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

If $e^{i \theta} = \frac{1 + i \sqrt{2}}{2}$, then find $\sin 3 \theta.$

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


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

Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \in [\frac{1}{2}, \frac{\sqrt{3}}{2}]$, find the maximum value of $a$.

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


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

Let $(b_1,b_2,b_3,\ldots,b_{10})$ be a permutation of $(1,2,3,\ldots,10)$ for which
$b_1>b_2>b_3>b_4 \mathrm{\ and \ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$
Find the number of such permutations.

Answer: $84$


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

Find \( g\left(\frac{1}{1996}\right) + g\left(\frac{2}{1996}\right) + g\left(\frac{3}{1996}\right) + \cdots + g\left(\frac{1995}{1996}\right) \) where \( g(x) = \frac{9x}{3 + 9x} \).

Answer: $997$


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

Given the function $f(x)=x^{2}+x-2$, determine the characteristics of the function $f(x)$ in the interval $[-1,1)$.

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


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

In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.

a) What is the minimum number of socks that can be in the box?

b) What is the minimum number of socks that can be in the box, given that the number of black socks is even?

Answer: $21$


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

Let the set $\mathbf{A}=\{1, 2, 3, 4, 5, 6\}$ and a bijection $f: \mathbf{A} \rightarrow \mathbf{A}$ satisfy the condition: for any $x \in \mathbf{A}$, $f(f(f(x)))=x$. Then the number of bijections $f$ satisfying the above condition is:

Answer: $81$


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

Grace is trying to solve the following equation by completing the square: $$36x^2-60x+25 = 0.$$ She successfully rewrites the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$?

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>.

The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one?

Answer: $108$


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

Evaluate the following expression:

$$\frac { \sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$

Answer: $-4 \sqrt {3}$


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

In a right triangle, side AB is 8 units, and side BC is 15 units. Calculate $\cos C$ and $\sin C$.

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


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

How many multiples of 15 are there between 40 and 240?

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>.

Given that the complex number $z$ satisfies the equation $\frac{1-z}{1+z}={i}^{2018}+{i}^{2019}$ (where $i$ is the imaginary unit), find the value of $|2+z|$.

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


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>.

Fill in the blanks with appropriate numbers.
4 liters 25 milliliters = ___ milliliters
6.09 cubic decimeters = ___ cubic centimeters
4.9 cubic decimeters = ___ liters ___ milliliters
2.03 cubic meters = ___ cubic meters ___ cubic decimeters.

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>.

Definition: If the line $l$ is tangent to the graphs of the functions $y=f(x)$ and $y=g(x)$, then the line $l$ is called the common tangent line of the functions $y=f(x)$ and $y=g(x)$. If the functions $f(x)=a\ln x (a>0)$ and $g(x)=x^{2}$ have exactly one common tangent line, then the value of the real number $a$ is ______.

Answer: $2e$


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

Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they arrived at the destination with the odometer showing another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum possible average speed (in kilometers per hour) that Chester's father could have driven?

Answer: $82.2$


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 triangle  $ABC$  it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray  $BC$  beyond the point  $C$  the point  $T$  is taken so that  $BC = CT$ . Let  $M$  be the midpoint of the segment  $AT$ . Find the measure of the  $\angle BMC$ .

(Anton Trygub)

Answer: $45$


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

Given the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9}{1+2+3+4+5+6+7+8+9+10}$, evaluate the expression.

Answer: $6608$


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

In triangle \(ABC\), the angle at vertex \(A\) is \(60^{\circ}\). A circle is drawn through points \(B\), \(C\), and point \(D\), which lies on side \(AB\), intersecting side \(AC\) at point \(E\). Find \(AE\) if \(AD = 3\), \(BD = 1\), and \(EC = 4\). Find the radius of the circle.

Answer: $\sqrt{7}$


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

Given a non-right triangle $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c=1$, also $$C= \frac {\pi}{3}$$, if $\sin C + \sin(A-B) = 3\sin 2B$, then the area of $\triangle ABC$ is \_\_\_\_\_\_.

