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

In the equation "中环杯是 + 最棒的 = 2013", different Chinese characters represent different digits. What is the possible value of "中 + 环 + 杯 + 是 + 最 + 棒 + 的"? (If there are multiple solutions, list them all).

Answer: $1250 + 763$


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

Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\in\mathbb{N}^*$), and that $\{a_{2n-1}\}$ is an increasing sequence, $\{a_{2n}\}$ is a decreasing sequence, find the limit $$\lim_{n\to\infty} \frac{a_{2n-1}}{a_{2n}} = \_\_\_\_\_\_.$$

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 sequence of distinct circles  $\omega_1, \omega_2, \cdots$  is inscribed in the parabola  $y=x^2$  so that  $\omega_n$  and  $\omega_{n+1}$  are tangent for all  $n$ .  If  $\omega_1$  has diameter  $1$  and touches the parabola at  $(0,0)$ , find the diameter of  $\omega_{1998}$ .

Answer: $3995$


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

In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$.

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 sum of all integer values of \( c \) for which the equation \( 15|p-1| + |3p - |p + c|| = 4 \) has at least one solution for \( p \).

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>.

On a circular keychain, I need to arrange six keys. Two specific keys, my house key and my car key, must always be together. How many different ways can I arrange these keys considering that arrangements can be rotated or reflected?

Answer: $48$


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

(1) Let the function $f(x) = |x - 2| + |x + a|$. If the inequality $f(x) \geq 3$ always holds for all $x \in \mathbb{R}$, find the range of values for the real number $a$.
(2) Given positive numbers $x, y, z$ that satisfy $x + 2y + 3z = 1$, find the minimum value of $\frac {3}{x} + \frac {2}{y} + \frac {1}{z}$.

Answer: $16 + 8\sqrt{3}$


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>.

A hexagonal lattice now consists of 13 points where each point is one unit distance from its nearest neighbors. Points are arranged with one center point, six points forming a hexagon around the center, and an additional outer layer of six points forming a larger hexagon. How many equilateral triangles have all three vertices in this new expanded lattice?

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>.

Let \( z \) be a complex number such that \( |z| = 2 \). Find the maximum value of
\[
|(z - 2)(z + 2)^2|.
\]

Answer: $16 \sqrt{2}$


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

What is the least positive integer with exactly $12$ positive factors?

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>.

Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line?

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


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

In triangle $\triangle ABC$, point $G$ satisfies $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\overrightarrow{AM}=m\overrightarrow{AB}$ $(m>0)$ and $\overrightarrow{AN}=n\overrightarrow{AC}$ $(n>0)$, then the minimum value of $3m+n$ is ______.

Answer: $\frac{4}{3}+\frac{2\sqrt{3}}{3}$


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>.

A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$?

Answer: $\frac{13}{16}$


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>.

Sarah multiplied an integer by itself. Which of the following could be the result?

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 a circle $C: (x-3)^2 + (y-4)^2 = 25$, the shortest distance from a point on circle $C$ to line $l: 3x + 4y + m = 0 (m < 0)$ is $1$. If point $N(a, b)$ is located on the part of line $l$ in the first quadrant, find the minimum value of $\frac{1}{a} + \frac{1}{b}$.

Answer: $\frac{7 + 4\sqrt{3}}{55}$


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

David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?

*Ray Li*

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>.

$ABCDEF$  is a regular hexagon. Let  $R$  be the overlap between  $\vartriangle ACE$  and  $\vartriangle BDF$ . What is the area of  $R$  divided by the area of  $ABCDEF$ ?

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>.

Suppose there is an octahedral die with the numbers 1, 2, 3, 4, 5, 6, 7, and 8 written on its eight faces. Each time the die is rolled, the chance of any of these numbers appearing is the same. If the die is rolled three times, and the numbers appearing on the top face are recorded in sequence, let the largest number be represented by $m$ and the smallest by $n$.
(1) Let $t = m - n$, find the range of values for $t$;
(2) Find the probability that $t = 3$.

Answer: $\frac{45}{256}$


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

Given the set of positive odd numbers $\{1, 3, 5, \ldots\}$, we are grouping them in order such that the $n$-th group contains $(2n-1)$ odd numbers, determine which group the number 2009 belongs to.

Answer: $31$


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

The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values?

Answer: $15,450$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 values of  $x$  such that the set  $\{107, 122,127, 137, 152,x\}$  has a mean that is equal to its median.

