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

Given the equation of a line is $Ax+By=0$, choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, and find the number of different lines obtained.

Answer: $18$


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

Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.

Answer: $5290000$


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

Given the function $f(x)=2\sin x( \sqrt {3}\cos x+\sin x)-2$.

1. If point $P( \sqrt {3},-1)$ is on the terminal side of angle $α$, find the value of $f(α)$.
2. If $x∈[0, \frac {π}{2}]$, find the minimum value of $f(x)$.

Answer: $-2$


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

Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{a} + \overrightarrow{b}|$, the cosine of the angle between $\overrightarrow{a}$ and $2\overrightarrow{a} - \overrightarrow{b}$ is ______.

Answer: $\frac{5\sqrt{7}}{14}$


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

Calculate the value of $15 \times 30 + 45 \times 15$.

Answer: $1125$


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

Xiao Ming attempts to remove all 24 bottles of beer from a box, with each attempt allowing him to remove either three or four bottles at a time. How many different methods are there for Xiao Ming to remove all the beer bottles?

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>.

Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river.

Answer: $1400$


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

Find the smallest positive integer $n$ such that for any $n$ mutually coprime integers greater than 1 and not exceeding 2009, there is at least one prime number among them.

Answer: $15$


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

Given the function $f(x)=x^{2}-2x$ where $x \in [-2,a]$. Find the minimum value of $f(x)$.

Answer: $-1$


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

Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?
- A) 35
- B) 42
- C) 48
- D) 53
- E) 82

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>.

Given a parallelogram with area  $1$  and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the  $8$  lines formed we have  a octagon inside of the parallelogram. Determine the area of this octagon

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>.

John has saved up $5555_8$ dollars for a new laptop. A laptop he desires costs $1500_{10}$ dollars. How many dollars will he have left after purchasing the laptop if all his money is in base eight and the laptop price is in base ten?

Answer: $1425$


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

As a result of five measurements of the rod's length with one device (without systematic errors), the following results (in mm) were obtained: $92, 94, 103, 105, 106$. Find: a) the sample mean length of the rod; b) the sample variance and the unbiased corrected variance of the measurement errors.

Answer: $42.5$


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

Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt{144}$ and $\angle BAC = 45^\circ,$ then compute the area of triangle $ABC.$

Answer: $72$


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

If $\{1, a, \frac{b}{a}\} = \{0, a^{2}, a+b\}$, find the value of $a^{2009} + b^{2009}$.

Answer: $-1$


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

Given an arithmetic-geometric sequence $\{a\_n\}$ with a sum of the first $n$ terms denoted as $S\_n$ and a common ratio of $\frac{3}{2}$.
(1) If $S\_4 = \frac{65}{24}$, find $a\_1$;
(2) If $a\_1=2$, $c\_n = \frac{1}{2}a\_n + bn$, and $c\_2$, $c\_4$, $c\_5$ form an arithmetic sequence, find $b$.

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 the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$.

Answer: $-6$


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>.

The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is calculated.

Answer: $266$


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

In Mr. Jacob's music class, 18 of the 27 students participated in the annual school musical. Mr. Jacob's ratio of participating to not participating students is applied to Mr. Steve's class for the regional music competition. If Mr. Steve has 45 students in total, how many students from Mr. Steve's class are expected to participate based on the given ratio?

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>.

Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped?

Answer: $\frac{1695}{4096}$


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

What is the greatest integer less than or equal to \[\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\]

Answer: $15624$


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

Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$.

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


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>.

Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction.

Answer: $\frac{1}{6}$


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

A total of 1000 senior high school students from a certain school participated in a mathematics exam. The scores in this exam follow a normal distribution N(90, σ²). If the probability of a score being within the interval (70, 110] is 0.7, estimate the number of students with scores not exceeding 70.

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>.

In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth  $1$ ,  $5$ ,  $25$ , and  $125$  quacks. How many ways can someone pay oﬀ  $125$  quacks using these coins?

*Proposed by Aaron Lin*

Answer: $82$


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

A positive integer is called *oneic* if it consists of only  $1$ 's. For example, the smallest three oneic numbers are  $1$ ,  $11$ , and  $111$ . Find the number of  $1$ 's in the smallest oneic number that is divisible by  $63$ .

