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

A bag contains four balls of identical shape and size, numbered $1$, $2$, $3$, $4$.
(I) A ball is randomly drawn from the bag, its number recorded as $a$, then another ball is randomly drawn from the remaining three, its number recorded as $b$. Find the probability that the quadratic equation $x^{2}+2ax+b^{2}=0$ has real roots.
(II) A ball is randomly drawn from the bag, its number recorded as $m$, then the ball is returned to the bag, and another ball is randomly drawn, its number recorded as $n$. If $(m,n)$ is used as the coordinates of point $P$, find the probability that point $P$ falls within the region $\begin{cases} x-y\geqslant 0 \\ x+y-5 < 0\end{cases}$.

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


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

Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$.

Answer: $-\frac{1}{\sqrt[3]{6}}$


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 vectors $\overrightarrow{m}=(1,3\cos \alpha)$ and $\overrightarrow{n}=(1,4\tan \alpha)$, where $\alpha \in (-\frac{\pi}{2}, \;\;\frac{\pi}{2})$, and their dot product $\overrightarrow{m} \cdot \overrightarrow{n} = 5$.
(I) Find the magnitude of $|\overrightarrow{m}+ \overrightarrow{n}|$;
(II) Let the angle between vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ be $\beta$, find the value of $\tan(\alpha + \beta)$.

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>.

The line \( L \) crosses the \( x \)-axis at \((-8,0)\). The area of the shaded region is 16. Find the slope of the line \( L \).

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>.

Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find the ratio of the speed of $A$ to the speed of $B$.

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


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

Determine, with proof, the smallest positive integer \( n \) with the following property: For every choice of \( n \) integers, there exist at least two whose sum or difference is divisible by 2009.

Answer: $1006$


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

A certain pharmaceutical company has developed a new drug to treat a certain disease, with a cure rate of $p$. The drug is now used to treat $10$ patients, and the number of patients cured is denoted as $X$.
$(1)$ If $X=8$, two patients are randomly selected from these $10$ people for drug interviews. Find the distribution of the number of patients cured, denoted as $Y$, among the selected patients.
$(2)$ Given that $p\in \left(0.75,0.85\right)$, let $A=\{k\left|\right.$ probability $P\left(X=k\right)$ is maximum$\}$, and $A$ contains only two elements. Find $E\left(X\right)$.

Answer: $\frac{90}{11}$


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

If $α∈(0, \dfrac{π}{2})$, $\cos ( \dfrac{π}{4}-α)=2 \sqrt{2}\cos 2α$, then $\sin 2α=$____.

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


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

Marla has a large white cube that has an edge of 8 feet. She also has enough green paint to cover 200 square feet. Marla uses all the paint to create a white circle centered on each face, surrounded by a green border. What is the area of one of the white circles, in square feet?
A) $15.34$ sq ft
B) $30.67$ sq ft
C) $45.23$ sq ft
D) $60.89$ sq ft

Answer: $30.67$


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>.

In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\).

Answer: $2177$


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

Consider all polynomials of the form
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \( b_i \in \{0,1\} \) for all \( 0 \le i \le 6 \). Find the number of such polynomials that have exactly two different integer roots, -1 and 0.

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>.

Let $x$ be Walter's age in 1996. If Walter was one-third as old as his grandmother in 1996, then his grandmother's age in 1996 is $3x$. The sum of their birth years is $3864$, and the sum of their ages in 1996 plus $x$ and $3x$ is $1996+1997$. Express Walter's age at the end of 2001 in terms of $x$.

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>.

An enterprise has four employees participating in a vocational skills assessment. Each employee can draw any one of the four available assessment projects to participate in. Calculate the probability that exactly one project is not selected.

Answer: $\frac{9}{16}$


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

There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?

Answer: $102$


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

A person bought a bond for 1000 yuan with a maturity of one year. After the bond matured, he spent 440 yuan and then used the remaining money to buy the same type of bond again for another year. After the bond matured the second time, he received 624 yuan. Calculate the annual interest rate of this bond.

Answer: $4\%$


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

50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table?

Answer: $32$


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

A batch of fragile goods totaling $10$ items is transported to a certain place by two trucks, A and B. Truck A carries $2$ first-class items and $2$ second-class items, while truck B carries $4$ first-class items and $2$ second-class items. Upon arrival at the destination, it was found that trucks A and B each broke one item. If one item is randomly selected from the remaining $8$ items, the probability of it being a first-class item is ______. (Express the result in simplest form)

Answer: $\frac{29}{48}$


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

Given unit vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ in the same plane, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, find the maximum value of $(\overrightarrow{a} - \overrightarrow{b}) \cdot (\overrightarrow{a} - 2\overrightarrow{c})$.

