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

Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers)
(Ⅰ) The two girls stand next to each other;
(Ⅱ) Girls cannot stand at the ends;
(Ⅲ) Girls are arranged from left to right from tallest to shortest;
(Ⅳ) Girl A cannot stand at the left end, and Girl B cannot stand at the right end.

Answer: $78$


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

Given triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, $c\cos A= \frac{4}{b}$, and the area of $\triangle ABC$, $S \geq 2$.
(1) Determine the range of possible values for angle $A$.
(2) Find the maximum value of the function $f(x) = \cos^2 A + \sqrt{3}\sin^2\left(\frac{\pi}{2}+ \frac{A}{2}\right) - \frac{\sqrt{3}}{2}$.

Answer: $\frac{1}{2} + \frac{\sqrt{6}}{4}$


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>.

Express $537_8 + 1C2E_{16}$ as a base 10 integer, where $C$ and $E$ denote the hexadecimal digits with values 12 and 14 respectively.

Answer: $7565$


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

A line $l$ is drawn through the fixed point $M(4,0)$, intersecting the parabola $y^{2}=4x$ at points $A$ and $B$. $F$ is the focus of the parabola. Calculate the minimum area of the triangle $\triangle AFB$.

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>.

A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59.

Answer: $24$


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

In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?

[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P, S);
draw(X--P--Y--Z--cycle);
[/asy]

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>.

Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.
$(1)$ Find the probability of school A winning the championship;
$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$.

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>.

Using the Horner's method, calculate the value of the polynomial $f(x)=2x^{4}-x^{3}+3x^{2}+7$ at $x=3$, and find the corresponding value of $v_{3}$.

Answer: $54$


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

Quadrilateral  $ALEX,$  pictured below (but not necessarily to scale!)
can be inscribed in a circle; with  $\angle LAX = 20^{\circ}$  and  $\angle AXE = 100^{\circ}:$

Answer: $80$


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

Given the set $A=\{x|0<x+a\leq5\}$, and the set $B=\{x|-\frac{1}{2}\leq x<6\}$
(Ⅰ) If $A\subseteq B$, find the range of the real number $a$;
(Ⅱ) If $A\cap B$ is a singleton set, find the value of the real number $a$.

Answer: $\frac {11}{2}$


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 set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a+b$.

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 $S = \{r_1, r_2, \ldots, r_n\} \subseteq \{1, 2, 3, \ldots, 50\}$, and the sum of any two numbers in $S$ cannot be divisible by 7. The maximum value of $n$ is ____.

Answer: $23$


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

To obtain the graph of the function $$y=2\sin(x+ \frac {\pi}{6})\cos(x+ \frac {\pi}{6})$$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$.

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


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>.

March 12th is Tree Planting Day. A school organizes 65 high school students and their parents to participate in the "Plant a tree, green a land" tree planting activity as a family unit. The activity divides the 65 families into two groups, A and B. Group A is responsible for planting 150 silver poplar seedlings, while group B is responsible for planting 160 purple plum seedlings. According to previous statistics, it takes $\frac{2}{5}h$ for a family to plant a silver poplar seedling and $\frac{3}{5}h$ to plant a purple plum seedling. Assuming that groups A and B start planting at the same time, in order to minimize the duration of the tree planting activity, the number of families in group A is \_\_\_\_\_\_, and the duration of the activity is \_\_\_\_\_\_ $(h)$.

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


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

Given that the vertices of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ lie on the surface of a sphere $O$, the lateral area of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ is $6$, and the base area is $\sqrt{3}$, calculate the surface area of the sphere $O$.

Answer: $\frac{19\pi}{3}$


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

Four planes divide space into $n$ parts at most. Calculate $n$.

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>.

Fold a piece of graph paper once so that the point (0, 2) coincides with the point (4, 0), and the point (9, 5) coincides with the point (m, n). The value of m+n is ______.

Answer: $10$


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

Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+m^2+n^2,$ for all positive integers $m$ and $n.$ Find $b_{12}.$

Answer: $160$


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

Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum value of $x\_3 x\_{18}$.

