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

Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually?

Answer: $16$


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

Find the smallest number  $n \geq 5$  for which there can exist a set of  $n$  people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.

*Bulgaria*

Answer: $11$


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

For how many integers  $a$  with  $|a| \leq 2005$ , does the system
 $x^2=y+a$
 $y^2=x+a$
have integer solutions?

Answer: $90$


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

A sphere with a radius of $1$ is placed inside a cone and touches the base of the cone. The minimum volume of the cone is \_\_\_\_\_\_.

Answer: $\dfrac{8\pi}{3}$


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>.

Given that there is a point P (x, -1) on the terminal side of ∠Q (x ≠ 0), and $\tan\angle Q = -x$, find the value of $\sin\angle Q + \cos\angle Q$.

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>.

A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many kilometers do we have left to ride to reach point C?" B answered, "We have a distance left to ride equivalent to one-third of the distance from here to point B." What is the distance between point A and point C? (Answer should be in fraction form.)

Answer: $\frac{40}{3}$


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

Determine the maximal size of a set of positive integers with the following properties:

(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).

(2) No digit occurs more than once in the same integer.

(3) The digits in each integer are in increasing order.

(4) Any two integers have at least one digit in common (possibly at different positions).

(5) There is no digit which appears in all the integers.

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>.

The orthocenter of triangle $DEF$ divides altitude $\overline{DM}$ into segments with lengths $HM = 10$ and $HD = 24.$ Calculate $\tan E \tan F.$

Answer: $3.4$


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

In triangle ABC, medians AD and BE intersect at centroid G. The midpoint of segment AB is F. Given that the area of triangle GFC is l times the area of triangle ABC, find the value of l.

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>.

Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=|\vec{b}|=2$, and $\vec{b}$ is perpendicular to $(2\vec{a}+\vec{b})$, find the angle between vector $\vec{a}$ and $\vec{b}$.

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>.

Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 3 & 7 \\ 4 & x & y \\ 4 & y & x+1 \end{vmatrix}
= 0.\]Find $x + y.$

Answer: $20$


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

What is the smallest positive integer \(n\) such that \(\frac{n}{n+51}\) is equal to a terminating decimal?

Answer: $74$


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

On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers that immediately precede it in a clockwise direction.

Determine the maximum possible number of positive numbers that can be among the 2018 numbers.

Answer: $1008$


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>.

In the quadratic equation $2x^{2}-1=6x$, the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______.

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 $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy the equation $a\sin B = \sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$.
$(2)$ Choose one set of conditions from the following three sets to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.
Set 1: $a = \sqrt{19}$, $c = 5$;
Set 2: The altitude $h$ on side $AB$ is $\sqrt{3}$, $a = 3$;
Set 3: $\cos C = \frac{1}{3}$, $c = 4\sqrt{2}$.

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


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

Determine how many "super prime dates" occurred in 2007, where a "super prime date" is defined as a date where both the month and day are prime numbers, and additionally, the day is less than or equal to the typical maximum number of days in the respective prime month.

Answer: $50$


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

Given the function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$.

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


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 cinema is showing four animated movies: Toy Story, Ice Age, Shrek, and Monkey King. The ticket prices are 50 yuan, 55 yuan, 60 yuan, and 65 yuan, respectively. Each viewer watches at least one movie and at most two movies. However, due to time constraints, viewers cannot watch both Ice Age and Shrek. Given that there are exactly 200 people who spent the exact same amount of money on movie tickets today, what is the minimum total number of viewers the cinema received today?

Answer: $1792$


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

Write the number in the form of a fraction (if possible):

$$
x=0.5123412341234123412341234123412341234 \ldots
$$

Can you generalize this method to all real numbers with a periodic decimal expansion? And conversely?