Answer: $234$


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

Using the digits 1 to 6 to form the equation shown below, where different letters represent different digits, the two-digit number represented by $\overline{A B}$ is what?
$$
\overline{A B} \times (\overline{C D} - E) + F = 2021
$$

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>.

What is $1010101_2 + 111000_2$? Write your answer in base $10$.

Answer: $141$


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

Find $\cos B$ and $\sin A$ in the following right triangle where side $AB = 15$ units, and side $BC = 20$ units.

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


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

Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exactly $3?
A) $\frac{1}{3}\quad$ B) $\frac{1}{2}\quad$ C) $\frac{9}{13}\quad$ D) $\frac{8}{13}\quad$ E) $\frac{3}{4}$

Answer: $\frac{8}{13}$


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>.

Given that \( n \) is a positive integer and \( n \leq 144 \), ten questions of the type "Is \( n \) smaller than \( a \)?" are allowed. The answers are provided with a delay such that the answer to the \( i \)-th question is given only after the \((i+1)\)-st question is asked for \( i = 1, 2, \ldots, 9 \). The answer to the tenth question is given immediately after it is asked. Find a strategy for identifying \( n \).

Answer: $144$


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

The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \in \left[ 0,\frac{\pi}{2} \right]$ such that the inequality $f(x) \leqslant m$ holds.

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 tetrahedron $PQRS$ has edges of lengths $8, 14, 19, 28, 37,$ and $42$ units. If the length of edge $PQ$ is $42$, determine the length of edge $RS$.

Answer: $14$


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

For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?

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>.

In the expansion of ${(6x+\frac{1}{3\sqrt{x}})}^{9}$, arrange the fourth term in ascending powers of $x$.

Answer: $\frac{224}{9}$


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>.

Evaluate the monotonic intervals of $F(x)=\int_{0}^{x}(t^{2}+2t-8)dt$ for $x > 0$.

(1) Determine the monotonic intervals of $F(x)$.

(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,3]$.

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


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

Find the time, in seconds, after 12 o'clock, when the area of $\triangle OAB$ will reach its maximum for the first time.

Answer: $\frac{15}{59}$


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>.

Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 12100$, where the signs change after each perfect square.

Answer: $1331000$


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

For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$.
$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$.
$(2)$ The inequality $t\leqslant \left(m-1\right)^{2}+\left(n-1\right)^{2}+\left(m-n\right)^{2}$ always holds. Find the maximum value of the real number $t$.

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


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>.

Determine $\sqrt[4]{105413504}$ without a calculator.

Answer: $101$


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

What is the largest integer that must divide the product of any $5$ consecutive integers?

Answer: $60$


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

The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are equal to 41 and 24 respectively, and its diagonals are perpendicular to each other. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).

Answer: $984$


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

Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying \( f(m+n) \geq f(m) + f(f(n)) - 1 \) for all \( m, n \in \mathbb{N} \). What values can \( f(2019) \) take?

Answer: $2019$


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

We choose 100 points in the coordinate plane. Let  $N$  be the number of triples  $(A,B,C)$  of distinct chosen points such that  $A$  and  $B$  have the same  $y$ -coordinate, and  $B$  and  $C$  have the same  $x$ -coordinate. Find the greatest value that  $N$  can attain considering all possible ways to choose the points.

Answer: $8100$


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

Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis?

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>.

In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board?

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>.

Let  $S$  be the locus of all points  $(x,y)$  in the first quadrant such that  $\dfrac{x}{t}+\dfrac{y}{1-t}=1$  for some  $t$  with  $0<t<1$ .  Find the area of  $S$ .

Answer: $1/6$


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>.

A rigid board with a mass of \(m\) and a length of \(l = 24\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \(2m\) is placed at its very edge. How far from the stone can a person with a mass of \(m\) walk on the board? Neglect the dimensions of the stone and the person compared to the dimensions of the board.

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 Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other.

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>.

In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. Given the line $l$: $ρ=- \frac{6}{3\cos θ +4\sin θ}$, and the curve $C$: $\begin{cases} \overset{x=3+5\cos α}{y=5+5\sin α} \end{cases}(α$ is the parameter$)$．
(I) Convert the line $l$ into rectangular equation and the curve $C$ into polar coordinate equation;
(II) If the line $l$ is moved up $m$ units and is tangent to the curve $C$, find the value of $m$.

