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

Let \( a, b, c, d, e \) be natural numbers with \( a < b < c < d < e \), and \( a + 2b + 3c + 4d + 5e = 300 \). Determine the maximum value of \( a + b \).

Answer: $35$


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

Given the parabola C: y² = 3x with focus F, and a line l with slope $\frac{3}{2}$ intersecting C at points A and B, and the x-axis at point P.
(1) If |AF| + |BF| = 4, find the equation of line l;
(2) If $\overrightarrow{AP}$ = 3$\overrightarrow{PB}$, find |AB|.

Answer: $\frac{4\sqrt{13}}{3}$


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

The maximum point of the function $f(x)=\frac{1}{3}x^3+\frac{1}{2}x^2-2x+3$ is ______.

Answer: $-2$


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

The king called two wise men. He gave the first one 100 blank cards and ordered him to write a positive number on each (the numbers do not have to be different) without showing them to the second wise man. Then, the first wise man can communicate several different numbers to the second wise man, each of which is either written on one of the cards or is the sum of the numbers on some of the cards (without specifying how each number is obtained). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be beheaded; otherwise, each will lose as many beard hairs as the numbers the first wise man communicated to the second. How can the wise men, without colluding, stay alive and lose the minimum number of hairs?

Answer: $101$


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

The sequence is defined as \( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \) for \( k=1,2, \cdots, n-1 \). Determine the value of \( n \) for which \( a_{n}=0 \).

Answer: $201$


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

Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$.

Answer: $3059$


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

In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. The value of \( x \) is

Answer: $80$


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

Let the original number be expressed as $x$. When the decimal point of $x$ is moved one place to the right, the resulting number can be expressed as $100x$. According to the given information, we have the equation $100x = x + 34.65$.

Answer: $3.85$


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

$\frac{\text{华杯赛}}{\text{少} \times \text{俊} + \text{金坛} + \text{论} \times \text{数}} = 15$

In the above equation, different Chinese characters represent different digits between $1$ and $9$. When the three-digit number "华杯赛" reaches its maximum value, please write a solution where the equation holds.

Answer: $975$


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

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

Answer: $606$


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

There are 8 seats in a row, and 3 people are sitting in the same row. If there are empty seats on both sides of each person, the number of different seating arrangements is \_\_\_\_\_\_\_\_\_.

Answer: $24$


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

You have four textbooks for mandatory courses numbered 1 to 4. They are randomly placed on the same shelf.

1. Calculate the probability that Textbook 2 is to the left of Textbook 4.
2. Calculate the probability that Textbook 2 is to the left of Textbook 3, and Textbook 3 is to the left of Textbook 4.

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


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

A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can travel to accomplish this?
A) $\sqrt{289} + 8$
B) $16 + \sqrt{185}$
C) $8 + \sqrt{545}$
D) $12 + \sqrt{400}$
E) $10 + \sqrt{365}$

Answer: $8 + \sqrt{545}$


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

Determine the base seven product of the numbers $321_7$ and $13_7$.

Answer: $4503_7$


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

Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction.

Answer: $\frac{23}{36}$


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

Determine the value of \( n \) if we know that

$$
\binom{n}{5}=\frac{n(n-1)(n-2)(n-3)(n-4)}{2 \cdot 3 \cdot 4 \cdot 5}
$$

(which, as we know, is an integer) in the decimal system is of the form \(\overline{ababa}\), where \( a \) and \( b \) represent digits.

Answer: $39$


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

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

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


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>.

A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled?

Answer: $665,280$


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

Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by 50. Determine the number of values of $n$ for which $q+r$ is divisible by 7.

Answer: $12600$


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

If $\lceil{\sqrt{x}}\rceil=17$, how many possible integer values of $x$ are there?

Answer: $33$


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

Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first 2 games, A and B each won 1 game.
(I) Calculate the probability of A winning the match;
(II) Let $\xi$ represent the number of games played from the 3rd game until the end of the match, calculate the distribution and the mathematical expectation of $\xi$.