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


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

The distance from the origin to the line $3x+2y-13=0$ is $\sqrt {13}$.

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>.

In $\triangle ABC$, with $D$ on $AC$ and $F$ on $BC$, given $AB \perp AC$, $AF \perp BC$, and $BD = DF = FC = 1$. If also $D$ is the midpoint of $AC$, find the length of $AC$.
A) 1
B) $\sqrt{2}$
C) $\sqrt{3}$
D) 2
E) $\sqrt[3]{4}$

Answer: $\sqrt{2}$


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 triangle $XYZ$, $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$.

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>.

Given the function $f(x)=2ax- \frac {3}{2}x^{2}-3\ln x$, where $a\in\mathbb{R}$ is a constant,
$(1)$ If $f(x)$ is a decreasing function on $x\in[1,+\infty)$, find the range of the real number $a$;
$(2)$ If $x=3$ is an extremum point of $f(x)$, find the maximum value of $f(x)$ on $x\in[1,a]$.

Answer: $\frac {33}{2}-3\ln 3$


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

The sum of the series $\frac{3}{4} + \frac{5}{8} + \frac{9}{16} + \frac{17}{32} + \frac{33}{64} + \frac{65}{128} - 3.5$.

Answer: $\frac{-1}{128}$


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

The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5.

Answer: $-24$


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

A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads.

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>.

Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room.

Answer: $360$


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

What is the $100^{\mathrm{th}}$ odd positive integer, and what even integer directly follows it?

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>.

Suppose you have an unlimited number of pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins.

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 function f(x) = 2sin(x - π/6)sin(x + π/3), x ∈ R.
(1) Find the smallest positive period of the function f(x) and the center of symmetry of its graph.
(2) In △ABC, if A = π/4, and acute angle C satisfies f(C/2 + π/6) = 1/2, find the value of BC/AB.

Answer: $\sqrt{2}$


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>.

Let $M \Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \Theta(A \bigodot 20) = 7$, find $A$.

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>.

Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms.

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>.

There are two equilateral triangles with a vertex at  $(0, 1)$ , with another vertex on the line  $y = x + 1$  and with the final vertex on the parabola  $y = x^2 + 1$ . Find the area of the larger of the two triangles.

Answer: $26\sqrt{3} + 45$


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>.

In $\triangle ABC$, the sides corresponding to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a=2$, $b=3$, $\cos B= \frac{1}{3}$.
(1) Find the value of side $c$;
(2) Find the value of $\cos (A-C)$.

Answer: $\frac{23}{27}$


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>.

There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.
$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find the number of ways to arrange them;
$(2)$ Take out $4$ balls from these $8$ balls, ensuring that balls of each color are taken. Find the number of ways to do so;
$(3)$ Divide these $8$ balls into three groups, each group having at least $2$ balls. Find the number of ways to divide them.

Answer: $490$


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

In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 18 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $G$ is the midpoint of $CD$. Determine the area of triangle $EFG$.

Answer: $52.5$


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

How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same?

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>.

How many positive 3-digit numbers are multiples of 25, but not of 60?

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>.

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

Answer: $-8$


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

Let $u,$ $v,$ and $w$ be the roots of the equation $x^3 - 18x^2 + 20x - 8 = 0.$ Find the value of $(2+u)(2+v)(2+w).$

Answer: $128$


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

Given the function $$f(x)= \begin{cases} \overset{2^{x},x\geq 3,}{f(x+1),x<3}\end{cases}$$, find the value of $f(\log_2 6)$.

Answer: $12$


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

Let  $m$  be the smallest positive integer such that  $m^2+(m+1)^2+\cdots+(m+10)^2$  is the square of a positive integer  $n$ . Find  $m+n$

Answer: $95$


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

Calculate the sum of $5.46$, $2.793$, and $3.1$ as a decimal.

Answer: $11.353$


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

The probability that the random variable $X$ follows a normal distribution $N\left( 3,{{\sigma }^{2}} \right)$ and $P\left( X\leqslant 4 \right)=0.84$ can be expressed in terms of the standard normal distribution $Z$ as $P(Z\leqslant z)=0.84$, where $z$ is the z-score corresponding to the upper tail probability $1-0.84=0.16$. Calculate the value of $P(2<X<4)$.