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>.

There are 3 different balls to be placed into 5 different boxes, with at most one ball per box. How many methods are there?

Answer: $60$


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

Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?

Answer: $1400$


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

Given a function $f(x)$ satisfies $f(x) + f(4-x) = 4$, $f(x+2) - f(-x) = 0$, and $f(1) = a$, calculate the value of $f(1) + f(2) + f(3) + \cdots + f(51)$.

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>.

Find all positive integers  $x$  such that the product of all digits of  $x$  is given by  $x^2 - 10 \cdot x - 22.$

Answer: $12$


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

Given the sums of the first n terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ denoted as $S_n$ and $T_n$, respectively, if $\frac {S_{n}}{T_{n}} = \frac {2n}{3n+1}$, calculate the value of $\frac {a_{6}}{b_{6}}$.

Answer: $\frac {11}{17}$


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

A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:

- It has  $3$  digits.
- It is a palindrome.
- The middle digit is a multiple of  $4$ .
- It is divisible by  $11$ .

What is the sum of all possible numbers of cows that the herder has?

Answer: $726$


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

In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 12$ and $CP = 4.$ If $\tan \angle APD = 2,$ then find $AB.$

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>.

Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.

Answer: $\frac{335}{2011}$


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

If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number).

Answer: $243$


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

A  $40$  feet high screen is put on a vertical wall  $10$  feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Answer: $10\sqrt{5}$


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

Given \( a, b > 0 \). Find the maximum values of \( \sqrt{\frac{a}{2a+b}} + \sqrt{\frac{b}{2b+a}} \) and \( \sqrt{\frac{a}{a+2b}} + \sqrt{\frac{b}{b+2a}} \).

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


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

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

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>.

Kim earned scores of 86, 82, and 89 on her first three mathematics examinations. She is expected to increase her average score by at least 2 points with her fourth exam. What is the minimum score Kim must achieve on her fourth exam to meet this target?

Answer: $94$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 greatest four-digit number whose digits have a product of $24$. Calculate the sum of the digits of $M$.

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>.

$n$ mushroom pickers went into the forest and brought back a total of 200 mushrooms (possibly, some of the pickers did not bring any mushrooms home). A boy named Petya, upon learning this, stated: "Some two of them must have brought the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly right? Don't forget to justify your answer.

Answer: $21$


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

If two points are randomly selected from the eight vertices of a cube, the probability that the line determined by these two points intersects each face of the cube is ______.

Answer: $\frac{1}{7}$


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

Calculate $\frac{1}{6} \cdot \frac{2}{7} \cdot \frac{3}{8} \cdot \frac{4}{9} \cdots \frac{94}{99} \cdot \frac{95}{100}$. Express your answer as a common fraction.

Answer: $\frac{1}{75287520}$


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

For each value of $x$, $g(x)$ is defined to be the minimum value of the three numbers $3x + 3$, $\frac{1}{3}x + 1$, and $-\frac{2}{3}x + 8$. Find the maximum value of $g(x)$.

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


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>.

Triangle $PQR$ has vertices $P(0, 6)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ cuts the area of $\triangle PQR$ in half. Find the sum of the slope and $y$-intercept of this line.

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>.

Given the infinite series $1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$, find the limiting sum of the series.

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


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

There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer  $ n$  in between 1 and 25. Betül draws  $ n$  balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the  $ n$  balls are blue and the second blue ball is drawn at  $ n^{th}$  order, Ahmet wins, otherwise Betül wins. To increase  the possibility to win,  Ahmet must choose

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 vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|= \sqrt {3}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}- \overrightarrow {b}) \perp \overrightarrow {a}$, find the projection of $\overrightarrow {a}$ on $\overrightarrow {b}$.

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


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

Given vectors $\overrightarrow{a} = (4\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\sin \beta, 4\cos \beta)$, and $\overrightarrow{c} = (\cos \beta, -4\sin \beta)$, where $\alpha, \beta \in \mathbb{R}$ and neither $\alpha$, $\beta$, nor $\alpha + \beta$ equals $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$:

1. Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$.
2. When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$ and perpendicular to $(\overrightarrow{b} - 2\overrightarrow{c})$, find the value of $\tan \alpha + \tan \beta$.