Answer: $15$


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

Given $a=(2,4,x)$ and $b=(2,y,2)$, if $|a|=6$ and $a \perp b$, then the value of $x+y$ is ______.

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>.

From the set of numbers \( 1, 2, 3, 4, \cdots, 1982 \), remove some numbers so that in the remaining numbers, no number is equal to the product of any two other numbers. What is the minimum number of numbers that need to be removed to achieve this? How can this be done?

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 that $|\vec{a}|=|\vec{b}|=1$ and $\vec{a} \perp \vec{b}$, find the projection of $2\vec{a}+\vec{b}$ in the direction of $\vec{a}+\vec{b}$.

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


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

If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?

Answer: $14$


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

To arrange a class schedule for one day with the subjects Chinese, Mathematics, Politics, English, Physical Education, and Art, where Mathematics must be in the morning and Physical Education in the afternoon, determine the total number of different arrangements.

Answer: $192$


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

For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$.

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>.

A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed by the one on the right. All possible sets of numbers corresponding to different positions of the 100-sided polygon are written down. Calculate the sum of the numbers that appear in the 13th place from the left in these sets.

Answer: $10100$


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

Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\]

Answer: $76$


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

According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:
$$
p= \begin{cases}
\frac{x^{2}+60}{540} & (0<x\leq 12) \\
\frac{1}{2} & (12<x\leq 20)
\end{cases}
$$
It is known that for each non-defective product produced, a profit of 2 yuan can be made, while producing a defective product results in a loss of 1 yuan. (The factory's daily profit $y$ = daily profit from non-defective products - daily loss from defective products).
(1) Express the daily profit $y$ (in 10,000 yuan) as a function of the daily output $x$ (in 10,000 pieces);
(2) At what daily output (in 10,000 pieces) is the daily profit maximized? What is the maximum daily profit in yuan?

Answer: $\frac{100}{9}$


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>.

The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table.
\[
\begin{tabular}{|l|c|}
\hline
Cyclone & 180 feet \\ \hline
Gravity Rush & 120 feet \\ \hline
Screamer & 150 feet \\ \hline
Sky High & 310 feet \\ \hline
Twister & 210 feet \\ \hline
Loop de Loop & 190 feet \\ \hline
\end{tabular}
\]
What is the positive difference between the mean and the median of these values?

Answer: $8.\overline{3}$


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

Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles.

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 that the first tank is $\tfrac{3}{4}$ full of oil and the second tank is empty, while the second tank becomes $\tfrac{5}{8}$ full after oil transfer, determine the ratio of the volume of the first tank to that of the second tank.

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>.

Suppose the polynomial  $f(x) = x^{2014}$  is equal to  $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$  for some real numbers  $a_0,... , a_{2014}$ . Find the largest integer  $m$  such that  $2^m$  divides  $a_{2013}$ .

Answer: $2004$


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

Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$.

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


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

Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square.

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>.

If the purchase price of a product is 8 yuan and it is sold for 10 yuan per piece, 200 pieces can be sold per day. If the selling price of each piece increases by 0.5 yuan, the sales volume will decrease by 10 pieces. What should the selling price be set at to maximize the profit? And calculate this maximum profit.

Answer: $720$


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

What is the result of adding 12.8 to a number that is three times more than 608?

Answer: $2444.8$


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

Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\left(ab+2\right)+\left(b+1\right)^{2}+2a$.

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>.

An equilateral triangle $ABC$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\overline{BC}$ and $\overline{AB}$, respectively. Determine the ratio $\frac{AC}{F_1 F_2}$ if $F_1 F_2 = 2$.

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


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>.

Solve for $y$: $50^4 = 10^y$

Answer: $6.79588$


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

Let  $ABC$  be a triangle with  $\angle BAC=60^\circ$ . Consider a point  $P$  inside the triangle having  $PA=1$ ,  $PB=2$  and  $PC=3$ . Find the maximum possible area of the triangle  $ABC$ .

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


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>.

Calculate the remainder when the sum $100001 + 100002 + \cdots + 100010$ is divided by 11.

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 an arithmetic sequence $\left\{a_{n}\right\}$ with the first term $a_{1}>0$, and the following conditions:
$$
a_{2013} + a_{2014} > 0, \quad a_{2013} a_{2014} < 0,
$$

find the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds true.