Answer: $2.48$


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

The diagram shows three touching semicircles with radius 1 inside an equilateral triangle, with each semicircle also touching the triangle. The diameter of each semicircle lies along a side of the triangle. What is the length of each side of the equilateral triangle?

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 a regular triangular pyramid \(P-ABC\), where points \(P\), \(A\), \(B\), and \(C\) all lie on the surface of a sphere with radius \(\sqrt{3}\), and \(PA\), \(PB\), and \(PC\) are mutually perpendicular, find the distance from the center of the sphere to the cross-section \(ABC\).

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


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

A boulevard has 25 houses on each side, for a total of 50 houses. The addresses on the east side of the boulevard follow an arithmetic sequence, as do the addresses on the west side. On the east side, the addresses start at 5 and increase by 7 (i.e., 5, 12, 19, etc.), while on the west side, they start at 2 and increase by 5 (i.e., 2, 7, 12, etc.). A sign painter charges $\$1$ per digit to paint house numbers. If he paints the house number on each of the 50 houses, how much will he earn?

Answer: $113$


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

In the diagram, points \( P_1, P_3, P_5, P_7 \) are on \( BA \) and points \( P_2, P_4, P_6, P_8 \) are on \( BC \) such that \( BP_1 = P_1P_2 = P_2P_3 = P_3P_4 = P_4P_5 = P_5P_6 = P_6P_7 = P_7P_8 \). If \(\angle ABC = 5^\circ\), what is the measure of \(\angle AP_7P_8\)?

Answer: $40$


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

20. Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ passes through point $M\left( 1,\frac{3}{2} \right)$, $F_1$ and $F_2$ are the two foci of ellipse $C$, and $\left| MF_1 \right|+\left| MF_2 \right|=4$, $O$ is the center of ellipse $C$.

(1) Find the equation of ellipse $C$;

(2) Suppose $P,Q$ are two different points on ellipse $C$, and $O$ is the centroid of $\Delta MPQ$, find the area of $\Delta MPQ$.

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


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

Given the ellipse \(3x^{2} + y^{2} = 6\) and the point \(P\) with coordinates \((1, \sqrt{3})\). Find the maximum area of triangle \(PAB\) formed by point \(P\) and two points \(A\) and \(B\) on the ellipse.

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>.

Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed?

Answer: $48$


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

It is known that $F_1$ and $F_2$ are the upper and lower foci of the ellipse $C: \frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$ ($a>b>0$), where $F_1$ is also the focus of the parabola $C_1: x^{2}=4y$. Point $M$ is the intersection of $C_1$ and $C_2$ in the second quadrant, and $|MF_{1}|= \frac {5}{3}$.
(1) Find the equation of the ellipse $C_1$.
(2) Given $A(b,0)$, $B(0,a)$, the line $y=kx$ ($k>0$) intersects $AB$ at point $D$ and intersects the ellipse $C_1$ at points $E$ and $F$. Find the maximum area of the quadrilateral $AEBF$.

Answer: $2 \sqrt {6}$


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

A point $P$ is randomly placed inside the right triangle $\triangle XYZ$ where $X$ is at $(0,6)$, $Y$ is at $(0,0)$, and $Z$ is at $(9,0)$. What is the probability that the area of triangle $PYZ$ is less than half of the area of triangle $XYZ$?
[asy]
size(7cm);
defaultpen(linewidth(0.7));
pair X=(0,6), Y=(0,0), Z=(9,0), P=(2,2);
draw(X--Y--Z--cycle);
draw(Y--P--Z);
label("$X$",X,NW);
label("$Y$",Y,SW);
label("$Z$",Z,E);
label("$P$",P,N);
draw((0,0.6)--(0.6,0.6)--(0.6,0));[/asy]

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


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

It is given that there exists a unique triple of positive primes  $(p,q,r)$  such that  $p<q<r$  and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find  $r$ .

Answer: $19$


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

In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles?

Answer: $10$


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

The number of unordered pairs of edges of a given rectangular cuboid that determine a plane.