Answer: $-30$


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

In the plane Cartesian coordinate system $xOy$, let two unit vectors in the directions of the $x$-axis and $y$-axis be $\overrightarrow{i}$ and $\overrightarrow{j}$, respectively. Given that $\overrightarrow{OA}=2\overrightarrow{i}+\overrightarrow{j}$ and $\overrightarrow{OB}=4\overrightarrow{i}-3\overrightarrow{j}$.
$(1)$ If point $P$ lies on the extension of line segment $AB$ and $|\overrightarrow{AP}|=\frac{3}{2}|\overrightarrow{PB}|$, find the coordinates of point $P$.
$(2)$ If point $P$ is the midpoint of line segment $AB$ and $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OA}+k\overrightarrow{OB}$, find the value of the real number $k$.

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


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

There are 7 people standing in a row. How many different arrangements are there according to the following requirements?
(1) Among them, A, B, and C cannot stand next to each other;
(2) Among them, A and B have exactly one person between them;
(3) A does not stand at the head of the row, and B does not stand at the end of the row.

Answer: $3720$


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

Given the sequence $\left\{ a_n \right\}$ such that $a_{n+1}+a_n={(-1)}^n\cdot n$ ($n\in \mathbb{N}^*$), find the sum of the first 20 terms of $\left\{ a_n \right\}$.

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>.

What is the product of the numerator and the denominator when $0.\overline{018}$ is expressed as a fraction in lowest terms?

Answer: $222$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 numbers are divisible by 5 and have 45 as their last two digits?

Answer: $90$


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

The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \_\_\_\_\_\_.

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


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

The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?

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


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

In $\triangle ABC$, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $C= \frac {\pi}{3}, b= \sqrt {2}, c= \sqrt {3}$, find the measure of angle A.

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


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

In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{8}.$  Find $\tan F.$

Answer: $12 + \sqrt{136}$


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

The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 630 degrees about point $C$ in a counterclockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?

Answer: $30$


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

Given that f(x) = |log₃x|, if a and b satisfy f(a - 1) = f(2b - 1), and a ≠ 2b, then the minimum value of a + b is ___.

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


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

Evaluate the value of $\frac{(2210-2137)^2 + (2137-2028)^2}{64}$.

Answer: $268.90625$


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

Choose one of the three conditions in $①$ $ac=\sqrt{3}$, $②$ $c\sin A=3$, $③$ $c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.
Question: Does there exist a $\triangle ABC$ where the internal angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$, and $\sin A=\sqrt{3}\sin B$, $C=\frac{π}{6}$, _______ $?$
Note: If multiple conditions are selected to answer separately, the first answer will be scored.

Answer: $2\sqrt{3}$


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

Find all solutions to
\[\sqrt{x + 2 - 2 \sqrt{x - 4}} + \sqrt{x + 12 - 8 \sqrt{x - 4}} = 4.\]

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 there is an extra $32.13 on the books due to a misplaced decimal point, determine the original amount of the sum of money.

Answer: $3.57$


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

Given right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, calculate the area of $\triangle ABD$.

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


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>.

Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y).

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>.

In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\frac{c}{b\sin B}$$.

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


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

Katherine makes Benj play a game called  $50$  Cent. Benj starts with  $\$ 0.50 $, and every century thereafter has a $ 50\% $ chance of doubling his money and a $ 50\% $ chance of having his money reset to $ \ $0.50$ . What is the expected value of the amount of money Benj will have, in dollars, after  $50$  centuries?

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 the function $f\left( x \right)=\sin \left( 2x+\phi \right)\left(\left| \phi \right| < \dfrac{\pi }{2} \right)$ whose graph is symmetric about the point $\left( \dfrac{\pi }{3},0 \right)$, and $f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)=0$ when ${{x}_{1}},{{x}_{2}}\in \left( \dfrac{\pi }{12},\dfrac{7\pi }{12} \right)$ $\left( {{x}_{1}}\ne {{x}_{2}} \right)$, find $f\left( {{x}_{1}}+{{x}_{2}} \right)$.