Answer: $4026$


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

Consider two right-angled triangles, ABC and DEF. Triangle ABC has a right angle at C with AB = 10 cm and BC = 7 cm. Triangle DEF has a right angle at F with DE = 3 cm and EF = 4 cm. If these two triangles are arranged such that BC and DE are on the same line segment and point B coincides with point D, what is the area of the shaded region formed between the two triangles?

Answer: $29$


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

In an isosceles right triangle $ABC$ with right angle at $C$ and an area of $18$ square units, the rays trisecting $\angle ACB$ intersect $AB$ at points $D$ and $E$. Calculate the area of triangle $CDE$.
A) 4.0
B) 4.5
C) 5.0
D) 5.5

Answer: $4.5$


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

The polynomial equation \[x^4 + dx^2 + ex + f = 0,\] where \(d\), \(e\), and \(f\) are rational numbers, has \(3 - \sqrt{5}\) as a root. It also has two integer roots. Find the fourth root.

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>.

The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$?

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>.

Find the sum of the distinct prime factors of $7^7 - 7^4$.

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>.

How many four-digit positive integers are multiples of 7?

Answer: $1286$


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

Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers.

Answer: $-530,784$


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

How many multiples of 15 are between 15 and 305?

Answer: $20$


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

Let  $2000 < N < 2100$  be an integer. Suppose the last day of year  $N$  is a Tuesday while the first day of year  $N+2$  is a Friday. The fourth Sunday of year  $N+3$  is the  $m$ th day of January. What is  $m$ ?

*Based on a proposal by Neelabh Deka*

Answer: $23$


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

Given \(a, b, c \in (0, 1]\) and \(\lambda \) is a real number such that
\[ \frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c), \]
find the maximum value of \(\lambda\).

Answer: $\frac{64}{27}$


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

An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the longer side of this frame?

Answer: $2:3$


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

Given a $4\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15$ and the fourth term being $45$.

Answer: $\frac{43}{3}$


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

Find the largest real number  $\lambda$  such that

\[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\]
for all real numbers  $a_1, \ldots, a_{2019}$ .  The coefficients on the right-hand side are  $1$  for all terms except  $a_{1009}a_{1010}$  and  $a_{1010}a_{1011}$ , which have coefficient  $\lambda$ .

Answer: $3/2$


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

Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time.

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>.

Place the terms of the sequence $\{2n-1\}$ ($n\in\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms, the fourth bracket contains the seventh term, and so on, in a cycle, such as: $(1)$, $(3, 5)$, $(7, 9, 11)$, $(13)$, $(15, 17)$, $(19, 21, 23)$, $(25)$, ... . What is the sum of the numbers in the 105th bracket?

Answer: $1251$


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

Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\frac{1}{2}m{x^2}+\sqrt{2}x+{m^2}=0$, and ${x_1}=\sqrt{a+2}-\sqrt{8-a}+\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation.

Answer: $2\sqrt{2}$


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

Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).

Answer: $1019$


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

During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item $x$ (yuan) (where $8\leqslant x\leqslant 15$, and $x$ is an integer). Some corresponding values are shown in the table below:

| Selling Price (yuan) | $9$ | $11$ | $13$ |
|----------------------|-----|------|------|
| Daily Sales Quantity (items) | $105$ | $95$ | $85$ |

$(1)$ Find the function relationship between $y$ and $x$.
$(2)$ If the store makes a profit of $425$ yuan per day selling this disinfectant product, what is the selling price per item?
$(3)$ Let the store's profit from selling this disinfectant product per day be $w$ (yuan). When the selling price per item is what amount, the daily sales profit is maximized? What is the maximum profit?

Answer: $525$


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

If $100^a = 7$ and $100^b = 11,$ then find $20^{(1 - a - b)/(2(1 - b))}.$

Answer: $\frac{100}{77}$


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

In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units.

Answer: $10 - 5 \sqrt{2+\sqrt{2}}$


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

Consider the set  $E = \{5, 6, 7, 8, 9\}$ . For any partition  ${A, B}$  of  $E$ , with both  $A$  and  $B$  non-empty, consider the number obtained by adding the product of elements of  $A$  to the product of elements of  $B$ . Let  $N$  be the largest prime number amonh these numbers. Find the sum of the digits of  $N$ .

Answer: $17$


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

For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube.