Answer: $66$


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

A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling.
(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development team;
(2) After a month of learning and discussion, the research team decided to select two employees for an experiment. The method is to first select one employee from the team to conduct the experiment, and after that, select another employee from the remaining team members to conduct the experiment. Calculate the probability that exactly one female employee is among the two selected employees;
(3) After the experiment, the first employee to conduct the experiment obtained the data 68, 70, 71, 72, 74, and the second employee obtained the data 69, 70, 70, 72, 74. Which employee's experiment is more stable? Explain your reasoning.

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


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

A metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into water that is initially at $80{ }^{\circ} \mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\circ} \mathrm{C}$. Without removing the first bar from the water, another metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into the water. What will the temperature of the water be after the new thermal equilibrium is reached?

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>.

In the diagram below, trapezoid $ABCD$ with $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$, it is given that $CD = 15$, $\tan C = 1.2$, and $\tan B = 1.8$. What is the length of $BC$?

Answer: $2\sqrt{106}$


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

Let $g$ be a function defined for all real numbers that satisfies $g(3+x) = g(3-x)$ and $g(8+x) = g(8-x)$ for all $x$. If $g(0) = 0$, determine the least number of roots $g(x) = 0$ must have in the interval $-1000 \leq x \leq 1000$.

Answer: $402$


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

There are 7 students standing in a row. How many different arrangements are there in the following situations?
(1) A and B must stand together;
(2) A is not at the head of the line, and B is not at the end of the line;
(3) There must be exactly one person between A and B.

Answer: $1200$


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

$A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After $n$ hours, B and C meet, and $n$ hours later, A catches up to D at location $C$. What is the distance between locations $B$ and $C$ in kilometers?

Answer: $30$


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

In the plane, fixed points A, B, C, D satisfy $|\overrightarrow{DA}| = |\overrightarrow{DB}| = |\overrightarrow{DC}| = 2$, $\overrightarrow{DA} \cdot \overrightarrow{BC} = \overrightarrow{DB} \cdot \overrightarrow{AC} = \overrightarrow{DC} \cdot \overrightarrow{AB} = 0$. For moving points P and M satisfying $|\overrightarrow{AP}| = 1$, $\overrightarrow{PM} = \overrightarrow{MC}$, the maximum value of $|\overrightarrow{BM}|^2$ is \_\_\_\_\_\_.

Answer: $\frac{49}{4}$


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>.

Suppose  $x,y$  and  $z$  are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate  $x^2+y^2+z^2.$

Answer: $245$


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

The diagram shows the two squares \( BCDE \) and \( FGHI \) inside the triangle \( ABJ \), where \( E \) is the midpoint of \( AB \) and \( C \) is the midpoint of \( FG \). What is the ratio of the area of the square \( BCDE \) to the area of the triangle \( ABJ \)?

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>.

Consider a square in the coordinate plane with vertices at $(2, 1)$, $(5, 1)$, $(5, 4)$, and $(2, 4)$. A line joining $(2, 3)$ and $(5, 1)$ divides the square shown into two parts. Determine the fraction of the area of the square that is above this line.

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


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

Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$, respectively. Find the ratio $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$.

Answer: $\frac{1}{16}$


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>.

Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?

*Ray Li.*

Answer: $52 - 2\sqrt{235}$


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

A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.).

Answer: $27.1$


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>.

What is the smallest possible sum of two consecutive integers whose product is greater than 420?

Answer: $43$


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

In triangle $ABC$, $AB=AC$, and $D$ is the midpoint of both $\overline{AB}$ and $\overline{CE}$. If $\overline{BC}$ is 14 units long, determine the length of $\overline{CD}$. Express your answer as a decimal to the nearest tenth.

Answer: $14.0$


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

What is the smallest positive integer with exactly 12 positive integer divisors?

Answer: $150$


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

Given α ∈ (0,π), β ∈ (-π/2,π/2) satisfies sin(α + π/3) = 1/3, cos(β - π/6) = √6/6, determine sin(α + 2β).

Answer: $\frac{2\sqrt{10}-2}{9}$


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>.

Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest point of the rotated square, to the line on which the bases of the original squares were placed.

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


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

A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

Answer: $12 + 8\sqrt{2}$


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>.

What is the total number of digits used when the first 2500 positive even integers are written?

Answer: $9444$


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

A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is at the back, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the truck; 5 minutes later, the car catches up with the bus; another $t$ minutes later, the truck catches up with the bus. Find the value of $t$.

Answer: $15$


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

A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value)

Answer: $\frac{3}{16}$


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

Given a line $l$ whose inclination angle $\alpha$ satisfies the condition $\sin \alpha +\cos \alpha = \frac{1}{5}$, determine the slope of $l$.

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 perimeter of the triangles that make up rectangle \(ABCD\) is 180 cm. \(BK = KC = AE = ED\), \(AK = KD = 17 \) cm. Find the perimeter of a rectangle, one of whose sides is twice as long as \(AB\), and the other side is equal to \(BC\).

Answer: $112$


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

Point  $P$  is located inside a square  $ABCD$  of side length  $10$ . Let  $O_1$ ,  $O_2$ ,  $O_3$ ,  $O_4$  be the circumcenters of  $P AB$ ,  $P BC$ ,  $P CD$ , and  $P DA$ , respectively. Given that  $P A+P B +P C +P D = 23\sqrt2$  and the area of  $O_1O_2O_3O_4$  is  $50$ , the second largest of the lengths  $O_1O_2$ ,  $O_2O_3$ ,  $O_3O_4$ ,  $O_4O_1$  can be written as  $\sqrt{\frac{a}{b}}$ , where  $a$  and  $b$  are relatively prime positive integers. Compute  $100a + b$ .

Answer: $5001$


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

Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm long. Calculate the time it takes Tia to reach the library in minutes.

Answer: $15.24$


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

The value of the quadratic polynomial $a(x^3 - x^2 + 3x) + b(2x^2 + x) + x^3 - 5$ when $x = 2$ is $-17$. What is the value of this polynomial when $x = -2$?

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>.

Calculate the value of $\log_{2}9 \cdot \log_{3}5 \cdot \log_{\sqrt{5}}8 = \_\_\_\_\_\_.$

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>.

Petya and Vasya are playing the following game. Petya chooses a non-negative random value  $\xi$  with expectation  $\mathbb{E} [\xi ] = 1$ , after which Vasya chooses his own value  $\eta$  with expectation  $\mathbb{E} [\eta ] = 1$  without reference to the value of  $\xi$ . For which maximal value  $p$  can Petya choose a value  $\xi$  in such a way that for any choice of Vasya's  $\eta$ , the inequality  $\mathbb{P}[\eta \geq \xi ] \leq p$  holds?

Answer: $1/2$


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

Let $x_1$, $x_2$, ..., $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is.

Answer: $61$


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

Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many pieces of string does Yuvraj obtain?

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>.

Throw a dice twice to get the numbers $a$ and $b$, respectively. What is the probability that the line $ax-by=0$ intersects with the circle $(x-2)^2+y^2=2$?

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


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>.

Kelly is attempting to unlock her electronic device with a four-digit password. She remembers that she only used digits from 1 to 6, each digit possibly being repeated, and that each odd digit must be followed by an even digit, with no specific rule for the sequences following even digits. How many combinations might Kelly need to consider?

Answer: $648$


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

A bug is on the edge of a ceiling of a circular room with a radius of 65 feet. The bug walks straight across the ceiling to the opposite edge, passing through the center of the circle. It next walks straight to another point on the edge of the circle but not back through the center. If the third part of its journey, back to the original starting point, was 100 feet long, how many total feet did the bug travel over the course of all three parts?

Answer: $313$


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

Given the function $f(x) = \sin(\omega x + \phi)$ ($\omega > 0$, $0 \leq \phi \leq \pi$) is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\sqrt{4+\pi^2}$.
(1) Find the analytical expression of the function $f(x)$.
(2) If $\sin\alpha + f(\alpha) = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan\alpha}$.

Answer: $-\frac{5}{9}$


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>.