Answer: $\frac{51229}{99990}$


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

Suppose that $\sec y - \tan y = \frac{15}{8}$ and that $\csc y - \cot y = \frac{p}{q},$ where $\frac{p}{q}$ is in lowest terms. Find $p+q.$

Answer: $30$


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

Find the absolute value of the difference of single-digit integers \( C \) and \( D \) such that in base \( 5 \):
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_5 \\
& & & \mathbf{3} & \mathbf{2} & D_5 \\
& & + & C & \mathbf{2} & \mathbf{4_5} \\
\cline{2-6}
& & C & \mathbf{2} & \mathbf{3} & \mathbf{1_5} \\
\end{array} $$

Answer: $1_5$


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

An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nine balls. What is the probability that there are exactly five red balls and four blue balls in the urn?
A) $\frac{1}{10}$
B) $\frac{2}{10}$
C) $\frac{3}{10}$
D) $\frac{4}{10}$
E) $\frac{5}{10}$

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>.

Consider  $7$  points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.

Answer: $127$


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

Let  $f(x)$  be the polynomial  $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ .  Let  $c$  be the coefficient of  $x^{48}$  in  $f(x)$ .  When  $c$  is divided by 101, what is the remainder?  (The remainder is an integer between 0 and 100.)

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 equation \(2008=1111+444+222+99+77+55\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.

i) Find a similar decomposition for the number 2009.

ii) Determine all possible such decompositions of the number 2009 that use the minimum number of terms (the order of terms does not matter).

Answer: $1111 + 777 + 66 + 55$


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

Calculate the largest prime factor of $18^4 + 12^5 - 6^6$.

Answer: $11$


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

For a positive real number  $a$ , let  $C$  be the cube with vertices at  $(\pm a, \pm a, \pm a)$  and let  $T$  be the tetrahedron with vertices at  $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of  $T$  and  $C$  has volume  $ka^3$  for some  $k$ , find  $k$ .

Answer: $4/3$


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

Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after this directional change will it take for Tom to catch up to Alice?

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>.

Point $D$ lies on side $AC$ of equilateral triangle $ABC$ such that the measure of angle $DBC$ is 30 degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$?

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>.

Concentric circles  $\Omega_1$  and  $\Omega_2$  with radii  $1$  and  $100$ , respectively, are drawn with center  $O$ . Points  $A$  and  $B$  are chosen independently at random on the circumferences of  $\Omega_1$  and  $\Omega_2$ , respectively. Denote by  $\ell$  the tangent line to  $\Omega_1$  passing through  $A$ , and denote by  $P$  the reflection of  $B$  across  $\ell$ . Compute the expected value of  $OP^2$ .

*Proposed by Lewis Chen*

Answer: $10004$


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

In the diagram, pentagon \( PQRST \) has \( PQ = 13 \), \( QR = 18 \), \( ST = 30 \), and a perimeter of 82. Also, \( \angle QRS = \angle RST = \angle STP = 90^\circ \). The area of the pentagon \( PQRST \) is:

Answer: $270$


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

Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \leqslant S_5$. The sum of the first $9$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$ is ______.

Answer: $- \frac{1}{9}$


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>.

Starting with the display "1," calculate the fewest number of keystrokes needed to reach "400".

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>.

To enhance and beautify the city, all seven streetlights on a road are to be changed to colored lights. If there are three colors available for the colored lights - red, yellow, and blue - and the installation requires that no two adjacent streetlights are of the same color, with at least two lights of each color, there are ____ different installation methods.

Answer: $114$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $202_4 + 330_4 + 1000_4$. Express your answer first in base 4, then convert that sum to base 10.

Answer: $158$


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

Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements.

Answer: $192$


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

A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___.

Answer: $600$


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

The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports.

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>.

Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7.

Answer: $27$


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

Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction.

Answer: $\frac{180}{37}$


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

Two circles, circle $A$ with radius 2 and circle $B$ with radius 1.5, are to be constructed with the following process: The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ to $(3,0)$. The center of circle $B$ is chosen uniformly and at random, and independently from the first choice, from the line segment joining $(1,2)$ to $(4,2)$. What is the probability that circles $A$ and $B$ intersect?
A) 0.90
B) 0.95
C) 0.96
D) 1.00

Answer: $0.96$


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>.