Answer: $18$


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

Let  $ABC$  be a triangle with side lengths  $5$ ,  $4\sqrt 2$ , and  $7$ .  What is the area of the triangle with side lengths  $\sin A$ ,  $\sin B$ , and  $\sin C$ ?

Answer: $\frac{7}{25}$


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

Given the proportion 3:5 = 6:10, if 3 is changed to 12, determine the new value of 5.

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>.

On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Determine the quotient when $N$ is divided by 10.

Answer: $400$


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

Pedrito's lucky number is  $34117$ . His friend Ramanujan points out that  $34117 = 166^2 + 81^2 = 159^2 + 94^2$  and  $166-159 = 7$ ,  $94- 81 = 13$ . Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is  $7$  and in the difference between the seconds it gives  $13$ . Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.

Answer: $545$


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>.

The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse?

Answer: $14$


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

Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$.

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>.

A group of $10$ students from Class 9(1) of a certain school are doing "pull-up" training. The number of times they have done it is recorded in the table below. Find the median of the number of times these $10$ students have done.

| Times | $4$ | $5$ | $6$ | $7$ | $8$ |
|-------|-----|-----|-----|-----|-----|
| Number of Students | $2$ | $3$ | $2$ | $2$ | $1$ |

Answer: $5.5$


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

Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2016} \right| = \frac{1}{2016}.\]

Answer: $\frac{1007}{504}$


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

A monic polynomial of degree $n$, with real coefficients, has its first two terms after $x^n$ as $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. It’s known that $a_{n-1} = 2a_{n-2}$. Determine the absolute value of the lower bound for the sum of the squares of the roots of this polynomial.

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>.

Find the difference between $1234_5$ and $432_5$ in base $5$.

Answer: $302_5$


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

How many perfect squares less than 5000 have a ones digit of 4, 5, or 6?

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>.

How many 9-digit numbers divisible by 2 can be formed by rearranging the digits of the number 131152152?

Answer: $3360$


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

On the number line, point $A$ represents $-4$, point $B$ represents $2$. Find the expression that represents the distance between points $A$ and $B$.

Answer: $2 - (-4)$


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

Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved.

Answer: $16.67\%$


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

Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y= \frac{1}{2}x+ \frac{11}{2}$. The sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\in{N}^*)$, and $b_3=11$, with the sum of the first $9$ terms being $153$.

$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

$(2)$ Let $c_n= \frac{3}{(2a_n-11)(2b_n-1)}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Find the maximum positive integer value of $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n\in{N}^*$;

Answer: $18$


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

Calculate the difference $(2001 + 2002 + 2003 + \cdots + 2100) - (51 + 53 + 55 + \cdots + 149)$.

Answer: $200050$


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

Given the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$, where $\overrightarrow{a} = (\sqrt{3}, 2\sin \frac{\omega x}{2})$ and $\overrightarrow{b} = (\sin \omega x, -\sin \frac{\omega x}{2})$, $\omega > 0$, and the smallest positive period of $f(x)$ is $\pi$.

(1) Find the value of $\omega$;
(2) Find the minimum value of $f(x)$ and the corresponding set of values of $x$;
(3) If the graph of $f(x)$ is translated to the left by $\varphi$ units, and the resulting graph is symmetric about the point $(\frac{\pi}{3}, 0)$, find the smallest positive value of $\varphi$.

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


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

Consider the function \( y = g(x) = \frac{x^2}{Ax^2 + Bx + C} \), where \( A, B, \) and \( C \) are integers. The function has vertical asymptotes at \( x = -1 \) and \( x = 2 \), and for all \( x > 4 \), it is true that \( g(x) > 0.5 \). Determine the value of \( A + B + C \).

Answer: $-4$


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

Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \in \mathbb{R}$).

$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.

$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$.

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


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

Compute
\[
\sum_{n = 1}^\infty \frac{1}{(n+1)(n + 3)}.
\]

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>.

Given the function $f(x)=x^{5}+ax^{3}+bx-8$, if $f(-2)=10$, find the value of $f(2)$.

Answer: $-26$


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

For how many integers \(n\) with \(1 \le n \le 2020\) is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?