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


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>.

In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\overrightarrow{CD}|=1$, then the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OD}|$ is ______.

Answer: $\sqrt{7}+1$


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

How many positive four-digit integers of the form $\_\_90$ are divisible by 90?

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>.

In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\dfrac{\pi }{3}$, and $\sin B=2\sin A$, find the area of $\triangle ABC$.

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


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

How many times does the digit 9 appear in the list of all integers from 1 to 1000?

Answer: $300$


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

Consider the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin 2x + 2m$, where $x \in \mathbb{R}$ and $m \in \mathbb{R}$.
(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.
(II) If $f(x)$ has a minimum value of 0 when $0 \leq x \leq \frac{\pi}{4}$, find the value of the real number $m$.

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 $(x+1)^4(x+4)^8 = a + a_1(x+3) + a_2(x+3)^2 + \ldots + a_{12}(x+3)^{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$.

Answer: $112$


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

Given vectors $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sin x, \sin x)$, and $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$

(1) If $x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the range of the function $f(x)$.

(2) Let the sides opposite the acute angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. If $f(B) = 1$, $b = \sqrt{2}$, and $c = \sqrt{3}$, find the value of $a$.

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


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>.

How many even divisors does $9!$ have?

Answer: $140$


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

The function $f(x)=\frac{1}{3}x^{3}-ax^{2}+4$, and $x=2$ is a local minimum point of the function $f(x)$.

(1) Find the value of the real number $a$;

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

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


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>.

Compute $156 + 44 + 26 + 74$.

Answer: $300$


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

If $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, calculate the value of $\tan \alpha$.

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


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 in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_.

Answer: $\frac{20\sqrt{5}\pi}{3}$


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

If the complex number $a^2 - 1 + (a - 1)i$ (where $i$ is the imaginary unit) is purely imaginary, calculate the real number $a$.

Answer: $-1$


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

Given the function $$f(x)= \begin{cases} ( \frac {1}{2})^{x} & ,x≥4 \\ f(x+1) & ,x<4\end{cases}$$, find the value of $f(\log_{2}3)$.

Answer: $\frac {1}{24}$


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

The Experimental Primary School football team has 42 members, and 17 new members joined. How many members are in the football team now?

Answer: $59$


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

Let  $a_n\ (n\geq 1)$  be the value for which  $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$  is maximal. Find  $\lim_{n\to\infty} \ln a_n.$

Answer: $-\ln 2$


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 graph of the function $y=\log_a(x+m)+n$ passes through the fixed point $(-1, -2)$, then find the value of $m \cdot n$.

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>.

Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - n \]for all \( n > 1 \). Find the smallest possible value of \( a_1 \).

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


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

Twelve million added to twelve thousand equals

Answer: $12012000$


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

Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy]

Answer: $150$


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

$44 \times 22$ is equal to

Answer: $88 \times 11$


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>.

On a $123 \times 123$ board, each cell is painted either purple or blue according to the following conditions:

- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.

Note: Two cells are neighbors if they share a side or a vertex.

(a) Consider a $3 \times 3$ sub-board within the $123 \times 123$ board. How many cells of each color can there be in this $3 \times 3$ sub-board?

(b) Calculate the number of purple cells on the $123 \times 123$ board.

Answer: $6724$


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

The coefficient sum of the expansion of the binomial ${{\left(\frac{1}{x}-2x^2\right)}^9}$, excluding the constant term, is $671$.

Answer: $671$


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

If three angles \(x, y, z\) form an arithmetic sequence with a common difference of \(\frac{\pi}{2}\), then \(\tan x \tan y + \tan y \tan z + \tan z \tan x = \) ______.

Answer: $-3$


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

Given point $A(3,0)$, $\overrightarrow{EA}=(2,1)$, $\overrightarrow{EF}=(1,2)$, and point $P(2,0)$ satisfies $\overrightarrow{EP}=λ \overrightarrow{EA}+μ \overrightarrow{EF}$, find the value of $λ+μ$.