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


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

Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$  Find $p(8).$

Answer: $40328$


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

Let the function \( f(x) = x^3 + a x^2 + b x + c \), where \( a \), \( b \), and \( c \) are non-zero integers. If \( f(a) = a^3 \) and \( f(b) = b^3 \), then the value of \( c \) is:

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  $ABCD$  be a unit square.  $E$  and  $F$  trisect  $AB$  such that  $AE<AF. G$  and  $H$  trisect  $BC$  such that  $BG<BH. I$  and  $J$  bisect  $CD$  and  $DA,$  respectively. Let  $HJ$  and  $EI$  meet at  $K,$  and let  $GJ$  and  $FI$  meet at  $L.$  Compute the length  $KL.$

Answer: $\frac{6\sqrt{2}}{35}$


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>.

The sum of four prime numbers $P,$ $Q,$ $P-Q,$ and $P+Q$ must be expressed in terms of a single letter indicating the property of the sum.

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>.

Assuming that the new demand is given by \( \frac{1}{1+ep} \), where \( p = 0.20 \) and \( e = 1.5 \), calculate the proportionate decrease in demand.

Answer: $0.23077$


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

Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a, b, c\) are positive real numbers.

Answer: $3 \cdot \frac{1}{\sqrt[3]{162}}$


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

Jun Jun is looking at an incorrect single-digit multiplication equation \( A \times B = \overline{CD} \), where the digits represented by \( A \), \( B \), \( C \), and \( D \) are all different from each other. Clever Jun Jun finds that if only one digit is changed, there are 3 ways to correct it, and if only the order of \( A \), \( B \), \( C \), and \( D \) is changed, the equation can also be corrected. What is \( A + B + C + D = \) ?

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>.

Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$.

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


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

In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A+B+C$?

Answer: $297$


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

The teacher gave each of her  $37$  students  $36$  pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.

Answer: $666$


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

Graphistan has  $2011$  cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer  $k$  such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than  $k.$

Answer: $1005$


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

A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to:

Answer: $2.8$


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

Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$.

Answer: $100$


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

Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads?

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


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>.

$ABCD$ is a square where each side measures 4 units. $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{CD},$ respectively. Find $\sin \phi$ where $\phi$ is the angle $\angle APQ$.

![Diagram](https://i.imgur.com/6T5xN1f.png)

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>.

If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$?

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>.

14 students attend the IMO training camp. Every student has at least  $k$  favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least  $k$ , such that this is always possible if: $a)$  The students can be arranged in a circle such that every two students sitting next to one another have different numbers. $b)$   $7$  of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number.

Answer: $k = 2$


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 $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal.

Answer: $14.0$


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

For  $n > 1$ , let  $a_n$  be the number of zeroes that  $n!$  ends with when written in base  $n$ . Find the maximum value of  $\frac{a_n}{n}$ .

Answer: $1/2$


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

What is the smallest positive value of $x$ such that $x + 8765$ results in a palindrome?

Answer: $13$


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

1. The converse of the proposition "If $x > 1$, then ${x}^{2} > 1$" is ________.
2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.
3. In a geometric sequence $\\{a\_{n}\\}$, if $a\_{3}$ and $a\_{15}$ are roots of the equation $x^{2}-6x+8=0$, then $\frac{{a}\_{1}{a}\_{17}}{{a}\_{9}} =$ ________.
4. Let $F$ be the left focus of the hyperbola $C$: $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{12} =1$. Let $A(1,4)$ and $P$ be a point on the right branch of $C$. When the perimeter of $\triangle APF$ is minimum, the distance from $F$ to the line $AP$ is ________.

Answer: $\frac{32}{5}$


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 absolute value of -1.2 is ____, and its reciprocal is ____.

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


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>.

Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd?

Answer: $37$


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

One dimension of a cube is increased by $2$, another is decreased by $2$, and the third is increased by $3$. The volume of the new rectangular solid is $7$ less than the volume of the cube. Find the original volume of the cube.

Answer: $27$


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>.

Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, calculate the value of $m$.

Answer: $-5$


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

Let  $\triangle ABC$  be equilateral with integer side length. Point  $X$  lies on  $\overline{BC}$  strictly between  $B$  and  $C$  such that  $BX<CX$ . Let  $C'$  denote the reflection of  $C$  over the midpoint of  $\overline{AX}$ . If  $BC'=30$ , find the sum of all possible side lengths of  $\triangle ABC$ .