Answer: $0.68$


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

Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $\frac{\sqrt{3}}{2}$, find the slope of the line that intersects the ellipse at points $A$ and $B$, where the midpoint of segment $AB$ is $M(-2, 1)$.

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>.

Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?

Answer: $32$


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

Given that $ab= \frac{1}{4}$, $a$, $b \in (0,1)$, find the minimum value of $\frac{1}{1-a}+ \frac{2}{1-b}$.

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


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

There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$.

Answer: $9999$


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

Given real numbers \( a, b, c \) satisfy
\[
a^{2}+b^{2}-4a \leqslant 1, \quad b^{2}+c^{2}-8b \leqslant -3, \quad c^{2}+a^{2}-12c \leqslant -26,
\]
what is the value of \( (a+b)^{c} \)?

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>.

How many distinct four-digit positive integers are there such that the product of their digits equals 18?

Answer: $48$


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

The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth.

Answer: $3.09$


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

Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined.

Answer: $20$


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

Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $-\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\in N^*$, the sum of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______.

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


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

An influenza outbreak occurred in three areas, $A$, $B$, and $C$, where $6\%$, $5\%$, and $4\%$ of the population in each area have the flu, respectively. Assuming the population ratios in these three areas are $6:5:4$, if a person is randomly selected from these three areas, the probability that this person has the flu is ______.

Answer: $\frac{77}{1500}$


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

If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$, find the smallest possible value of $k$ greater than 100.

Answer: $111$


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

(1) Compute: $\sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ$
(2) Given that $\alpha$ is an angle in the second quadrant, and $\sin \alpha = \frac{\sqrt{15}}{4}$, find the value of $\frac{\sin(\alpha + \frac{\pi}{4})}{\sin 2\alpha + \cos 2\alpha + 1}$.

Answer: $-\sqrt{2}$


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>.

The largest prime factor of  $999999999999$  is greater than  $2006$ . Determine the remainder obtained when this prime factor is divided by  $2006$ .

Answer: $1877$


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

The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$.

Answer: $625$


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

Simplify $$\frac{13!}{11! + 3 \cdot 9!}$$

Answer: $\frac{17160}{113}$


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 the real roots of the equation $4x^4-3x^2+7x-3=0$?

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>.

Five people, named A, B, C, D, and E, stand in a row. If A and B must be adjacent, and B must be to the left of A, what is the total number of different arrangements?

Answer: $24$


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

Evaluate the ratio $\frac{10^{3000} + 10^{3004}}{10^{3001} + 10^{3003}}$ and determine which whole number it is closest to.

Answer: $10$


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

A hall is organizing seats in rows for a lecture. Each complete row must contain $13$ chairs. Initially, the hall has $169$ chairs arranged. To maintain fully occupied rows with minimal empty seats, if $100$ students are expected to attend, how many chairs should be removed or added?

Answer: $65$


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

Given that Nayla has an index card measuring $5 \times 7$ inches, and she shortens the length of one side by $2$ inches, resulting in a card with an area of $21$ square inches, determine the area of the card if instead, she shortens the length of the other side by the same amount.

Answer: $25$


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

In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin(C+\frac{π}{3})-c\sin B=0$.
$(1)$ Find the value of angle $C$.
$(2)$ If the area of $\triangle ABC$ is $10\sqrt{3}$ and $D$ is the midpoint of $AC$, find the minimum value of $BD$.

Answer: $2\sqrt{5}$


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

The robot vacuum cleaner is programmed to move on the floor according to the law:

$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$

where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute.

Answer: $\sqrt{445}$


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

If \( e^{i \theta} = \frac{3 + i \sqrt{2}}{4}, \) then find \( \cos 3\theta. \)

Answer: $\frac{9}{64}$


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 lines $l_{1}$: $\rho\sin(\theta-\frac{\pi}{3})=\sqrt{3}$ and $l_{2}$: $\begin{cases} x=-t \\ y=\sqrt{3}t \end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B$, and $C$ lie on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, with $O$ being the coordinate origin. If $\angle{AOB}=\angle{BOC}=\angle{COA}=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+\frac{1}{|OB|^{2}}+\frac{1}{|OC|^{2}}$.

