer: $\frac{199}{100}$


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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 - x - 1 = 0$, find the value of $\frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}$.

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>.

Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________.

Answer: $330$


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

Lucy surveyed a group of people about their knowledge of mosquitoes. To the nearest tenth of a percent, she found that $75.3\%$ of the people surveyed thought mosquitoes transmitted malaria. Of the people who thought mosquitoes transmitted malaria, $52.8\%$ believed that mosquitoes also frequently transmitted the common cold. Since mosquitoes do not transmit the common cold, these 28 people were mistaken. How many total people did Lucy survey?

Answer: $70$


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

Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest.

Answer: $19:59$


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

Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a "buy two, get one free" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy?

Answer: $18$


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

The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\circ$ and the common difference being $5^\circ$. Determine the number of sides $n$ for the polygon.

Answer: $n = 9$


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 three points $A$, $B$, $C$ on a straight line in the Cartesian coordinate system, satisfying $\overrightarrow{OA}=(-3,m+1)$, $\overrightarrow{OB}=(n,3)$, $\overrightarrow{OC}=(7,4)$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$, where $O$ is the origin.
$(1)$ Find the values of the real numbers $m$, $n$;
$(2)$ Let $G$ be the centroid of $\triangle AOC$, and $\overrightarrow{OG}= \frac{2}{3} \overrightarrow{OB}$, find the value of $\cos \angle AOC$.

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


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 that \( x \) is a multiple of \( 7200 \), what is the greatest common divisor of \( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \) and \( x \)?

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>.

There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is also used in the seventh week? (Express your answer as a simplified fraction.)

Answer: $61/243$


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

If $x^2 + 3x - 1 = 0$, then $x^3 + 5x^2 + 5x + 18 =$ ?

Answer: $20$


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

In the sequence \(\left\{a_{n}\right\}\), it is known that \(a_{1}=1\) and \(a_{n+1}>a_{n}\), and that \(a_{n+1}^{2}+a_{n}^{2}+1=2\left(a_{n+1}+a_{n}+2 a_{n+1} a_{n}\right)\). Find \(\lim \limits_{n \rightarrow \infty} \frac{S_{n}}{n a_{n}}\).

Answer: $1/3$


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

Simplify $\sqrt{18} \times \sqrt{32} \times \sqrt{2}$.

Answer: $24\sqrt{2}$


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>.

There are 19 weights with values $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all the iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight.

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 $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix.

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


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

How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?

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>.

What is the smallest integer larger than $(\sqrt{5}+\sqrt{3})^4$?

Answer: $248$


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

Given the function $f(x)=\begin{cases}f(x+1), & x < 4 \\ 2^{x}, & x\geq 4\end{cases}$, evaluate $f(2+\log_{2}3)$.

Answer: $24$


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

In triangle $\triangle ABC$, with angles A, B, and C, the sides opposite these angles are labeled as $a, b,$ and $c$ respectively. Given the vectors $\overrightarrow{m}=\left( \frac{a}{2}, \frac{c}{2} \right)$ and $\overrightarrow{n}=(\cos C, \cos A)$, it is also known that $\overrightarrow{n} \cdot \overrightarrow{m} = b\cos B$.
1. Determine the value of angle B;
2. If $\cos \left( \frac{A-C}{2} \right) = \sqrt{3}\sin A$ and $|\overrightarrow{m}|=\sqrt{5}$, find the area of $\triangle ABC$.

Answer: $2\sqrt{3}$


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

Given that $\frac{x}{3} = y^2$ and $\frac{x}{6} = 3y$, solve for the value of $x$.

Answer: $108$


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

João, Jorge, José, and Jânio are good friends. One time, João was out of money, but his friends had some. So Jorge gave João a fifth of his money, José gave João a fourth of his money, and Jânio gave João a third of his money. If all of them gave the same amount of money to João, what fraction of the group's money did João end up with?

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>.

Let $D$ be the circle with equation $x^2 - 10y - 7 = -y^2 - 8x + 4$. Find the center $(a, b)$ and radius $r$ of $D$, and determine the value of $a + b + r$.