*Proposed by Connor Gordon*

Answer: $130$


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

In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) and \(\angle BLN = 45^{\circ}\). In what ratio does point \( N \) divide the side \( AB \)?

Answer: $1:2$


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

If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\]

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


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

What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?

Answer: $16.67\%$


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

The diagram shows the miles traveled by cyclists Clara and David. After five hours, how many more miles has Clara cycled than David?
[asy]
/* Modified AMC8 1999 #4 Problem */
draw((0,0)--(6,0)--(6,4.5)--(0,4.5)--cycle);
for(int x=0; x <= 6; ++x) {
    for(real y=0; y <=4.5; y+=0.9) {
        dot((x, y));
    }
}
draw((0,0)--(5,3.6));  // Clara's line
draw((0,0)--(5,2.7));  // David's line
label(rotate(36)*"David", (3,1.35));
label(rotate(36)*"Clara", (3,2.4));
label(scale(0.75)*rotate(90)*"MILES", (-1, 2.25));
label(scale(0.75)*"HOURS", (3, -1));

label(scale(0.85)*"90", (0, 4.5), W);
label(scale(0.85)*"72", (0, 3.6), W);
label(scale(0.85)*"54", (0, 2.7), W);
label(scale(0.85)*"36", (0, 1.8), W);
label(scale(0.85)*"18", (0, 0.9), W);
label(scale(0.86)*"1", (1, 0), S);
label(scale(0.86)*"2", (2, 0), S);
label(scale(0.86)*"3", (3, 0), S);
label(scale(0.86)*"4", (4, 0), S);
label(scale(0.86)*"5", (5, 0), S);
label(scale(0.86)*"6", (6, 0), S);
[/asy]

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>.

If an integer $a$ ($a \neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is.

Answer: $16$


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

Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\frac{\sqrt{5}}{3}$.
(1) Find the equation of ellipse $C$;
(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle $x^{2}+y^{2}=1$, with the tangent points being $A$ and $B$. When the line $AB$ intersects the $x$-axis and $y$-axis at points $N$ and $M$, respectively, find the minimum value of $|MN|$.

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


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 set of 10 distinct integers  $S$  is chosen. Let  $M$  be the number of nonempty subsets of  $S$  whose elements have an even sum. What is the minimum possible value of  $M$ ?
<details><summary>Clarifications</summary>

- $S$  is the ``set of 10 distinct integers'' from the first sentence.
</details>

*Ray Li*

Answer: $511$


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

Given the function $f(x)=\cos ωx \cdot \sin (ωx- \frac {π}{3})+ \sqrt {3}\cos ^{2}ωx- \frac{ \sqrt {3}}{4}(ω > 0,x∈R)$, and the distance from one symmetry center of the graph of the function $y=f(x)$ to the nearest symmetry axis is $\frac {π}{4}$.
(I) Find the value of $ω$ and the equation of the symmetry axis of $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $f(A)= \frac { \sqrt {3}}{4}, \sin C= \frac {1}{3}, a= \sqrt {3}$, find the value of $b$.

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


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 constant term in the expansion of $\left(1+x+\frac{1}{x^{2}}\right)^{10}$.

Answer: $4351$


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

Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$.

Answer: $-5$


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

Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.

Answer: $25$


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

Given the broadcast time of the "Midday News" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20.

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>.

Given the set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set.

Answer: $53$


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

If the function $f(x)=\tan (\omega x+ \frac {\pi}{4})$ ($\omega > 0$) has a minimum positive period of $2\pi$, then $\omega=$ ______ ; $f( \frac {\pi}{6})=$ ______ .

Answer: $\sqrt {3}$


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

Find the maximum value of the function $y=\sin^2x+3\sin x\cos x+4\cos^2x$ for $0 \leqslant x \leqslant \frac{\pi}{2}$ and the corresponding value of $x$.

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


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

Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton?

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>.

In the Nanjing area, the weather in July and August is relatively hot. Xiaohua collected the highest temperature for ten consecutive days, obtaining the following set of data in sequence: 34, 35, 36, 34, 36, 37, 37, 36, 37, 37 (unit: ℃). The mode of this set of data is ▲, and the median is ▲.