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


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

Find the last three digits of $7^{103}.$

Answer: $343$


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

Two people are playing a game. One person thinks of a ten-digit number, and the other can ask questions about which digits are in specific sets of positions in the number's sequence. The first person answers the questions without indicating which digits are in which exact positions. What is the minimum number of questions needed to reliably guess the number?

Answer: $10$


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

Given the function $f(x)=2\sin ωx\cos ωx-2\sqrt{3} \cos ^{2}ωx+\sqrt{3} (ω > 0)$, and the distance between two adjacent symmetry axes of the graph of $y=f(x)$ is $\frac{π}{2}$.

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

(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$. Find the area of $\triangle ABC$.

Answer: $3\sqrt{3}$


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>.

Jamie King invested some money in real estate and mutual funds. The total amount he invested was $\$200,\!000$. If he invested 5.5 times as much in real estate as he did in mutual funds, what was his total investment in real estate?

Answer: $169,230.77$


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

A number in the set $\{50, 51, 52, 53, ... , 500\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction.

Answer: $\frac{17}{451}$


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

Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\cos B=b\cos C$, and the dot product $\vec{AB}\cdot \vec{BC} = -3$,
1. Find the area of $\triangle ABC$;
2. Find the minimum value of side $AC$.

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>.

Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket?

Answer: $15$


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

Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$.

Answer: $506$


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

Given $f(\alpha)= \dfrac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\cos (-\alpha+ \dfrac {3}{2}\pi)}{\cos ( \dfrac {\pi}{2}-\alpha)\sin (-\pi-\alpha)}$
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \dfrac {3}{2}\pi)= \dfrac {1}{5}$, find the value of $f(\alpha)$;
$(3)$ If $\alpha=- \dfrac {31}{3}\pi$, find the value of $f(\alpha)$.

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


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

Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$.

Answer: $\frac{6\sqrt{6}}{11}$


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>.

Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$.

Answer: $-\sqrt{2}$


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>.

The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
Determine the value of \(a + c + f\).

Answer: $73$


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

What is the area of the smallest square that can contain a circle of radius 5?

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>.

Simplify the expression: $(a-\frac{3a}{a+1}$) $÷\frac{{a}^{2}-4a+4}{a+1}$, then choose a number you like from $-2$, $-1$, and $2$ to substitute for $a$ and calculate the value.

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


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

A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in simplest radical form.

Answer: $3\sqrt{2}$


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

If the line $2x+my=2m-4$ is parallel to the line $mx+2y=m-2$, find the value of $m$.

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>.

The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$.

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


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

A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display.

Answer: $24$


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

Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 4, 5, 7\}$.

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>.

What is the value of $a^3 - b^3$ given that $a+b=12$ and $ab=20$?

Answer: $992$


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

Given \\(\cos \left(\alpha+ \frac{\pi}{6}\right)-\sin \alpha= \frac{2 \sqrt{3}}{3} \\), determine the value of \\(\sin \left(\alpha- \frac{7\pi}{6}\right) \\).

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


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

Given the function $f(x)=4\cos (ωx- \frac {π}{6})\sin (π-ωx)-\sin (2ωx- \frac {π}{2})$, where $ω > 0$.
(1) Find the range of the function $f(x)$.
(2) If $y=f(x)$ is an increasing function in the interval $[- \frac {3π}{2}, \frac {π}{2}]$, find the maximum value of $ω$.

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 two lines $$l_{1}: \sqrt {3}x+y-1=0$$ and $$l_{2}: ax+y=1$$, and $l_{1}$ is perpendicular to $l_{2}$, then the slope angle of $l_{1}$ is \_\_\_\_\_\_, and the distance from the origin to $l_{2}$ is \_\_\_\_\_\_.

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


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

How many integer quadruples  $a,b,c,d$  are there such that  $7$  divides  $ab-cd$  where  $0\leq a,b,c,d < 7$ ?

Answer: $385$


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

Let $P = (\sqrt{2007}+\sqrt{2008}),$ $Q = (-\sqrt{2007}-\sqrt{2008}),$ $R = (\sqrt{2007}-\sqrt{2008}),$ and $S = (-\sqrt{2008}+\sqrt{2007}).$ Calculate $PQRS.$

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>.