Answer: $1 + 2\sqrt{13}$


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  $x$ ,  $y$ ,  $z$  be positive integers satisfying  $x<y<z$  and  $x+xy+xyz=37$ . Find the greatest possible value of  $x+y+z$ .

Answer: $20$


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

Given \(\lg 2 = 0.30103\), calculate the number of digits in \( M = 1 + 10^4 + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + \frac{10^4 (10^4-1)(10^4-2)}{1 \cdot 2 \cdot 3} + \cdots + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + 10^4 + 1\).

Answer: $3011$


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

Simplify \(\left(\cos 42^{\circ}+\cos 102^{\circ}+\cos 114^{\circ}+\cos 174^{\circ}\right)^{2}\) into a rational number.

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


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

The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\triangle OAB$, where $O$ is the origin.

Answer: $\frac{1}{16}$


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>.

There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)?

Answer: $337$


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

Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \) if \( BD \cdot BC = 5 \).

Answer: $\sqrt{5}$


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

In the diagram, \(BD\) is perpendicular to \(BC\) and to \(AD\). If \(AB = 52\), \(BC = 21\), and \(AD = 48\), what is the length of \(DC\)?

Answer: $29$


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

A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth if necessary.

Answer: $5.97$


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>.

Find the smallest 6-digit palindrome in base 2, that can be expressed as a 4-digit palindrome in a different base. Provide your response in base 2.

Answer: $100001_2$


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>.

There are 49 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then he 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 49 minutes?

Answer: $1176$


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

How many triangles are there with all sides being integers and the longest side being 11?

Answer: $36$


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

Given the function $f(x)=4\sin x\sin \left(x+ \frac {\pi}{3}\right)-1$.
$(1)$ Calculate the value of $f\left( \frac {5\pi}{6}\right)$:
$(2)$ Let $A$ be the smallest angle in $\triangle ABC$, and $f(A)= \frac {8}{5}$, find the value of $f\left(A+ \frac {\pi}{4}\right)$.

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


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

Let \( P \) be the parabola with equation \( y = x^2 \) and let \( Q = (10, 6) \). There are real numbers \( r \) and \( s \) such that the line through \( Q \) with slope \( m \) does not intersect \( P \) if and only if \( r < m < s \). What is \( r + s \)?

Answer: $40$


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

If \\(\alpha \in \left( 0, \frac{\pi}{2} \right)\\), and \\(\cos \left( \frac{\pi}{4} - \alpha \right) = 2 \sqrt{2} \cos 2\alpha\\), then \\(\sin 2\alpha = \)_______.

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


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

Find \( k \) such that, for all \( n \), the following expression is a perfect square:

$$
4 n^{2} + k n + 9
$$

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>.

Let $f(x) = \sqrt{-x^2 + 5x + 6}$.
$(1)$ Find the domain of $f(x)$.
$(2)$ Determine the intervals where $f(x)$ is increasing or decreasing.
$(3)$ Find the maximum and minimum values of $f(x)$ on the interval $[1,5]$.

Answer: $\sqrt{6}$


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

Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)?

Answer: $2998.5$


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 \( x^{2} + y^{2} - 2x - 2y + 1 = 0 \) where \( x, y \in \mathbb{R} \), find the minimum value of \( F(x, y) = \frac{x + 1}{y} \).

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 \(\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+}\) and \(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi\), find the minimum value of \(\left(2 \sin ^{2} \theta_{1}+\frac{1}{\sin ^{2} \theta_{1}}\right)\left(2 \sin ^{2} \theta_{2}+\frac{1}{\sin ^{2} \theta_{2}}\right)\left(2 \sin ^{2} \theta_{3}+\frac{1}{\sin ^{2} \theta_{3}}\right)\left(2 \sin ^{2} \theta_{4}+\frac{1}{\sin ^{2} \theta_{4}}\right)\).

Answer: $81$


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

Express $0.3\overline{206}$ as a common fraction.

Answer: $\frac{5057}{9990}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$.