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>.

The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex?

Answer: $3795$


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

From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____.

Answer: $5.5$


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

Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$.

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


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

In the Cartesian coordinate system $xOy$, point $P\left( \frac{1}{2},\cos^2\theta\right)$ is on the terminal side of angle $\alpha$, and point $Q(\sin^2\theta,-1)$ is on the terminal side of angle $\beta$, and $\overrightarrow{OP}\cdot \overrightarrow{OQ}=-\frac{1}{2}$.
$(1)$ Find $\cos 2\theta$;
$(2)$ Find the value of $\sin(\alpha+\beta)$.

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


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>.

Let \\(f(x)\\) be defined on \\((-∞,+∞)\\) and satisfy \\(f(2-x)=f(2+x)\\) and \\(f(7-x)=f(7+x)\\). If in the closed interval \\([0,7]\\), only \\(f(1)=f(3)=0\\), then the number of roots of the equation \\(f(x)=0\\) in the closed interval \\([-2005,2005]\\) is      .

Answer: $802$


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

What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?

Answer: $16.67\%$


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

Given the function $f(x) = \sin^2(wx) - \sin^2(wx - \frac{\pi}{6})$ ($x \in \mathbb{R}$, $w$ is a constant and $\frac{1}{2} < w < 1$), the graph of function $f(x)$ is symmetric about the line $x = \pi$.

(I) Find the smallest positive period of the function $f(x)$;

(II) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a = 1$ and $f(\frac{3}{5}A) = \frac{1}{4}$, find the maximum area of $\triangle ABC$.

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


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>.

How many positive four-digit integers are divisible by both 7 and 6?

Answer: $215$


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

There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.

(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.

(2) Given that among the 3 selected questions, there are 2 Type A questions and 1 Type B question, and the probability of Xiao Ming correctly answering each Type A question is $\dfrac{3}{5}$, and the probability of correctly answering each Type B question is $\dfrac{4}{5}$. Assuming the correctness of answers to different questions is independent, calculate the probability that Xiao Ming correctly answers at least 2 questions.

Answer: $\dfrac{93}{125}$


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

Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$, when the perimeter of $\triangle APF$ is minimized, the ordinate of point $P$ is ______.

Answer: $2 \sqrt {6}$


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

How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-10^2)(x-11^3)(x-12^3)\dotsm(x-13^3)}
\]
have for \(x\)?

Answer: $190$


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

A total of  $731$  objects are put into  $n$  nonempty bags where  $n$  is a positive integer. These bags can be distributed into  $17$  red boxes and also into  $43$  blue boxes so that each red and each blue box contain  $43$  and  $17$  objects, respectively. Find the minimum value 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>.

A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts?

Answer: $27.325\%$


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

In triangle ABC below, find the length of side AB.

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18\sqrt{2}$",C/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy]

Answer: $18\sqrt{2}$


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

Convert the complex number \(1 + i \sqrt{3}\) into its exponential form \(re^{i \theta}\) and find \(\theta\).

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


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

In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

Answer: $13$


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

Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum possible value of \(A\).

Answer: $4648$


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

A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle.

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>.

If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ 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>.

Rolling a die twice, let the points shown the first and second times be \(a\) and \(b\) respectively. Find the probability that the quadratic equation \(x^{2} + a x + b = 0\) has two distinct real roots both less than -1. (Answer with a number).

Answer: $1/12$


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

Given the functions $f(x)=x-\frac{1}{x}$ and $g(x)=2a\ln x$.

(1) When $a\geqslant -1$, find the monotonically increasing interval of $F(x)=f(x)-g(x)$;

(2) Let $h(x)=f(x)+g(x)$, and $h(x)$ has two extreme values $({{x}_{1}},{{x}_{2}})$, where ${{x}_{1}}\in (0,\frac{1}{3}]$, find the minimum value of $h({{x}_{1}})-h({{x}_{2}})$.

Answer: $\frac{20\ln 3-16}{3}$


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

The value of $1.000 + 0.101 + 0.011 + 0.001$ is:

Answer: $1.113$


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

What is the sum of all the integers between -25.4 and 15.8, excluding the integer zero?

