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

Let  $ABCD$  be a unit square. For any interior points  $M,N$  such that the line  $MN$  does not contain a vertex of the square, we denote by  $s(M,N)$  the least area of the triangles having their vertices in the set of points  $\{ A,B,C,D,M,N\}$ . Find the least number  $k$  such that  $s(M,N)\le k$ , for all points  $M,N$ .

*Dinu Șerbănescu*

Answer: $1/8$


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

Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).
(I) Find the equation of the parabola C;
(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distances from points A and B to line l.

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


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

There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?

Answer: $20$


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

Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse.

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


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

Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'?

Answer: $\frac{11}{36}$


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

Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).

Find the maximum number \(n\) of such sets.

Answer: $871$


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

China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly bill for Package $B$ (in yuan), and $x$ represent the monthly call duration in minutes.
$(1)$ Express the functions of $y_{1}$ with respect to $x$ and $y_{2}$ with respect to $x$.
$(2)$ For how long should the monthly call duration be such that the charges for Packages $A$ and $B$ are the same?
$(3)$ In what scenario is Package $A$ more cost-effective?

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>.

The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\sqrt{37}$. Find the cosine of the angle at the vertex.

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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^2 + b^2 = 2017c^2$, calculate the value of $\frac{\tan C}{\tan A} + \frac{\tan C}{\tan B}$.

Answer: $\frac{1}{1008}$


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

Given that Mary is 30% older than Sally, and Sally is 50% younger than Danielle, and the sum of their ages is 45 years, determine Mary's age on her next birthday.

Answer: $14$


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

If each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a "modest number". For example, for the four-digit number $2163$, since $5\times 2=1+6+3$, $2163$ is a "modest number". Another example is the four-digit number $3147$, since $5\times 3\neq 1+4+7$, $3147$ is not a "modest number". If the four-digit number $\overline{467x}$ is a "modest number", then $x=$______. If a "modest number" $M=\overline{abcd}$ (where $d$ is even), rearrange the tens and units digits of the "modest number" $M$ to the front of the thousand and hundred digits to form a new four-digit number $M'=\overline{cdab}$. Define $G(M)=\frac{M-M'}{99}$. If $G(M)$ is divisible by $11$, and $\overline{abc}$ is divisible by $3$, then the maximum value of $M$ is ______.

Answer: $3816$


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

Given the parametric equations of curve C as $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta \end{cases}(\theta\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\begin{cases} x'= \frac {1}{2}x \\ y'= \frac {1}{ \sqrt {3}}y \end{cases}$$ to obtain curve C'. With the origin as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established.
(Ⅰ) Find the polar equation of curve C';
(Ⅱ) If a line l passing through point $$A\left( \frac {3}{2},\pi\right)$$ (in polar coordinates) with a slope angle of $$\frac {\pi}{6}$$ intersects curve C' at points M and N, and the midpoint of chord MN is P, find the value of $$\frac {|AP|}{|AM|\cdot |AN|}$$.

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


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

Add $123.4567$ to $98.764$ and round your answer to the nearest hundredth. Then, subtract $0.02$ from the rounded result.

Answer: $222.20$


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

The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$.

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>.

Let the number  $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent  of the number  $x$  (for example,  $x\cdot x=x^{2}$ ,  $x^{2}\cdot x^{2}=x^{4}$ ,  $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating  $x^{2006}$ .

Answer: $17$


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

Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \(E\), and the point of intersection of the diagonals is labeled as \(F\). Similar right triangles \(BFC\) and \(DFA\) have corresponding sides \(x, y\) in the first triangle and \(4x, 4y\) in the second triangle.