Answer: $\frac{10}{29}$


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

Given the following four propositions:
(1) Two lines parallel to the same plane are parallel to each other.
(2) Two lines perpendicular to the same line are parallel to each other.
(3) Through a point outside a known plane, there exists exactly one plane parallel to the given plane.
(4) Through a line outside a known plane, a plane can always be constructed parallel to the given plane.
The sequence numbers of the true propositions are ____. (List all the numbers of the true propositions.)

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>.

150 people were surveyed and asked: "Do you think teal is more green or blue?" Of them, 90 believe teal is "more green," and 50 believe it's "more blue." Additionally, 40 believe it's both "more green" and "more blue." Another 20 think teal is neither "more green" nor "more blue."

How many of those 150 people believe that teal is "more blue"?

Answer: $80$


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

Find the sum of all positive integers \( p \) such that the expression \((x-p)(x-13)+4\) can be expressed in the form \((x+q)(x+r)\) for distinct integers \( q \) and \( r \).

Answer: $26$


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

Given the parabola $y^{2}=2px(p > 0)$ and the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ have the same focus $F$, and point $A$ is an intersection point of the two curves, and $AF$ is perpendicular to the x-axis, calculate the eccentricity of the hyperbola.

Answer: $\sqrt {2} + 1$


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 that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers.

Answer: $09:38$


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

In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other?

Answer: $\frac{7}{9}$


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>.

Among the rye seeds, there are $0.4\%$ weed seeds. What is the probability of detecting 5 weed seeds when randomly selecting 5000 seeds?

Answer: $0.000055$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 sums of the first \( n \) terms of two arithmetic sequences is \(\frac{9n+2}{n+7}\). Find the ratio of their 5th terms.

Answer: $\frac{83}{16}$


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>.

35 times 61,000 unit cubes are combined to form a large cube with an edge length of 10 units. After being painted, the large cube is then separated back into the original unit cubes. How many of these unit cubes have at least one face painted?

Answer: $488$


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

In a competition with five participants A, B, C, D, and E, determine the probability that neither B nor C appears adjacent to A.

Answer: $\frac{3}{10}$


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

There are 203 students in the third grade, which is 125 fewer than the fourth grade. How many students are there in total in the third and fourth grades?

Answer: $531$


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

There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat M traveled for 50 minutes from A towards B and covered $\frac{1}{3}$ of the river's length. At this moment, the reservoir started releasing water, and the speedboat took only 20 minutes to travel another $\frac{1}{3}$ of the river's length. The driver then turned off the speedboat's engine and allowed it to drift with the current, taking $\quad$ minutes for the speedboat to reach B.

Answer: $100/3$


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

Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers?

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>.

Consider the sequence of six real numbers 60, 10, 100, 150, 30, and  $x$ .  The average (arithmetic mean) of this sequence is equal to the median of the sequence.  What is the sum of all the possible values of  $x$ ?  (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)

Answer: $135$


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

Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \in N^{*}$, find the value of $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.

Answer: $2015$


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

Given vectors $\overrightarrow{m}=(\cos \alpha- \frac{\sqrt{2}}{3},-1)$ and $\overrightarrow{n}=(\sin \alpha,1)$, vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ are collinear, and $\alpha \in [-\frac{\pi}{2},0]$.
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;
$(2)$ Find the value of $\frac{\sin 2\alpha}{\sin \alpha - \cos \alpha}$.

Answer: $\frac{7}{12}$


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

Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-2$, $b=1$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$.

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


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

Let \( a, b \in \{2, 3, \cdots, 8\} \). Find the maximum value of \(\frac{a}{10b + a} + \frac{b}{10a + b}\).

Answer: $\frac{89}{287}$


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

Given that \(\alpha, \beta \in \left(\frac{3\pi}{4}, \pi \right)\), \(\cos (\alpha + \beta) = \frac{4}{5}\), and \(\sin \left(\alpha - \frac{\pi}{4}\right) = \frac{12}{13}\), find \(\cos \left(\beta + \frac{\pi}{4}\right)\).

Answer: $-\frac{56}{65}$