Answer: $-200$


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

Rhombus $ABCD$ has side length $3$ and $\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
**A)** $0.81$
**B)** $1.62$
**C)** $2.43$
**D)** $2.16$
**E)** $3.24$

Answer: $2.16$


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>.

The average yield per unit area of a rice variety for five consecutive years was 9.4, 9.7, 9.8, 10.3, and 10.8 (unit: t/hm²). Calculate the variance of this sample data.

Answer: $0.244$


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

Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?

Answer: $33$


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

Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$.

Answer: $81$


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

In order to cultivate students' financial management skills, Class 1 of the second grade founded a "mini bank". Wang Hua planned to withdraw all the money from a deposit slip. In a hurry, the "bank teller" mistakenly swapped the integer part (the amount in yuan) with the decimal part (the amount in cents) when paying Wang Hua. Without counting, Wang Hua went home. On his way home, he spent 3.50 yuan on shopping and was surprised to find that the remaining amount of money was twice the amount he was supposed to withdraw. He immediately contacted the teller. How much money was Wang Hua supposed to withdraw?

Answer: $14.32$


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>.

A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\frac{1}{2 \times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares.

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


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

Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle.

Answer: $16\sqrt{3}$


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>.

Consider equilateral triangle  $ABC$  with side length  $1$ . Suppose that a point  $P$  in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant  $\kappa$ . Compute the sum of all possible values of  $\kappa$ .

*2018 CCA Math Bonanza Lightning Round #3.4*

Answer: $\frac{18\sqrt{3}}{5}$


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>.

Given the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$.

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>.

As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum?

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>.

There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. How many minutes earlier did the Type I car start running compared to the Type II car?

Answer: $36$


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

In the polar coordinate system, circle $C$ is centered at point $C\left(2, -\frac{\pi}{6}\right)$ with a radius of $2$.

$(1)$ Find the polar equation of circle $C$;

$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\theta = -\frac{5\pi}{12} (\rho \in \mathbb{R})$.

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>.

Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less than what person B bought for $3000$ yuan.
$(1)$ Using fractional equations, find the quantity of Xibing that person A purchased;
$(2)$ When person A and person B went to purchase the same kind of gift box filled with Xibing again, they coincidentally encountered a store promotion where the unit price was $20$ yuan less per box compared to the previous purchase. Person A spent the same total amount on Xibing as before, while person B bought the same quantity as before. Then, the average unit price of Xibing for person A over the two purchases is ______ yuan per box, and for person B is ______ yuan per box (write down the answers directly).

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>.

$30$ same balls are put into four boxes $A$, $B$, $C$, $D$ in such a way that sum of number of balls in $A$ and $B$ is greater than sum of in $C$ and $D$. How many possible ways are there?

Answer: $2600$


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

Perform the calculations:
3.21 - 1.05 - 1.95
15 - (2.95 + 8.37)
14.6 × 2 - 0.6 × 2
0.25 × 1.25 × 32

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>.

Let  $a_1=1$  and  $a_n=n(a_{n-1}+1)$  for all  $n\ge 2$  . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute  $\lim_{n\to \infty} P_n$

Answer: $e$


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

A sweater costs 160 yuan, it was first marked up by 10% and then marked down by 10%. Calculate the current price compared to the original.

Answer: $0.99$


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

Determine the probability that two edges selected at random from the twelve edges of a cube with side length 1 are skew lines (i.e., non-intersecting and not in the same plane).

Answer: $\frac{4}{11}$


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

Let the internal angles $A$, $B$, $C$ of $\triangle ABC$ be opposite to the sides $a$, $b$, $c$ respectively, and $c\cos B= \sqrt {3}b\sin C$.
$(1)$ If $a^{2}\sin C=4 \sqrt {3}\sin A$, find the area of $\triangle ABC$;
$(2)$ If $a=2 \sqrt {3}$, $b= \sqrt {7}$, and $c > b$, the midpoint of side $BC$ is $D$, find the length of $AD$.

Answer: $\sqrt {13}$


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

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. What is the probability that the bug is at vertex $A$ after crawling exactly 10 meters?