The height of the trapezoid \(h\) is given by the sum of the heights of triangles \(BFC\) and \(AFD\):

\[ h = x y + \frac{4x \cdot 4y}{4} = 5xy \]

The area of the trapezoid is \(\frac{15}{16}\) of the area of triangle \(AED\):

\[ \frac{1}{2} AC \cdot BD = \frac{15}{16} \cdot \frac{1}{2} \cdot AE \cdot ED \cdot \sin 60^\circ = \frac{15 \sqrt{3}}{64} \cdot \frac{4}{3} AB \cdot \frac{4}{3} CD = \frac{15 \sqrt{3}}{36} AB \cdot CD \]

From this, we have:

\[ \frac{25}{2} xy = \frac{15 \sqrt{3}}{36} \sqrt{x^2 + 16y^2} \cdot \sqrt{y^2 + 16x^2} \]

Given that \(x^2 + y^2 = 1\), find:

\[ \frac{30}{\sqrt{3}} xy = \sqrt{1 + 15y^2} \cdot \sqrt{1 + 15x^2} \]

By simplifying:

\[ 300 x^2 y^2 = 1 + 15(x^2 + y^2) + 225 x^2 y^2 \]
\[ 75 x^2 y^2 = 16 \]
\[ 5xy = \frac{4}{\sqrt{3}} \]

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


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

Isosceles trapezoid  $ABCD$  has side lengths  $AB = 6$  and  $CD = 12$ , while  $AD = BC$ . It is given that  $O$ , the circumcenter of  $ABCD$ , lies in the interior of the trapezoid. The extensions of lines  $AD$  and  $BC$  intersect at  $T$ . Given that  $OT = 18$ , the area of  $ABCD$  can be expressed as  $a + b\sqrt{c}$  where  $a$ ,  $b$ , and  $c$  are positive integers where  $c$  is not divisible by the square of any prime. Compute  $a+b+c$ .

*Proposed by Andrew Wen*

Answer: $84$


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

Compute: $\sin 187^{\circ}\cos 52^{\circ}+\cos 7^{\circ}\sin 52^{\circ}=\_\_\_\_\_\_ \cdot$

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


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

Let \(ABC\) be a non-degenerate triangle and \(I\) the center of its incircle. Suppose that \(\angle A I B = \angle C I A\) and \(\angle I C A = 2 \angle I A C\). What is the value of \(\angle A B C\)?

Answer: $60$


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

The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:

(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left( \frac{2x}{5} \right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$

Find $g\left( \frac{3}{5} \right).$

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


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

Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagonal $D B_1$?

Answer: $3:7$


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

Calculate the value of $\cos 96^\circ \cos 24^\circ - \sin 96^\circ \sin 66^\circ$.

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>.

Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$?

Answer: $86.4$


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>.

A mole has chewed a hole in a carpet in the shape of a rectangle with sides of 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (a patch covers the hole if all points of the rectangle lie inside the square or on its boundary).

Answer: $\sqrt{58}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$.

Answer: $20$


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

Given an integer \( n \) with \( n \geq 2 \), determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i \leq j \leq n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq c\left(\sum_{i=1}^{n} x_{i}\right)^{4}\) holds for all non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \).

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


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

A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence.

Answer: $729$


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

A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 5$?

Answer: $\dfrac{17}{18}$


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

Given the function $f(x) = \sin x + \cos x$, where $x \in \mathbb{R}$,
- (I) Find the value of $f\left(\frac{\pi}{2}\right)$.
- (II) Determine the smallest positive period of the function $f(x)$.
- (III) Calculate the minimum value of the function $g(x) = f\left(x + \frac{\pi}{4}\right) + f\left(x + \frac{3\pi}{4}\right)$.

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>.

What is the value of \(1.90 \frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}\)?

Answer: $-2$


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

Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square.

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


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

In a row of 10 chairs, Mary and James each choose their seats at random but are not allowed to sit in the first or the last chair (chairs #1 and #10). What is the probability that they do not sit next to each other?

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


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

Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$?

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>.

A person was asked how much he paid for a hundred apples and he answered the following:

- If a hundred apples cost 4 cents more, then for 1 dollar and 20 cents, he would get five apples less.

How much did 100 apples cost?

Answer: $96$


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

Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black.

Answer: $\frac{1}{65536}$


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

Four cats, four dogs, and four mice are placed in 12 cages. If a cat and a mouse are in the same column, the cat will meow non-stop; if a mouse is surrounded by two cats on both sides, the mouse will squeak non-stop; if a dog is flanked by a cat and a mouse, the dog will bark non-stop. In other cases, the animals remain silent. One day, the cages numbered 3, 4, 6, 7, 8, and 9 are very noisy, while the other cages are quiet. What is the sum of the cage numbers that contain the four dogs?