Let  $ABC$  be a triangle with  $AB = 5$ ,  $AC = 8$ , and  $BC = 7$ . Let  $D$  be on side  $AC$  such that  $AD = 5$  and  $CD = 3$ . Let  $I$  be the incenter of triangle  $ABC$  and  $E$  be the intersection of the perpendicular bisectors of  $\overline{ID}$  and  $\overline{BC}$ .  Suppose  $DE = \frac{a\sqrt{b}}{c}$  where  $a$  and  $c$  are relatively prime positive integers, and  $b$  is a positive integer not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Ray Li*

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>.

An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$?

Answer: $\frac{13}{70}$


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

Suppose  $ABCD$  is a rectangle whose diagonals meet at  $E$ . The perimeter of triangle  $ABE$  is  $10\pi$  and the perimeter of triangle  $ADE$  is  $n$ . Compute the number of possible integer values of  $n$ .

Answer: $47$


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

The graph of the function $f(x)=\sin({ωx-\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\left(0,\frac{π}{ω}\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum value of $F(x)$.

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>.

Let \( p \) and \( q \) be positive integers such that
\[
\frac{6}{11} < \frac{p}{q} < \frac{5}{9}
\]
and \( q \) is as small as possible. What is \( p+q \)?

Answer: $31$


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

The line $y=2b$ intersects the left and right branches of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ at points $B$ and $C$ respectively, with $A$ being the right vertex and $O$ the origin. If $\angle AOC = \angle BOC$, then calculate the eccentricity of the hyperbola.

Answer: $\frac{\sqrt{19}}{2}$


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 function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two axes of symmetry of the graph of the function $y=f(x)$, evaluate the value of $f(-\frac{{5π}}{{12}})$.

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


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

A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
- **A)** $\sqrt{84}$
- **B)** $\sqrt{85}$
- **C)** $\sqrt{89}$
- **D)** $\sqrt{91}$
- **E)** $\sqrt{95}$

Answer: $\sqrt{91}$


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

The coefficient of the $x^2$ term in the expansion of $(1+x)(1+ \sqrt{x})^5$ is \_\_\_\_\_\_.

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>.

Let \( f \) be a function taking positive integers to positive integers, such that:

(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) = m \)

Find all possible values of \( f(60) \).

Answer: $3600$


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

A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the "Creating a Civilized City" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an additional 20 helmets are sold per month. It is known that the cost price of the helmets is $50 each.
$(1)$ If the price of each helmet is reduced by $10, the store can sell ______ helmets per month, and the monthly profit from sales is ______ dollars.
$(2)$ If the store plans to reduce the price of these helmets to reduce inventory while ensuring a monthly profit of $7500, find the selling price of the helmets.

Answer: $65$


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

The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$?

Answer: $41$


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

The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers.

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>.

The largest and smallest possible three-digit numbers that can be formed using the digits 5, 1, and 9 are found by maximizing and minimizing the order of the digits, respectively. Calculate the difference between these two numbers.

Answer: $792$


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

There are 5 people standing in a row, where A and B must stand next to each other, and C and D cannot stand next to each other. How many different arrangements are there?

Answer: $24$


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

Given that the central angle of a sector is $\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \_\_\_\_\_\_ cm, and the area of the sector is \_\_\_\_\_\_ cm2.

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>.

As shown in the diagram, rectangle \(ABCD\) is inscribed in a semicircle, with \(EF\) as the diameter of the semicircle. Given that \(DA = 16\), and \(FD = AE = 9\), find the area of rectangle \(ABCD\).

Answer: $240$


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

A wholesaler gives different discounts based on the size of the customer's order. The steps to calculate the amount payable by the customer are as follows:
S1 Input the order amount x (unit: pieces); input the unit price A (unit: yuan);
S2 If x < 250, then the discount rate d=0;
If 250 ≤ x < 500, then the discount rate d=0.05;
If 500 ≤ x < 1000, then the discount rate d=0.10;
If x ≥ 1000, then the discount rate d=0.15;
S3 Calculate the payable amount T=Ax(1-d) (unit: yuan);
S4 Output the payable amount T.
It is known that a customer pays 38000 yuan for 400 pieces, then the order amount is     pieces when the payable amount is 88200 yuan.

Answer: $980$


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

A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.

(1) If the shop buys 16 roses in one day, find the profit function \( y \) (in yuan) with respect to the demand \( n \) for that day (in roses, \( n \in \mathbf{N} \)).

(2) The shop recorded the daily demand for roses (in roses) for 100 days and summarized the data in Table 1.