Answer: $1998$


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

Ellie and Sam run on the same circular track but in opposite directions, with Ellie running counterclockwise and Sam running clockwise. Ellie completes a lap every 120 seconds, while Sam completes a lap every 75 seconds. They both start from the same starting line simultaneously. Ten to eleven minutes after the start, a photographer located inside the track takes a photo of one-third of the track, centered on the starting line. Determine the probability that both Ellie and Sam are in this photo.
A) $\frac{1}{4}$
B) $\frac{5}{12}$
C) $\frac{1}{3}$
D) $\frac{7}{18}$
E) $\frac{1}{5}$

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


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 that $f(x)$ is an odd function defined on $\mathbb{R}$, and for $x \geqslant 0$, $f(x) = \begin{cases} \log_{\frac{1}{2}}(x+1), & 0 \leqslant x < 1 \\ 1-|x-3|, & x \geqslant 1 \end{cases}$, determine the sum of all zeros of the function $y = f(x) + \frac{1}{2}$.

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>.

Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $6$ miles per hour. Find $r$.
A) 10
B) 11
C) 12
D) 13
E) 14

Answer: $12$


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

Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$?

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


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

Define the sequence \( b_1, b_2, b_3, \ldots \) by \( b_n = \sum\limits_{k=1}^n \cos{k} \), where \( k \) represents radian measure. Find the index of the 50th term for which \( b_n < 0 \).

Answer: $314$


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

Given that the odometer initially showed 35,400 miles and the driver filled the gas tank with 8 gallons of gasoline, and later filled the tank again with 15 gallons when the odometer showed 35,680 miles, and finally filled the tank with 18 gallons of gasoline when the odometer read 36,000 miles, calculate the car's average miles-per-gallon for the entire trip.

Answer: $14.6$


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

A certain store in Hefei plans to sell a newly launched stationery item, with a purchase price of 20 yuan per item. During the trial marketing phase, it was found that when the selling price is 25 yuan per item, the daily sales volume is 150 items; for every 1 yuan increase in the selling price, the daily sales volume decreases by 10 items.
(1) Find the function relationship between the daily sales profit $w$ (in yuan) and the selling price $x$ (in yuan) for this stationery item;
(2) At what selling price will the daily sales profit for this stationery item be maximized?
(3) The store now stipulates that the daily sales volume of this stationery item must not be less than 120 items. To maximize the daily sales profit for this stationery item, at what price should it be set to achieve the maximum daily profit?

Answer: $960$


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

Given a seminar recording of 495 minutes that needs to be divided into multiple USB sticks, each capable of holding up to 65 minutes of audio, and the minimum number of USB sticks is used, calculate the length of audio that each USB stick will contain.

Answer: $61.875$


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

Given that a shop advertises everything as "half price in today's sale," and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both items that the customer ultimately pays when the second item's original price is the same as the first's.

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>.

Two circles of radius $s$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8.$ Find $s.$

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


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

Given Orvin goes to a store with just enough money to buy 40 balloons, and the store has a special promotion: for every balloon bought at full price, a second one can be bought at 1/2 off. Find the maximum number of balloons Orvin can buy.

Answer: $52$


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

On an algebra test, there were $7x$ problems. Lucky Lacy missed $2x$ of them. What percent of the problems did she get correct?

Answer: $71.43\%$


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>.

Points $P$ and $Q$ are on a circle of radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of the line segment $PR$.

Answer: $\sqrt{98 - 14\sqrt{33}}$


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

Calculate the product: $500 \times 2019 \times 0.02019 \times 5.$

Answer: $0.25 \times 2019^2$


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 average of the numbers $1, 2, 3,\dots, 49, 50,$ and $x$ is $80x$. What is $x$?

Answer: $\frac{1275}{4079}$


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

In the trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are 8 and 18 respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to the lines \(BC\) and \(CD\). Find the perimeter of the trapezoid.

Answer: $56$


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

What is the largest value of $n$ less than 100,000 for which the expression $10(n-3)^5 - n^2 + 20n - 30$ is a multiple of 7?

Answer: $99999$


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

Given the parametric equation of curve $C_{1}$ is $\begin{cases} x=-2+2\cos \theta \\ y=2\sin \theta \end{cases}$ (with $\theta$ as the parameter), and establishing a coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{2}$ is $\rho=4\sin \theta$.
(Ⅰ) Find the coordinates of the intersection points of curves $C_{1}$ and $C_{2}$.
(Ⅱ) Points $A$ and $B$ are on curves $C_{1}$ and $C_{2}$, respectively. When $|AB|$ is maximized, find the area of $\triangle OAB$ (where $O$ is the origin).