Answer: $28$


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

Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method)

Answer: $-3-\sqrt{14}$


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 $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly after the $4$th person draws is $\_\_\_\_\_\_$.

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


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

The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981)

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>.

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

Answer: $18$


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

Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained.

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 a regular triangular prism \(ABC-A_1B_1C_1\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \(A_1ABC\), \(B_1ABC\), and \(C_1ABC\).

Answer: $\frac{\sqrt{3}}{36}$


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>.

Solve the equations:
$(1)x^{2}-10x-10=0$;
$(2)3\left(x-5\right)^{2}=2\left(5-x\right)$.

Answer: $\frac{13}{3}$


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

Let set $P=\{0, 2, 4, 6, 8\}$, and set $Q=\{m | m=100a_1+10a_2+a_3, a_1, a_2, a_3 \in P\}$. Determine the 68th term of the increasing sequence of elements in set $Q$.

Answer: $464$


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

In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a  $1$  on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins  $3$  games before Jacob wins  $3$  games can be written as  $\tfrac{p}{2^q}$ , where  $p$  and  $q$  are positive integers, and  $p$  is odd. Find the remainder when  $p+q$  is divided by  $1000$ .

*Proposed by firebolt360*

Answer: $360$


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

The function
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+ \\
& |4x-1|+|5x-1|
\end{aligned}
$$

achieves its minimum value when the variable $x$ equals ______.

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>.

When any two numbers are taken from the set {0, 1, 2, 3, 4, 5} to perform division, calculate the number of different sine values that can be obtained.

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>.

The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.

Calculate $({-15})÷({\frac{1}{3}-3-\frac{3}{2}})×6$.

Solution: Original expression $=({-15})÷({-\frac{{25}}{6}})×6\ldots \ldots $ First step $=\left(-15\right)\div \left(-25\right)\ldots \ldots $ Second step $=-\frac{3}{5}\ldots \ldots $ Third step

$(1)$ There are two errors in the solution process. The first error is in the ______ step, the mistake is ______. The second error is in the ______ step, the mistake is ______.

$(2)$ Please write down the correct solution process.

Answer: $\frac{108}{5}$


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 angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,)$, $|\overrightarrow{b}|=1$, calculate $|\overrightarrow{a}+2\overrightarrow{b}|$.

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>.

Add $12_8 + 157_8.$ Express your answer in base 8.

Answer: $171_8$


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

24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \( a_k \) based on their ranking \( k \) in that round, where \( a_{k} \in \mathbb{N}_+, k = 1, 2, \ldots, n, a_1 > a_2 > \cdots > a_n \). After all the rounds are completed, the overall ranking is determined based on the total score each person has accumulated over the 12 rounds. Find the smallest positive integer \( n \) such that no matter the ranking in the penultimate round, at least 2 participants have the potential to win the championship.

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>.

Find the number of sets  $A$  that satisfy the three conditions: $\star$   $A$  is a set of two positive integers $\star$  each of the numbers in  $A$  is at least  $22$  percent the size of the other number $\star$   $A$  contains the number  $30.$

Answer: $129$


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

Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\frac{(x-a)(x-b)}{x}=\frac{{x}^{2}-(a+b)x+ab}{x}=x+\frac{ab}{x}-\left(a+b\right)$, the solutions to the equation $x+\frac{ab}{x}=a+b$ with respect to $x$ are $x_{1}=a$ and $x_{2}=b$.
$(1)$ Understanding and application: The solutions to the equation $\frac{{x}^{2}+2}{x}=5+\frac{2}{5}$ are: $x_{1}=$______, $x_{2}=$______;
$(2)$ Knowledge transfer: If the solutions to the equation $x+\frac{3}{x}=7$ are $x_{1}=a$ and $x_{2}=b$, find the value of $a^{2}+b^{2}$;
$(3)$ Extension and enhancement: If the solutions to the equation $\frac{6}{x-1}=k-x$ are $x_{1}=t+1$ and $x_{2}=t^{2}+2$, find the value of $k^{2}-4k+4t^{3}$.

Answer: $32$


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

Given the function $f(x)=ax+b\sin x\ (0 < x < \frac {π}{2})$, if $a\neq b$ and $a, b\in \{-2,0,1,2\}$, the probability that the slope of the tangent line at any point on the graph of $f(x)$ is non-negative is ___.