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>.

Five packages are delivered to five different houses, with each house receiving one package. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? Express your answer as a common fraction.

Answer: $\frac{1}{12}$


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

Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the sum of the first \\(n\) terms denoted as \\(S_n\) and \\(T_n\) respectively. If \\( \dfrac {S_n}{T_n}= \dfrac {2n}{3n+1}\), then \\( \dfrac {a_2}{b_3+b_7}+ \dfrac {a_8}{b_4+b_6}=\) ______.

Answer: $\dfrac {9}{14}$


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>.

There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?

Answer: $98$


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

Given $$(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{7}(x-1)^{7}$$, find the value of $a_{3}$.

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>.

For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$

Answer: $63$


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

If $\tan (\alpha+\beta)= \frac {3}{4}$ and $\tan (\alpha- \frac {\pi}{4})= \frac {1}{2}$, find the value of $\tan (\beta+ \frac {\pi}{4})$.

Answer: $\frac {2}{11}$


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 real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.

Answer: $1+3\sqrt{2}$


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>.

Let  $n$  be an integer, and let  $\triangle ABC$  be a right-angles triangle with right angle at  $C$ . It is given that  $\sin A$  and  $\sin B$  are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of  $n$

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>.

In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that $(a+c)^2 = b^2 + 2\sqrt{3}ac\sin C$.
1. Find the measure of angle B.
2. If $b=8$, $a>c$, and the area of triangle ABC is $3\sqrt{3}$, find the value of $a$.

Answer: $5 + \sqrt{13}$


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>.

Find the volume of the region in space defined by
\[|x + 2y + z| + |x + 2y - z| \le 12\]
and \(x, y, z \ge 0.\)

Answer: $54$


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

Calculate $7 \cdot 9\frac{2}{5}$.

Answer: $65\frac{4}{5}$


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>.

Given a quadratic function $f(x) = ax^2 - 4bx + 1$.

(1) Let set $P = \{-1,1,2,3,4,5\}$ and set $Q = \{-2,-1,1,2,3,4\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.

(2) Suppose the point $(a, b)$ is a random point within the region defined by $\begin{cases} x+y-8\leqslant 0, \\ x > 0, \\ y > 0 \end{cases}$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.

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


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

What is the sum of all integer solutions to $4<(x-3)^2<64$?

Answer: $30$


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

Xiao Wang's scores in three rounds of jump rope were 23, 34, and 29, respectively. Xiao Wang's final score is.

Answer: $86$


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

Given the function $f(x)=2\sin ωx (ω > 0)$, find the minimum value of $ω$ such that the minimum value in the interval $[- \frac {π}{3}, \frac {π}{4}]$ is $(-2)$.

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


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy the vectors $\overrightarrow{m}=(\cos A,\cos B)$, $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(I) Find the measure of angle $A$;
(II) If $a=2 \sqrt {5}$, find the maximum area of $\triangle ABC$.

Answer: $5 \sqrt {3}$


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>.

From a deck of cards, 5 spades, 4 clubs, and 6 hearts, totaling 15 cards, are drawn. If drawing $m$ cards such that all three suits are present is a certain event, then the minimum value of $m$ is \_\_\_\_\_\_\_\_\_.

Answer: $12$


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

Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?

The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with numbers 16 and 5, is considered the same way.

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>.

Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book?

Answer: $298$


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

Given that $θ$ is a real number, if the complex number $z=\sin 2θ-1+i( \sqrt {2}\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______.

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>.

How many three-digit numbers are there in which the hundreds digit is greater than the tens digit which is in turn greater than the ones digit?

Answer: $120$


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

In a quadrilateral $WXYZ$, the angles satisfy $\angle W = 3\angle X = 4\angle Y = 6\angle Z$. Determine the exact degree measure of $\angle W$.

Answer: $\frac{1440}{7}$


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

Two hundred people were surveyed. Of these, 150 indicated they liked Beethoven, and 120 indicated they liked Chopin. Additionally, it is known that of those who liked both Beethoven and Chopin, 80 people also indicated they liked Vivaldi. What is the minimum number of people surveyed who could have said they liked both Beethoven and Chopin?