Answer: $\frac{20}{81}$


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

In $\triangle ABC$, it is known that $AB=2$, $AC=3$, and $A=60^{\circ}$.
$(1)$ Find the length of $BC$;
$(2)$ Find the value of $\sin 2C$.

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


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

Given vectors $\overrightarrow{a}=(\sin x, \sqrt{3}\cos x)$, $\overrightarrow{b}=(-1,1)$, and $\overrightarrow{c}=(1,1)$, where $x \in [0, \pi]$.

(1) If $(\overrightarrow{a} + \overrightarrow{b}) \parallel \overrightarrow{c}$, find the value of $x$;

(2) If $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{2}$, find the value of the function $\sin \left(x + \frac{\pi}{6}\right)$.

Answer: $\frac{\sqrt{15}}{4}$


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

Given a geometric sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\) such that \(S_n = 2^n + r\) (where \(r\) is a constant), let \(b_n = 2(1 + \log_2 a_n)\) for \(n \in \mathbb{N}^*\).

1. Find the sum of the first \(n\) terms of the sequence \(\{a_n b_n\}\), denoted as \(T_n\).
2. If for any positive integer \(n\), the inequality \(\frac{1 + b_1}{b_1} \cdot \frac{1 + b_2}{b_2} \cdots \cdot \frac{1 + b_n}{b_n} \geq k \sqrt{n + 1}\) holds, determine \(k\).

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


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

In right triangle $XYZ$, we have $\angle Y = \angle Z$ and $XY = 8\sqrt{2}$. What is the area of $\triangle XYZ$?

Answer: $64$


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

Given the function $f(x)= \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y=f(x)$ has a tangent at the point $(1,f(1))$ which is perpendicular to the line $y=\frac{1}{2}x$.
(i) Find the value of $a$;
(ii) Determine the intervals of monotonicity and the extreme values for the function $f(x)$.

Answer: $-\ln 5$


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 line $2x+ay-2=0$ is parallel to the line $ax+(a+4)y-4=0$. Find the value of $a$.

Answer: $-2$


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

Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ passing through the point $E(\sqrt{3}, 1)$, with an eccentricity of $\frac{\sqrt{6}}{3}$, and $O$ as the coordinate origin.
(I) Find the equation of the ellipse $C$;
(II) If point $P$ is a moving point on the ellipse $C$, and the perpendicular bisector of segment $AP$, where $A(3, 0)$, intersects the $y$-axis at point $B$, find the minimum value of $|OB|$.

Answer: $\sqrt{6}$


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

If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____.

Answer: $-1$


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

For any number $y$, define the operations $\&y = 2(7-y)$ and $\&y = 2(y-7)$. What is the value of $\&(-13\&)$?

Answer: $66$


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

Given that the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extreme value of 10 at $x = 1$, find the slope of the tangent to the function at $x = 2$.

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>.

Given two vectors in space, $\overrightarrow{a} = (x - 1, 1, -x)$ and $\overrightarrow{b} = (-x, 3, -1)$. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of $x$.

Answer: $-1$


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

How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?

Answer: $100$


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

The number of students in James' graduating class is greater than 100 but fewer than 200 and is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6. How many students are in James' graduating class?

Answer: $183$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 the three smallest prime factors of 180.

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>.

A total of 6 letters are used to spell the English word "theer". Calculate the probability that the person spells this English word incorrectly.

Answer: $\frac{59}{60}$


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 number of elements in a finite set $P$ is denoted as $\text{card}(P)$. It is known that $\text{card}(M) = 10$, $A \subseteq M$, $B \subseteq M$, $A \cap B = \emptyset$, and $\text{card}(A) = 2$, $\text{card}(B) = 3$. If the set $X$ satisfies $A \subseteq X \subseteq M$, then the number of such sets $X$ is ____. (Answer with a number)

Answer: $256$


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

Given that out of 6 products, 2 are defective and the rest are qualified, calculate the probability of selecting exactly one defective product from these 6 products.

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


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

Given the geometric sequence $\{a_{n}\}$, $3a_{5}-a_{3}a_{7}=0$. If $\{b_{n}\}$ is an arithmetic sequence where $b_{5}=a_{5}$, find the sum of the first 9 terms, $S_{9}$, of $\{b_{n}\}$.

Answer: $27$


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>.

A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$.

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