The positive integers  $x_1, x_2, ... , x_7$  satisfy  $x_6 = 144$ ,  $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$  for  $n = 1, 2, 3, 4$ . Find  $x_7$ .

Answer: $3456$


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

Find the largest positive integer  $k$  such that  $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$  denotes the number of positive integers that are smaller than  $n$  and relatively prime to  $n$ , and  $\sigma(n)$  denotes the sum of divisors of  $n$ ). As a hint, you are given that  $641|2^{32}+1$ .

Answer: $31$


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

Let \\(f(x)=3\sin (\omega x+ \frac {\pi}{6})\\), where \\(\omega > 0\\) and \\(x\in(-\infty,+\infty)\\), and the function has a minimum period of \\(\frac {\pi}{2}\\).

\\((1)\\) Find \\(f(0)\\).

\\((2)\\) Find the expression for \\(f(x)\\).

\\((3)\\) Given that \\(f( \frac {\alpha}{4}+ \frac {\pi}{12})= \frac {9}{5}\\), find the value of \\(\sin \alpha\\).

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


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

Homer started peeling a pile of 60 potatoes at a rate of 4 potatoes per minute. Five minutes later, Christen joined him peeling at a rate of 6 potatoes per minute. After working together for 3 minutes, Christen took a 2-minute break, then resumed peeling at a rate of 4 potatoes per minute. Calculate the total number of potatoes Christen peeled.

Answer: $23$


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

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(4,3)$, respectively. What is its area?

Answer: $37.5\sqrt{3}$


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

Find all integer values of the parameter \(a\) for which the system
\[
\begin{cases}
x - 2y = y^2 + 2, \\
ax - 2y = y^2 + x^2 + 0.25a^2
\end{cases}
\]
has at least one solution. In the answer, indicate the sum of the found values of the parameter \(a\).

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>.

Five people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Determine the probability that no two adjacent people stand up.

Answer: $\frac{11}{32}$


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>.

If we exchange a 10-dollar bill into dimes and quarters, what is the total number \( n \) of different ways to have two types of coins?

Answer: $20$


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

Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\div y$ is ______.

Answer: $-2$


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

Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.

Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$?

Answer: $40 \sqrt{1925}$


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

The sum of three numbers \( a \), \( b \), and \( c \) is 150. If we increase \( a \) by 10, decrease \( b \) by 5, and multiply \( c \) by 7, the three resulting numbers are equal. What is the value of \( b \)?

Answer: $77.\overline{3}$


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 the function $f(x)$ is defined on $\mathbb{R}$ and is not identically zero, and for any real numbers $x$, $y$, it satisfies: $f(2)=2$, $f(xy)=xf(y)+yf(x)$, $a_{n}= \dfrac {f(2^{n})}{2^{n}}(n\in\mathbb{N}^{*})$, $b_{n}= \dfrac {f(2^{n})}{n}(n\in\mathbb{N}^{*})$, consider the following statements:
$(1)f(1)=1$; $(2)f(x)$ is an odd function; $(3)$ The sequence $\{a_{n}\}$ is an arithmetic sequence; $(4)$ The sequence $\{b_{n}\}$ is a geometric sequence.
The correct statements are \_\_\_\_\_\_.

Answer: $(2)(3)(4)$


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

Given 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>.

What is the smallest positive value of $x$ such that $x + 8901$ results in a palindrome?

Answer: $108$


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

Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - 9 a b}{\lambda^3} \).

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>.

Let  $ABC$  be a triangle with  $\angle BAC = 90^o$  and  $D$  be the point on the side  $BC$  such that  $AD \perp BC$ . Let $ r, r_1$ , and  $r_2$  be the inradii of triangles  $ABC, ABD$ , and  $ACD$ , respectively. If  $r, r_1$ , and  $r_2$  are positive integers and one of them is  $5$ , find the largest possible value of    $r+r_1+ r_2$ .