Identical matches of length 1 are used to arrange the following pattern. If \( c \) denotes the total length of matches used, find \( c \).

Answer: $700$


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

Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______.

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>.

An entrepreneur invested $\$20,\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $\$22,\!446.40$. If the annual interest rate of the second term deposit is $s\%$, what is $s?$

Answer: $7.840$


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

Given that the sum of three numbers, all equally likely to be $1$, $2$, $3$, or $4$, drawn from an urn with replacement, is $9$, calculate the probability that the number $3$ was drawn each time.

Answer: $\frac{1}{13}$


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>.

Monsieur Dupont remembered that today is their wedding anniversary and invited his wife to dine at a fine restaurant. Upon leaving the restaurant, he noticed that he had only one fifth of the money he initially took with him. He found that the centimes he had left were equal to the francs he initially had (1 franc = 100 centimes), while the francs he had left were five times less than the initial centimes he had.

How much did Monsieur Dupont spend at the restaurant?

Answer: $7996$


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

Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$.

Answer: $11$


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

A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shandong Province Mathematics Competition, 1979)

Answer: $1979$


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

Consider the function $y=a\sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x}$ ($a\in\mathbb{R}$), and let $t= \sqrt{1+x} + \sqrt{1-x}$ ($\sqrt{2} \leq t \leq 2$).

(1) Express $y$ as a function of $t$, denoted as $m(t)$.

(2) Let the maximum value of the function $m(t)$ be $g(a)$. Find $g(a)$.

(3) For $a \geq -\sqrt{2}$, find all real values of $a$ that satisfy $g(a) = g\left(\frac{1}{a}\right)$.

Answer: $a = 1$


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>.

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe is opened, the key inside the safe can be used to open another safe.)

Answer: $1/47$


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

Let $A$, $B$, $R$, $M$, and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AB) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RB) &= 5.
\end{align*}
Compute the value of the product $ABRML$.

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>.

Determine the smallest integer $B$ such that there exist several consecutive integers, including $B$, that add up to 2024.

Answer: $-2023$


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

For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{2520}$ terminate?

Answer: $47$


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

In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is

Answer: $67.5$


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

Let  $ABCD$  be a cyclic quadrilateral, and  $E$  be the intersection of its diagonals. If  $m(\widehat{ADB}) = 22.5^\circ$ ,  $|BD|=6$ , and  $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral  $ABCD$ .

Answer: $9\sqrt{2}$


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>.

John earned scores of 92, 85, and 91 on his first three physics examinations. If John receives a score of 95 on his fourth exam, then by how much will his average increase?

Answer: $1.42$


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>.

If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$.

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>.

For any positive integer  $a$ , define  $M(a)$  to be the number of positive integers  $b$  for which  $a+b$  divides  $ab$ . Find all integer(s)  $a$  with  $1\le a\le 2013$  such that  $M(a)$  attains the largest possible value in the range of  $a$ .

Answer: $1680$


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

The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?

Answer: $12$


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

Given the sequence \(\left\{a_{n}\right\}\) that satisfies
\[ a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad \text{for} \quad n \in \mathbf{Z}_{+}, \]
where \([x]\) denotes the greatest integer less than or equal to the real number \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C. \]
Find the minimum value of \(C\).

Answer: $\frac{\sqrt{5} - 2}{4}$


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

Given that in the rectangular coordinate system $(xOy)$, the origin is the pole and the positive semi-axis of $x$ is the polar axis to establish a polar coordinate system, the polar coordinate equation of the conic section $(C)$ is $p^{2}= \frac {12}{3+\sin ^{2}\theta }$, the fixed point $A(0,- \sqrt {3})$, $F\_{1}$, $F\_{2}$ are the left and right foci of the conic section $(C)$, and the line $l$ passes through point $F\_{1}$ and is parallel to the line $AF\_{2}$.
(I) Find the rectangular coordinate equation of conic section $(C)$ and the parametric equation of line $l$;
(II) If line $l$ intersects conic section $(C)$ at points $M$ and $N$, find $|F\_{1}M|⋅|F\_{1}N|$.