Answer: $80$


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

A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green?

Answer: $45\%$


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

Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\parallel l_{2}$, then $m=$_______.

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>.

Calculate $6^6 \cdot 3^6$.

Answer: $34012224$


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

Let \(a\), \(b\), \(c\) be distinct complex numbers such that
\[
\frac{a+1}{2 - b} = \frac{b+1}{2 - c} = \frac{c+1}{2 - a} = k.
\]
Find the sum of all possible values of \(k\).

Answer: $1.5$


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

Given real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-3)^{2}+y^{2}=9\\), find the minimum value of \\(-2y-3x\\) \_\_\_\_\_\_.

Answer: $-3\sqrt{13}-9$


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

Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$

Answer: $27$


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

Given an arithmetic sequence $\{a_n\}$, it is known that $\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$.

Answer: $19$


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

How many pages does the book "Folk Tales" have if from the first page to the last page, a total of 357 digits were used to number the pages?

Answer: $155$


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

Given $$\overrightarrow {m} = (\sin \omega x + \cos \omega x, \sqrt {3} \cos \omega x)$$, $$\overrightarrow {n} = (\cos \omega x - \sin \omega x, 2\sin \omega x)$$ ($\omega > 0$), and the function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, if the distance between two adjacent axes of symmetry of $f(x)$ is not less than $\frac {\pi}{2}$.
(1) Find the range of values for $\omega$;
(2) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=2$, when $\omega$ is at its maximum, $f(A) = 1$, find the maximum area of $\triangle ABC$.

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>.

Given that $[x]$ represents the greatest integer less than or equal to $x$, if
$$
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
$$

then the smallest value of $x$ is ( ).

Answer: $11.5$


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>.

A certain machine has a display showing an integer $x$, and two buttons, $\mathrm{A}$ and $\mathrm{B}$. When button $\mathrm{A}$ is pressed, the number $x$ on the display is replaced by $2x + 1$. When button $\mathrm{B}$ is pressed, the number $x$ on the display is replaced by $3x - 1$. What is the largest two-digit number that can be obtained by pressing some sequence of buttons $\mathrm{A}$ and $\mathrm{B}$ starting from the number 5 on the display?

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>.

Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.

$(1)$ Find the intervals of monotonic increase for the function $f(x)$;

$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=\sqrt{7}$, $f(B)=0$, $\sin A=3\sin C$, find the values of $a$, $c$ and the area of $\triangle ABC$.

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


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

Given the parametric equation of line $l$ as $\begin{cases} x=m+\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t \end{cases} (t \text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho^{2}\cos ^{2}\theta+3\rho^{2}\sin ^{2}\theta=12$, and the left focus $F$ of the curve $C$ is on line $l$.

(1) If line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $|FA| \cdot |FB|$;

(2) Find the maximum value of the perimeter of the inscribed rectangle of curve $C$.

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>.

Add $A85_{12}$ and $2B4_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.

Answer: $1179_{12}$


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>.

Expand and find the sum of the coefficients of the expression $-(2x - 5)(4x + 3(2x - 5))$.

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 that $S_{n}$ is the sum of the first $n$ terms of a positive geometric sequence $\{a_{n}\}$, and $S_{3}=10$, find the minimum value of $2S_{9}-3S_{6}+S_{3}$.

Answer: $-\frac{5}{4}$


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>.

For all positive integers $n$ greater than 2, the greatest common divisor of $n^5 - 5n^3 + 4n$ is.

Answer: $120$


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

Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side.

Answer: $2\sqrt{127}$


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

The sum of three numbers \(x, y,\) and \(z\) is 120. If we decrease \(x\) by 10, we get the value \(M\). If we increase \(y\) by 10, we also get the value \(M\). If we multiply \(z\) by 10, we also get the value \(M\). What is the value of \(M\)?

Answer: $\frac{400}{7}$


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

How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect cube factor other than one?

Answer: $15$