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


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

A point  $E$  on side  $CD$  of a rectangle  $ABCD$  is such that  $\triangle DBE$  is isosceles and  $\triangle ABE$  is right-angled. Find the ratio between the side lengths of the rectangle.

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


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

Find the largest positive integer  $n$  such that for each prime  $p$  with  $2<p<n$  the difference  $n-p$  is also prime.

Answer: $10$


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

A cauldron has the shape of a paraboloid of revolution. The radius of its base is \( R = 3 \) meters, and the depth is \( H = 5 \) meters. The cauldron is filled with a liquid, the specific weight of which is \( 0.8 \Gamma / \text{cm}^3 \). Calculate the work required to pump the liquid out of the cauldron.

Answer: $294300\pi$


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>.

Define the operation: \(a \quad b = \frac{a \times b}{a + b}\). Calculate the result of the expression \(\frac{20102010 \cdot 2010 \cdots 2010 \cdot 201 \text{ C}}{\text{ 9 " "}}\).

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 second quadrant and $\cos (α+π)= \frac {3}{13}$.
(1) Find the value of $\tan α$;
(2) Find the value of $\sin (α- \frac {π}{2}) \cdot \sin (-α-π)$.

Answer: $-\frac{12\sqrt{10}}{169}$


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

Rectangle $EFGH$ has sides $\overline {EF}$ of length 5 and $\overline {FG}$ of length 4. Divide $\overline {EF}$ into 196 congruent segments with points $E=R_0, R_1, \ldots, R_{196}=F$, and divide $\overline {FG}$ into 196 congruent segments with points $F=S_0, S_1, \ldots, S_{196}=G$. For $1 \le k \le 195$, draw the segments $\overline {R_kS_k}$. Repeat this construction on the sides $\overline {EH}$ and $\overline {GH}$, and then draw the diagonal $\overline {EG}$. Find the sum of the lengths of the 389 parallel segments drawn.

Answer: $195 \sqrt{41}$


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

Given that $a$, $b$, $c$, $d$, $e$, and $f$ are all positive numbers, and $\frac{bcdef}{a}=\frac{1}{2}$, $\frac{acdef}{b}=\frac{1}{4}$, $\frac{abdef}{c}=\frac{1}{8}$, $\frac{abcef}{d}=2$, $\frac{abcdf}{e}=4$, $\frac{abcde}{f}=8$, find $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}$.

Answer: $\frac{119}{8}$


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>.

What is the largest integer that must divide the product of any $5$ consecutive integers?

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 triangle ABC, where sides $a$, $b$, and $c$ correspond to angles A, B, and C respectively, and $a=4$, $\cos{B}=\frac{4}{5}$.
(1) If $b=6$, find the value of $\sin{A}$;
(2) If the area of triangle ABC, $S=12$, find the values of $b$ and $c$.

Answer: $2\sqrt{13}$


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>.

A certain high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:

| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | 200 | 100 | 100 |

From these 1000 students, one student is randomly selected. Let:
- $A=$ "The student chose Physics"
- $B=$ "The student chose Chemistry"
- $C=$ "The student chose Biology"
- $D=$ "The student chose Politics"
- $E=$ "The student chose History"
- $F=$ "The student chose Geography"

$(Ⅰ)$ Find $P(B)$ and $P(DEF)$.

$(Ⅱ)$ Find $P(C \cup E)$ and $P(B \cup F)$.

$(Ⅲ)$ Are events $A$ and $D$ independent? Please explain your reasoning.

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>.

A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering.

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>.

How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers?

Answer: $50$


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

The graph of the function in the form \( y=\frac{b}{|x|-a} \) (where \( a, b > 0 \)) resembles the Chinese character "唄". It is referred to as the "唄 function", and the point symmetric to its intersection with the y-axis about the origin is called the "目 point". A circle with its center at the 明 point that intersects the 唄 function is called the "唄 circle". For \( a=b=1 \), the minimum area of all 唄 circles is .

Answer: $3\pi$


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

Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Assume the shaded region is the smaller region formed inside the triangle including points $D$, $E$, $F$, $G$, and $H$.

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>.

There are 21 different pairs of digits (a, b) such that $\overline{5a68} \times \overline{865b}$ is divisible by 824.

Answer: $19$


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

Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.

Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter.

Answer: $157$


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

What is the product of the prime numbers less than 20?

Answer: $9699690$