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


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

Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.

1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$.

Answer: $2016$


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

Given the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $\pi$ units and coincides with the original graph. When ${x}_{1}$, ${x}_{2} \in \left( -\frac{17\pi }{12},-\frac{2\pi }{3} \right)$ and ${x}_{1} \ne {x}_{2}$, if $f\left( {x}_{1} \right)=f\left( {x}_{2} \right)$, find $f({x}_{1}+{x}_{2})$.

Answer: $-1$


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

A circle with radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through?

Answer: $800$


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

In a circle with a radius of 5 units, \( CD \) and \( AB \) are mutually perpendicular diameters. A chord \( CH \) intersects \( AB \) at \( K \) and has a length of 8 units, calculate the lengths of the two segments into which \( AB \) is divided.

Answer: $8.75$


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

Real numbers \(a, b, c\) and a positive number \(\lambda\) satisfy \(f(x) = x^3 + a x^2 + b x + c\), which has 3 real roots \(x_1, x_2, x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).

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>.

You are given the numbers $0$, $2$, $3$, $4$, $6$. Use these numbers to form different combinations and calculate the following:
   $(1)$ How many unique three-digit numbers can be formed?
   $(2)$ How many unique three-digit numbers that can be divided by $3$ can be formed? (Note: Write the result of each part in data form)

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>.

I live on a very short street with 14 small family houses. The odd-numbered houses from 1 are on one side of the street, and the even-numbered houses from 2 are on the opposite side (e.g., 1 and 2 are opposite each other).

On one side of the street, all families have surnames that are colors, and on the other side, the surnames indicate professions.

Szabó and Fazekas live opposite to Zöld and Fehér, respectively, who are both neighbors of Fekete.

Kovács is the father-in-law of Lakatos.

Lakatos lives in a higher-numbered house than Barna. The sum of the house numbers of Lakatos and Barna is equal to the sum of the house numbers of Fehér and Fazekas. Kádárné's house number is twice the house number of her sister, Kalaposné.

Sárga lives opposite Pék.

If Bordóné's house number is two-digit and she lives opposite her sister, Kádárné, what is the house number of Mr. Szürke?

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>.

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

Answer: $41$


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

In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 1500$ and $PS = 3000$. Let $\angle P = 37^\circ$, $\angle S = 53^\circ$, and $X$ and $Y$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $XY$.

Answer: $750$


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

In the textbook, students were once asked to explore the coordinates of the midpoint of a line segment: In a plane Cartesian coordinate system, given two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, the midpoint of the line segment $AB$ is $M$, then the coordinates of $M$ are ($\frac{{x}_{1}+{x}_{2}}{2}$, $\frac{{y}_{1}+{y}_{2}}{2}$). For example, if point $A(1,2)$ and point $B(3,6)$, then the coordinates of the midpoint $M$ of line segment $AB$ are ($\frac{1+3}{2}$, $\frac{2+6}{2}$), which is $M(2,4)$. Using the above conclusion to solve the problem: In a plane Cartesian coordinate system, if $E(a-1,a)$, $F(b,a-b)$, the midpoint $G$ of the line segment $EF$ is exactly on the $y$-axis, and the distance to the $x$-axis is $1$, then the value of $4a+b$ is ____.

Answer: $4 \text{ or } 0$


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

Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
\[f(x) f(-x) = f(x^3).\]

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>.

Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?