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


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

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

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


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>.

Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB.

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


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

In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\begin{cases}x=-\sqrt{3}+\cos\varphi\\y=-2+\sin\varphi\end{cases}$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis.

(I) Find the polar equation of $C_1$ and the rectangular equation of $C_2$;
(II) Rotate line $C_1$ counterclockwise around the coordinate origin by an angle of $\frac{\pi}{3}$ to obtain line $C_3$, which intersects curve $C_2$ at points $A$ and $B$. Find the length $|AB|$.

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>.

In isosceles trapezoid $ABCD$ where $AB$ (shorter base) is 10 and $CD$ (longer base) is 20. The non-parallel sides $AD$ and $BC$ are extended to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$?

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>.

Square $PQRS$ has sides of length 1. Points $T$ and $U$ are on $\overline{QR}$ and $\overline{RS}$, respectively, so that $\triangle PTU$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PT}.$ The length of a side of this smaller square is $\frac{a-\sqrt{b}}{c},$ where $a, b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c.$

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>.

Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{\sqrt{3}}{k}\rceil - \lfloor\log_{\sqrt{3}}{k} \rfloor).$$

Answer: $1999907$


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

Given the function $f(x) = x + a\ln x$ has its tangent line at $x = 1$ perpendicular to the line $x + 2y = 0$, and the function $g(x) = f(x) + \frac{1}{2}x^2 - bx$,
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Let $x_1$ and $x_2$ ($x_1 < x_2$) be two extreme points of the function $g(x)$. If $b \geq \frac{7}{2}$, find the minimum value of $g(x_1) - g(x_2)$.

Answer: $\frac{15}{8} - 2\ln 2$


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>.

In the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2} - a_n = 1 + (-1)^n$ ($n \in \mathbb{N}^*$), then the sum $a_1 + a_2 + \ldots + a_{51} =$ ?

Answer: $676$


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

Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60

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>.

Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$.

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>.

As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices?

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>.

Express $0.6\overline{03}$ as a common fraction.

Answer: $\frac{104}{165}$


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

A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 16$. With the exception of the bottom row, each square rests on two squares in the row immediately below. In each square of the sixteenth row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $5$?

Answer: $16384$


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

A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and $\theta$ varies from $0^\circ$ to $90^\circ$. As $\theta$ varies within this range, the highest points of the projectile paths trace out a curve. Calculate the area enclosed by this curve, which can be expressed as $c \cdot \frac{v^4}{g^2}$. Find the value of $c$.

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


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

The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Express your answer as a common fraction.

Answer: $\frac{60}{7}$


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>.

Harry Potter can do any of the three tricks arbitrary number of times: $i)$  switch  $1$  plum and  $1$  pear with  $2$  apples  $ii)$  switch  $1$  pear and  $1$  apple with  $3$  plums  $iii)$  switch  $1$  apple and  $1$  plum with  $4$  pears
In the beginning, Harry had  $2012$  of plums, apples and pears, each. Harry did some tricks and now he has  $2012$  apples,  $2012$  pears and more than  $2012$  plums. What is the minimal number of plums he can have?

Answer: $2025$


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

A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments measuring 6 cm and 7 cm. Calculate the area of the triangle.

Answer: $42$


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

Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. For $x \in [-3,-1)$, $f(x)=-(x+2)^{2}$, and for $x \in [-1,3)$, $f(x)=x$. Calculate the sum $f(1)+f(2)+f(3)+\ldots+f(2015)$.

Answer: $336$


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

Suppose that  $x$  is real number such that  $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of  $2^{-(1+\log_23)x}$

Answer: $216$


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

Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers?

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>.

Find the root of the following equation to three significant digits:

$$
(\sqrt{5}-\sqrt{2})(1+x)=(\sqrt{6}-\sqrt{3})(1-x)
$$

Answer: $-0.068$


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

Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{a}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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>.

Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?
A) 8
B) 10
C) 12
D) 14
E) 16

Answer: $10$


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

A high school math preparation group consists of six science teachers and two liberal arts teachers. During a three-day period of smog-related class suspensions, they need to arrange teachers to be on duty for question-answering sessions. The requirement is that each day, there must be one liberal arts teacher and two science teachers on duty. Each teacher should be on duty for at least one day and at most two days. How many different arrangements are possible?