Answer: $30$


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

Given an infinite geometric sequence $\{a_n\}$, the product of its first $n$ terms is $T_n$, and $a_1 > 1$, $a_{2008}a_{2009} > 1$, $(a_{2008} - 1)(a_{2009} - 1) < 0$, determine the maximum positive integer $n$ for which $T_n > 1$.

Answer: $4016$


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

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

Answer: $666$


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

A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

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


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

For any positive integer  $x$ , let  $f(x)=x^x$ . Suppose that  $n$  is a positive integer such that there exists a positive integer  $m$  with  $m  \neq 1$  such that  $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of  $n$ .

*Proposed by Luke Robitaille*

Answer: $13611$


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

China's space station has entered the formal construction phase. The Tianhe core module, Wentian experimental module, and Mengtian experimental module will all dock in 2022, forming a "T" shaped structure. During the construction phase of the Chinese space station, there are 6 astronauts staying in the space station. It is expected that in a certain construction task, 6 astronauts need to work simultaneously in the Tianhe core module, Wentian experimental module, and Mengtian experimental module. Due to space limitations, each module must have at least 1 person and at most 3 people. The total number of different arrangement plans is ______.

Answer: $450$


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

Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and centroid $G$. If $2\sin A\overrightarrow{GA}+\sqrt{3}\sin B\overrightarrow{GB}+3\sin C\cdot \overrightarrow{GC}=\vec{0}$, then $\cos B=$_______.

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


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

Given a set $T = \{a, b, c, d, e, f\}$, determine the number of ways to choose two subsets of $T$ such that their union is $T$ and their intersection contains exactly three elements.

Answer: $80$


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

In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have?

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>.

For a finite sequence $B = (b_1, b_2, \dots, b_{50})$ of numbers, the Cesaro sum of $B$ is defined as
\[\frac{T_1 + \cdots + T_{50}}{50},\]
where $T_k = b_1 + \cdots + b_k$ for $1 \leq k \leq 50$.

If the Cesaro sum of the 50-term sequence $(b_1, \dots, b_{50})$ is 200, what is the Cesaro sum of the 51-term sequence $(2, b_1, \dots, b_{50})$?

Answer: $198.078431372549$


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

In the diagram, $\triangle ABF$, $\triangle BCF$, and $\triangle CDF$ are right-angled, with $\angle ABF=\angle BCF = 90^\circ$ and $\angle CDF = 45^\circ$, and $AF=36$. Find the length of $CF$.
[asy]
pair A, B, C, D, F;
A=(0,25);
B=(0,0);
C=(0,-12);
D=(12, -12);
F=(24,0);
draw(A--B--C--D--F--A);
draw(B--F);
draw(C--F);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, SE);
label("F", F, NE);
[/asy]

Answer: $36$


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

A certain scenic area has two attractions that require tickets for visiting. The three ticket purchase options presented at the ticket office are as follows:
Option 1: Visit attraction A only, $30$ yuan per person;
Option 2: Visit attraction B only, $50$ yuan per person;
Option 3: Combined ticket for attractions A and B, $70$ yuan per person.
It is predicted that in April, $20,000$ people will choose option 1, $10,000$ people will choose option 2, and $10,000$ people will choose option 3. In order to increase revenue, the ticket prices are adjusted. It is found that when the prices of options 1 and 2 remain unchanged, for every $1$ yuan decrease in the price of the combined ticket (option 3), $400$ people who originally planned to buy tickets for attraction A only and $600$ people who originally planned to buy tickets for attraction B only will switch to buying the combined ticket.
$(1)$ If the price of the combined ticket decreases by $5$ yuan, the number of people buying tickets for option 1 will be _______ thousand people, the number of people buying tickets for option 2 will be _______ thousand people, the number of people buying tickets for option 3 will be _______ thousand people; and calculate how many tens of thousands of yuan the total ticket revenue will be?
$(2)$ When the price of the combined ticket decreases by $x$ (yuan), find the functional relationship between the total ticket revenue $w$ (in tens of thousands of yuan) in April and $x$ (yuan), and determine at what price the combined ticket should be to maximize the total ticket revenue in April. What is the maximum value in tens of thousands of yuan?