Answer: $500$


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

Given the function $f(x)=\begin{cases} x+2 & (x\leqslant -1) \\ x^{2} & (-1< x < 2) \\ 2x & (x\geqslant 2) \end{cases}$
$(1)$ Find $f(2)$, $f\left(\dfrac{1}{2}\right)$, $f[f(-1)]$;
$(2)$ If $f(a)=3$, find the value of $a$.

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>.

What is the greatest prime factor of $15! + 18!$?

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>.

In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\triangle ABC$.

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>.

Given an ellipse $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}$ and $F_{2}$ respectively, and the top vertex is $B$. If the perimeter of $\triangle BF_{1}F_{2}$ is $6$, and the distance from point $F_{1}$ to the line $BF_{2}$ is $b$.

$(1)$ Find the equation of ellipse $C$;
$(2)$ Let $A_{1}, A_{2}$ be the two endpoints of the major axis of ellipse $C$, and point $P$ is any point on ellipse $C$ other than $A_{1}, A_{2}$. The line $A_{1}P$ intersects the line $x = m$ at point $M$. If the circle with diameter $MP$ passes through point $A_{2}$, find the value of the real number $m$.

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>.

How many of the natural numbers from 1 to 700, inclusive, contain the digit 3 at least once?

Answer: $214$


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

The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \in [-2,0]$, $f(x)=2x+1$. If there exist $x\_1$, $x\_2$, $…x\_n$ satisfying $0 \leqslant x\_1 < x\_2 < … < x\_n$, and $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_1)|+…+|f(x\_{n-1}-f(x\_n))|=2016$, then the minimum value of $n+x\_n$ is \_\_\_\_\_\_.

Answer: $1513$


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

How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?

Answer: $511$


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

The solution to the inequality

$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$

is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.

Recall that $[x]$ denotes the greatest integer less than or equal to $x$.

Answer: $78.00$


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

What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));

xaxis(0,8,Ticks(1.0,NoZero));
yaxis(0,8,Ticks(1.0,NoZero));

fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,7)--(0,6)--(6,6)--cycle);
label("$E$",(0,0),SW);
label("$F$",(0,7),N);
label("$G$",(7,7),NE);
label("$H$",(7,0),E);
[/asy]

Answer: $67\%$


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

What is the smallest five-digit positive integer congruent to $2 \pmod{17}$?

Answer: $10013$


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

A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag?

Answer: $\frac{91}{169}$


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

Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7.

Answer: $60718293$


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

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(10,2)$, respectively. Calculate its area.

Answer: $52\sqrt{3}$


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

A certain intelligence station has four different passwords A, B, C, and D. Each week, one of the passwords is used, and the password for each week is equally likely to be randomly selected from the three passwords not used in the previous week. If password A is used in the first week, what is the probability that password A is also used in the seventh week? (Express your answer as a simplest fraction.)

Answer: $1/3$


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

How many positive integers \( n \) satisfy \[ (n + 9)(n - 4)(n - 13) < 0 \]?

Answer: $11$


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

Jia participates in a shooting practice with 4 bullets, among which one is a blank (a "blank" means a bullet without a projectile).
(1) If Jia shoots only once, calculate the probability of the shot being a blank;
(2) If Jia shoots a total of 3 times, calculate the probability of a blank appearing in these three shots;
(3) If an equilateral triangle PQR with a side length of 10 is drawn on the target, and Jia, using live rounds, aims and randomly shoots at the area of triangle PQR, calculate the probability that all bullet holes are more than 1 unit away from the vertices of △PQR (ignoring the size of the bullet holes).

Answer: $1 - \frac{\sqrt{3}\pi}{150}$


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

A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x + 7 + t \right)^2\). Find the coordinates of points \(M\) and \(N\) that minimize \(\rho^2\).