Answer: $540$


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

Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$

Answer: $234 \times 345$


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

If the space vectors $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ satisfy $|\overrightarrow{{e_1}}|=|2\overrightarrow{{e_1}}+\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\overrightarrow{{e_1}}$ in the direction of $\overrightarrow{{e_2}}$.

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


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

Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$.

Answer: $2889$


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

Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express your answer as a common fraction.

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


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

Given that the lateral surface of a cone is the semicircle with a radius of $2\sqrt{3}$, find the radius of the base of the cone. If the vertex of the cone and the circumference of its base lie on the surface of a sphere $O$, determine the volume of the sphere.

Answer: $\frac{32\pi}{3}$


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>.

An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism?

Answer: $12\sqrt{3}$


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

Let the polynomial be defined as $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{20}.$$ Calculate the sum of the first five distinct $\alpha_k$ values where each zero of $Q(x)$ can be expressed in the complex form $z_k = r_k [\cos(2\pi \alpha_k) + i\sin(2\pi \alpha_k)]$, with $\alpha_k \in (0, 1)$ and $r_k > 0$.

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


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

The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people.

Answer: $\frac{258}{7}$


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>.

Suppose  $\mathcal{T}=A_0A_1A_2A_3$  is a  tetrahedron with  $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$ ,  $A_0A_1=5, A_0A_2=12$  and  $A_0A_3=9$ . A cube  $A_0B_0C_0D_0E_0F_0G_0H_0$  with side length  $s$  is inscribed inside  $\mathcal{T}$  with  $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$ , and  $G_0\in \triangle A_1A_2A_3$ ; what is  $s$ ?

Answer: $180/71$


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

In the expansion of the binomial ${({(\frac{1}{x}}^{\frac{1}{4}}+{{x}^{2}}^{\frac{1}{3}})}^{n})$, the coefficient of the third last term is $45$. Find the coefficient of the term containing $x^{3}$.

Answer: $210$


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

Line $m$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $30^{\circ}$ counterclockwise about the point $(10, 10)$ to form line $n$. Find the $x$-coordinate of the $x$-intercept of line $n$.

Answer: $\frac{20\sqrt{3} + 20}{2\sqrt{3} + 3}$


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>.

Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\left( 1,\frac{1}{2} \right)$.

(Ⅰ) Find the standard equation of the ellipse;

(Ⅱ) If a line passing through the origin $O$ intersects the ellipse at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$.

Answer: $\sqrt {2}$


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

In recent years, the food delivery industry in China has been developing rapidly, and delivery drivers shuttling through the streets of cities have become a beautiful scenery. A certain food delivery driver travels to and from $4$ different food delivery stores (numbered $1, 2, 3, 4$) every day. The rule is: he first picks up an order from store $1$, called the first pick-up, and then he goes to any of the other $3$ stores for the second pick-up, and so on. Assuming that starting from the second pick-up, he always goes to one of the other $3$ stores that he did not pick up from last time. Let event $A_{k}=\{$the $k$-th pick-up is exactly from store $1\}$, $P(A_{k})$ is the probability of event $A_{k}$ occurring. Obviously, $P(A_{1})=1$, $P(A_{2})=0$. Then $P(A_{3})=$______, $P(A_{10})=$______ (round the second answer to $0.01$).

Answer: $0.25$


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>.

$D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\angle ABC=\alpha$ and $\angle CAB=\beta$.
$(1)$ When $CD\perp AB$ and $\beta =2\alpha$, find the value of $CD$;
$(2)$ If $α+\beta=\frac{π}{4}$, find the maximum area of triangle $ACD$.

Answer: $5(\sqrt{2} - 1)$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|= \sqrt {2}$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}-\overrightarrow{b})\perp \overrightarrow{a}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______

Answer: $1985$


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

Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches.

Answer: $20$


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

Given $ \frac {\pi}{2} < \alpha < \pi$ and $0 < \beta < \frac {\pi}{2}$, with $\tan \alpha= -\frac {3}{4}$ and $\cos (\beta-\alpha)= \frac {5}{13}$, find the value of $\sin \beta$.