Using the frequencies of the demands recorded over the 100 days as probabilities for each demand:

(i) If the shop buys 16 roses in one day, let \( X \) represent the profit (in yuan) for that day. Find the distribution, expected value, and variance of \( X \).

(ii) If the shop plans to buy either 16 or 17 roses in one day, which would you recommend they buy? Please explain your reasoning.

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 that $\dfrac{\pi}{2} < \alpha < \beta < \dfrac{3\pi}{4}, \cos(\alpha - \beta) = \dfrac{12}{13}, \sin(\alpha + \beta) = -\dfrac{3}{5}$, find the value of $\sin 2\alpha$.

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


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

Eight people are sitting around a circular table for a meeting, and the recorder is sitting between the leader and the deputy leader. Calculate the total number of different seating arrangements possible, considering arrangements that can be made identical through rotation as the same.

Answer: $240$


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

A digital watch displays hours and minutes in a 24-hour format. Find the largest possible sum of the digits in the display.

Answer: $24$


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

Positive real numbers  $a$ ,  $b$ ,  $c$  satisfy  $a+b+c=1$ . Find the smallest possible value of   $$ E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}. $$

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


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

Given an angle measuring $54^{\circ}$, use only a compass to divide it into three equal parts (that is, find such points that rays passing through the vertex of the given angle and these points divide the angle into three equal parts).

Answer: $18$


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

The school's boys basketball team has 16 players, including a set of twins, Bob and Bill, and a set of triplets, Chris, Craig, and Carl. In how many ways can we choose 7 starters if all three triplets must be in the starting lineup and both twins must either both be in the lineup or both not be in the lineup?

Answer: $385$


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

As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} \) at point \( B \), and circle \( \odot O_{4} \) is internally tangent to \( \odot O_{2} \) at point \( C \). The common external tangent of \( \odot O_{2} \) and \( \odot O_{3} \) passes through point \( A \), tangent to \( \odot O_{3} \) at point \( E \) and tangent to \( \odot O_{2} \) at point \( F \). The common external tangent of \( \odot O_{1} \) and \( \odot O_{4} \) passes through point \( D \). If the radius of circle \( \odot O_{3} \) is 1.2, what is the radius of circle \( \odot O_{4} \)?

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>.

Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\rho\cos^2\theta-4\sin\theta=0$, and the polar coordinates of point $P$ are $(3, \frac{\pi}{2})$, in the Cartesian coordinate system, line $l$ passes through point $P$ with a slope of $\sqrt{3}$.

(Ⅰ) Write the Cartesian coordinate equation of curve $C$ and the parametric equation of line $l$;

(Ⅱ) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{1}{|PA|}+ \frac{1}{|PB|}$.

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


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

Compute the sum of $302^2 - 298^2$ and $152^2 - 148^2$.

Answer: $3600$


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

Find the value of $x$ in the following expressions:
(1) $8x^3 = 27$; (2) $(x-2)^2 = 3$.

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


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

A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\frac {1}{96-x} & (1\leq x\leq 94, x\in \mathbb{N}) \\
\frac {2}{3} & (x>94, x\in \mathbb{N})
\end{cases}
$$
It is known that for every qualified instrument produced, a profit of $A$ yuan can be made, but for every defective product produced, a loss of $\frac {A}{2}$ yuan will be incurred. The factory wishes to determine an appropriate daily output.
(1) Determine whether producing this instrument can be profitable when the daily output (pieces) exceeds 94 pieces, and explain the reason;
(2) When the daily output $x$ pieces does not exceed 94 pieces, try to express the daily profit $T$ (yuan) of producing this instrument as a function of the daily output $x$ (pieces);
(3) To obtain the maximum profit, how many pieces should the daily output $x$ be?

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>.

Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region.

Answer: $\frac{225\pi - 450\sqrt{2}}{4}$


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

Let $\triangle XYZ$ be a right triangle with $\angle Y$ as the right angle. A circle with a diameter of $YZ$ intersects side $XZ$ at point $W$. Given that $XW = 2$ and $YW = 3$, find the length of $ZW$.

Answer: $4.5$


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

If the inequality system about $x$ is $\left\{\begin{array}{l}{\frac{x+3}{2}≥x-1}\\{3x+6＞a+4}\end{array}\right.$ has exactly $3$ odd solutions, and the solution to the equation about $y$ is $3y+6a=22-y$ is a non-negative integer, then the product of all integers $a$ that satisfy the conditions is ____.

Answer: $-3$