When the point \(M\) is fixed, \(\rho^2(t)\) depends on \(t\), and at the point of minimum, its derivative with respect to \(t\) is zero: \[-2(x - t) + 2\left( x^2 - 4x + 7 + t \right) = 0.\]

When point \(N\) is fixed, the function \(\rho^2(x)\) depends on \(x\), and at the point of minimum, its derivative with respect to \(x\) is zero:
\[2(x - t) + 2\left( x^2 - 4x + 7 + t \right)(2x - 4) = 0.\]

We solve the system:
\[
\begin{cases}
2 \left( x^2 - 4x + 7 + t \right) (2x - 3) = 0 \\
4(x - t) + 2 \left( x^2 - 4x + 7 + t \right) (2x - 5) = 0
\end{cases}
\]

**Case 1: \(x = \frac{3}{2}\)**

Substituting \(x = \frac{3}{2}\) into the second equation, we get \(t = -\frac{7}{8}\). Critical points are \(N^* \left( -\frac{7}{8}, -\frac{9}{8} \right)\) and \(M^* \left( \frac{3}{2}, \frac{5}{4} \right)\). The distance between points \(M^*\) and \(N^*\) is \(\frac{19\sqrt{2}}{8}\). If \(2r \leq \frac{19\sqrt{2}}{8}\), then the circle does not intersect the parabola (it touches at points \(M^*\) and \(N^*\)).

**Case 2: \(x \neq \frac{3}{2}\)**

Then \(\left( x^2 - 4x + 7 + t \right) = 0\) and from the second equation \(x = t\). Substituting \(x = t\) into the equation \(\left( x^2 - 4x + 7 + t \right) = 0\), we get the condition for \(t\), i.e. \(t^2 - 3t + 7 = 0\). Since this equation has no solutions, case 2 is not realizable. The maximum radius \(r_{\max} = \frac{19\sqrt{2}}{16}\).

Answer: $\frac{19\sqrt{2}}{8}$


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

If \(A\ \clubsuit\ B\) is defined as \(A\ \clubsuit\ B = 3A^2 + 2B + 7\), what is the value of \(A\) for which \(A\ \clubsuit\ 7 = 61\)?

Answer: $\frac{2\sqrt{30}}{3}$


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

Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms.

Answer: $180$


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

Given a regular 2017-sided polygon \( A_{1} A_{2} \cdots A_{2017} \) inscribed in a unit circle \(\odot O\), choose any two distinct vertices \( A_{i} \) and \( A_{j} \). What is the probability that \( \overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2} \)?

Answer: $2/3$


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

Evaluate the absolute value of the expression $|7 - \sqrt{53}|$.
A) $7 - \sqrt{53}$
B) $\sqrt{53} - 7$
C) $0.28$
D) $\sqrt{53} + 7$
E) $-\sqrt{53} + 7$

Answer: $\sqrt{53} - 7$


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

Given $x+x^{-1}=3$, calculate the value of $x^{ \frac {3}{2}}+x^{- \frac {3}{2}}$.

Answer: $\sqrt{5}$


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

Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.

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>.

Determine the smallest positive integer $n$, different from 2004, such that there exists a polynomial $f(x)$ with integer coefficients for which the equation $f(x) = 2004$ has at least one integer solution and the equation $f(x) = n$ has at least 2004 different integer solutions.

Answer: $(1002!)^2 + 2004$


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

Let  $f(n)=1 \times 3 \times 5 \times \cdots \times (2n-1)$ . Compute the remainder when  $f(1)+f(2)+f(3)+\cdots +f(2016)$  is divided by  $100.$ *Proposed by James Lin*

Answer: $74$


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

Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \).

Answer: $9 - 8\sqrt{2}$


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>.

All the complex roots of $(z + 2)^6 = 64z^6$, when plotted in the complex plane, lie on a circle. Find the radius of this circle.

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


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>.

How many ways are there to arrange the letters of the word $\text{CA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{T}_1\text{T}_2$, where there are three A's, two N's, and two T's, with each A, N, and T considered distinct?

Answer: $5040$


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

Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.
$(1)$ Find $p$;
$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$.