Answer: $\frac {63}{65}$


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

A printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word "JANUARY", 10,000 posters with the word "FEBRUARY", 10,000 posters with the word "MARCH", and so on. The typefaces used to print the month names were made on special order and were very expensive. Therefore, the printer wanted to buy as few typefaces as possible, with the idea that some of the typefaces used for one month's name could be reused for the names of other months. The goal is to have a sufficient stock of typefaces to print the names of all months throughout the year.

How many different typefaces does the printer need to buy? All words are printed in uppercase letters, as shown above.

Answer: $22$


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

Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.

Answer: $1023$


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

What is the value of \(b\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = b\] represents a non-degenerate ellipse?

Answer: $-\frac{129}{4}$


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

In the right triangle $ABC$, where $\angle B = \angle C$, the length of $AC$ is $8\sqrt{2}$. Calculate the area of triangle $ABC$.

Answer: $64$


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

Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements.

Answer: $18$


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

Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? Express your answer in simplest radical form.

Answer: $\sqrt{34 + 15\sqrt{2}}$


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>.

Given the function $f\left(x\right)=\left(x+1\right)e^{x}$.
$(1)$ Find the intervals where the function $f\left(x\right)$ is monotonic.
$(2)$ Find the maximum and minimum values of $f\left(x\right)$ on the interval $\left[-4,0\right]$.

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


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 belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, find the distance between the centers of the two pulleys.

Answer: $26$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $30^{\circ}$ between them, and $|\overrightarrow{a}|=\sqrt{3}$, $|\overrightarrow{b}|=1$,

$(1)$ Find the value of $|\overrightarrow{a}-2\overrightarrow{b}|$

$(2)$ Let vector $\overrightarrow{p}=\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{q}=\overrightarrow{a}-2\overrightarrow{b}$, find the projection of vector $\overrightarrow{p}$ in the direction of $\overrightarrow{q}$

Answer: $-1$


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

In the diagram, each of the two circles has center \(O\). Also, \(O P: P Q = 1:2\). If the radius of the larger circle is 9, what is the area of the shaded region?

Answer: $72 \pi$


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

The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the  $2014^{\text{th}}$  number in that sequence?

Answer: $62048$


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

Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king?

Answer: $\frac{17}{884}$


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

Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______.

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


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>.

Cynthia and Lynnelle are collaborating on a problem set. Over a  $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous  $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.

Answer: $4/9$


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>.

There are 5 people seated at a circular table. What is the probability that Angie and Carlos are seated directly opposite each other?

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


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

The line $y = a$ intersects the curves $y = 2(x + 1)$ and $y = x + \ln x$ at points $A$ and $B$, respectively. Find the minimum value of $|AB|$.

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>.

Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?

Answer: $1333$


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

Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form.

Answer: $\sqrt{113 + 56\sqrt{2}}$


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

A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52

Answer: $38$


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

Suppose that  $n$  persons meet in a meeting, and that each of the persons is acquainted to exactly  $8$  others. Any two acquainted persons have exactly  $4$  common acquaintances, and any two non-acquainted persons have exactly  $2$  common acquaintances. Find all possible values of  $n$ .

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>.

The height of the pyramid $P-ABCD$ with a square base of side length $2\sqrt{2}$ is $1$. If the radius of the circumscribed sphere of the pyramid is $2\sqrt{2}$, then the distance between the center of the square $ABCD$ and the point $P$ is ______.

Answer: $2\sqrt{2}$


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

Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\]

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 the rectangular coordinate system $(xOy)$, the slope angle of line $l$ passing through point $M(2,1)$ is $\frac{\pi}{4}$. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, using the same unit length for both coordinate systems. The polar equation of circle $C$ is $\rho = 4\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$.

(I) Find the parametric equations of line $l$ and the rectangular form of the equation of circle $C$.

(II) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. Find the value of $\frac{1}{|MA|} + \frac{1}{|MB|}$.

Answer: $\frac{\sqrt{30}}{7}$


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>.

From 50 products, 10 are selected for inspection. The total number of items is \_\_\_\_\_\_\_, and the sample size is \_\_\_\_\_\_.

Answer: $10$