Answer: $188.1$


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

Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$.

Answer: $4045$


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

Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 chicks were sitting with their beaks open and 11 were growing feathers.

a) How many chicks were sitting with their beaks open two weeks ago?

b) How many chicks will be growing feathers next week?

Record the product of these numbers as the answer.

Answer: $165$


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

Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the length of the minor axis being $2$ and the eccentricity being $\frac{\sqrt{2}}{2}$, the line $l: y = kx + m$ intersects the ellipse $C$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ passes through the point $(0, -\frac{1}{2})$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Find the maximum area of $\triangle AOB$ ($O$ is the origin).

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


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

Two cells in a \(20 \times 20\) board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a \(20 \times 20\) board such that every cell is adjacent to at most one marked cell?

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>.

James borrows $2000$ dollars from Alice, who charges an interest of $3\%$ per month (which compounds monthly). What is the least integer number of months after which James will owe more than three times as much as he borrowed?

Answer: $37$


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

If the community center has 8 cans of soup and 2 loaves of bread, with each can of soup feeding 4 adults or 7 children and each loaf of bread feeding 3 adults or 4 children, and the center needs to feed 24 children, calculate the number of adults that can be fed with the remaining resources.

Answer: $22$


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

In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \).

Answer: $6\pi$


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

What is the total number of digits used when the first 4500 positive even integers are written?

Answer: $19444$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = 1$, $| \overrightarrow{b}| = 6$, and $\overrightarrow{a} \cdot (\overrightarrow{b} - \overrightarrow{a}) = 2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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  $(x,y,z)$  be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\end{align*} If  $m$  is the minimum possible value of  $\lfloor x^3+y^3+z^3\rfloor$ , find the modulo  $2007$  residue of  $m$ .

Answer: $2004$


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>.

The pentagon \( A B C D E \) is inscribed around a circle.

The angles \( \angle A B C \), \( \angle B A E \), and \( \angle C D E \) each measure \( 104^\circ \). Find \( \angle A D B \). Provide the answer in degrees (only the number, without units).

Answer: $38$


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

A spinner with seven congruent sectors numbered from 1 to 7 is used. If Jane and her brother each spin the spinner once, and Jane wins if the absolute difference of their numbers is less than 4, what is the probability that Jane wins? Express your answer as a common fraction.

Answer: $\frac{37}{49}$


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

In $\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \sqrt{2}$, $b=2$, and $\sin B - \cos B = \sqrt{2}$, find the measure of angle $A$.

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


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

Suppose $a$, $b$, and $c$ are real numbers, and the roots of the equation \[x^4 - 10x^3 + ax^2 + bx + c = 0\] are four distinct positive integers. Compute $a + b + c.$

Answer: $109$


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

How many multiples of 4 are between 100 and 350?

Answer: $62$


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

A circle with a radius of 6 is inscribed around the trapezoid \(ABCD\). The center of this circle lies on the base \(AD\), and \(BC = 4\). Find the area of the trapezoid.

Answer: $24\sqrt{2}$


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

The Intermediate Maths Challenge has 25 questions with the following scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks.
Where no answer is given 0 marks are scored.
Fiona scored 80 marks in total. What possible answers are there to the number of questions Fiona answered correctly?

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>.

Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product by at least 2? Express your answer as a common fraction.

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


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

What is the area of the circle defined by \(x^2 - 8x + y^2 - 16y + 48 = 0\) that lies above the line \(y = 4\)?

Answer: $24\pi$


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

Given $\cos(α-β)= \frac{3}{5}$, $\sin β= -\frac{5}{13}$, and $α∈(0, \frac{π}{2})$, $β∈(-\frac{π}{2},0)$, find $\sin α$.

Answer: $\frac{33}{65}$


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

In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.

Answer: $\sqrt{3}$


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

Let  $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$.

Answer: $47$


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

The union of sets \( A \) and \( B \) is \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), \((A, B)\) and \((B, A)\) are considered different pairs. How many such pairs \((A, B)\) exist?

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>.

Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer.

Answer: $63$