Answer: $20\sqrt{5}$


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

Biejia and Vasha are playing a game. Biejia selects 100 non-negative numbers \(x_1, x_2, \cdots, x_{100}\) (they can be the same), whose sum equals 1. Vasha then pairs these numbers into 50 pairs in any way he chooses, computes the product of the two numbers in each pair, and writes the largest product on the blackboard. Biejia wants the number written on the blackboard to be as large as possible, while Vasha wants it to be as small as possible. What will be the number written on the blackboard under optimal strategy?

Answer: $1/396$


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

In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD = 17^3$, and $\cos B = m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: $162$


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

The surface of a 3x3x3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked such that no two marked cells share a common vertex?

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>.

Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{3}{7}.\] Find the product of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\]

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


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

The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1, calculate the sum of these numbers.

Answer: $4436$


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

A conical glass is in the form of a right circular cone. The slant height is  $21$  and the radius of the top rim of the glass is  $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If  $d$  the shortest distance it should crawl to reach the honey drop, what is the integer part of  $d$  ?

[center]![Image](https://i.imgur.com/T1Y3zwR.png)[/center]

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>.

A table  $110\times 110$  is given, we define the distance between two cells  $A$  and  $B$  as the least quantity of moves to move a chess king from the cell  $A$  to cell  $B$ . We marked  $n$  cells on the table  $110\times 110$  such that the distance between any two cells is not equal to  $15$ . Determine the greatest value of  $n$ .

Answer: $6050$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, an ellipse C passes through points A $(\sqrt{3}, 0)$ and B $(0, 2)$.
(I) Find the equation of ellipse C;
(II) Let P be any point on the ellipse, find the maximum area of triangle ABP, and find the coordinates of point P when the area of triangle ABP is maximum.

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


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

Given $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\cos 2C+2c\cos A\cos C+a+b=0$.

$(1)$ Find the size of angle $C$;

$(2)$ If $b=4\sin B$, find the maximum value of the area $S$ of $\triangle ABC$.

Answer: $\sqrt{3}$


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

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

Answer: $1286$


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

In acute triangle  $ABC$ , points  $D$  and  $E$  are the feet of the angle bisector and altitude from  $A$  respectively.  Suppose that  $AC - AB = 36$  and  $DC - DB = 24$ .  Compute  $EC - EB$ .

Answer: $54$


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

Let $x = \frac{\sum\limits_{n=1}^{30} \cos n^\circ}{\sum\limits_{n=1}^{30} \sin n^\circ}$. What is the smallest integer that does not fall below $100x$?

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>.

Given the function $f(x)=\frac{1}{2}x^{2}+(2a^{3}-a^{2})\ln x-(a^{2}+2a-1)x$, and $x=1$ is its extreme point, find the real number $a=$ \_\_\_\_\_\_.

Answer: $-1$


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

The real numbers \(x_{1}, x_{2}, \cdots, x_{2001}\) satisfy \(\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001\). Let \(y_{k}=\frac{1}{k}\left(x_{1}+ x_{2} + \cdots + x_{k}\right)\) for \(k=1, 2, \cdots, 2001\). Find the maximum possible value of \(\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|\).

Answer: $2000$


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

A travel agency conducted a promotion: "Buy a trip to Egypt, bring four friends who also buy trips, and get your trip cost refunded." During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of these customers each brought exactly four new clients, while the remaining 100 brought no one. How many tourists went to the Land of the Pyramids for free?

Answer: $29$


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

Find the smallest positive integer which cannot be expressed in the form \(\frac{2^{a}-2^{b}}{2^{c}-2^{d}}\) where \(a, b, c, d\) are non-negative integers.

Answer: $11$


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

A regular hexagon with side length 1 has an arbitrary interior point that is reflected over the midpoints of its six sides. Calculate the area of the hexagon formed in this way.

Answer: $\frac{9\sqrt{3}}{2}$


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 that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____.

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