Answer: $337$


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

Given that \(\triangle ABC\) has sides \(a\), \(b\), and \(c\) corresponding to angles \(A\), \(B\), and \(C\) respectively, and knowing that \(a + b + c = 16\), find the value of \(b^2 \cos^2 \frac{C}{2} + c^2 \cos^2 \frac{B}{2} + 2bc \cos \frac{B}{2} \cos \frac{C}{2} \sin \frac{A}{2}\).

Answer: $64$


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

In the sequence $\{a\_n\}$, $a\_1=1$, $a\_{n+1}=3a\_n (n∈N^{})$，then $a\_3=$ _______ , $S\_5=$ _______ .

Answer: $121$


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

Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma?

Answer: $\frac{5}{24}$


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

Given the function $f\left(x\right)=\frac{2×202{3}^{x}}{202{3}^{x}+1}$, if the inequality $f(ae^{x})\geqslant 2-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.

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


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

Given that $\dfrac {\pi}{2} < \alpha < \pi$ and $\sin (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}$, find the value of $\cos (\alpha- \dfrac {\pi}{6})$.

Answer: $\dfrac {3\sqrt {3}-4}{10}$


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>.

During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each type of fruit must be transported by at least 3 cars. According to the information provided in the table below, answer the following questions:

|          | $A$  | $B$  | $C$  |
|----------|------|------|------|
| Cargo Capacity per Car (tons) | 7 | 6 | 5 |
| Profit per ton of Fruit (yuan) | 1200 | 1800 | 1500 |

$(1)$ Let the number of cars transporting fruit $A$ be $x$, and the number of cars transporting fruit $B$ be $y$. Find the functional relationship between $y$ and $x$, and determine how many arrangements of cars are possible.

$(2)$ Let $w$ represent the profit obtained from sales. How should the cars be arranged to maximize the profit from this sale? Determine the maximum value of $w$.

Answer: $198900$


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

A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southward moves, from $C$ to $D$ have 2 eastward and 1 southward move, and from $D$ to $B$ have 1 eastward and 2 southward moves. If at each intersection where the student has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south, find the probability that through any given morning, he goes through $C$ and then $D$.
A) $\frac{15}{77}$
B) $\frac{90}{462}$
C) $\frac{10}{21}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$

Answer: $\frac{15}{77}$


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

Remove all perfect squares from the sequence of positive integers $1, 2, 3, \ldots$ to obtain a new sequence, and find the 2003rd term of this new sequence.

Answer: $2048$


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

In triangle $ABC$, $AB=3$, $BC=4$, and $\angle B=60^{\circ}$. Find the length of $AC$.

Answer: $\sqrt {13}$


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

Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers.

Answer: $2201$


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

In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam?

Answer: $200$


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

Given a function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(-x) = 8 - f(4 + x)$, and a function $g(x) = \frac{4x + 3}{x - 2}$, determine the value of $(x_1 + y_1) + (x_2 + y_2) + \ldots + (x_{168} + y_{168})$ where $P_i(x_i, y_i)$ ($i = 1, 2, \ldots, 168$) are the common points of the graphs of functions $f(x)$ and $g(x)$.

Answer: $1008$


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

To obtain the graph of the function $y=2\cos \left(2x-\frac{\pi }{6}\right)$, all points on the graph of the function $y=2\sin 2x$ need to be translated $\frac{\pi }{6}$ units to the left.

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>.

Find the maximum and minimum values of the function $f(x)=\frac{1}{3}x^3-4x$ on the interval $\left[-3,3\right]$.

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


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

For non-zero real numbers \( x, y, z, w \), if
$$
\frac{6 x y + 5 y z + 6 z w}{x^{2} + y^{2} + z^{2} + w^{2}} \leq f,
$$

find the minimum value of \( f \).

Answer: $9/2$


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 $\cos (\pi+\alpha)=- \frac { \sqrt {10}}{5}$, and $\alpha\in(- \frac {\pi}{2},0)$, determine the value of $\tan \alpha$.

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


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

Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$.

Answer: $1- \dfrac { \sqrt {3}\pi}{24}$


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 sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?

Answer: $400$


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

Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}= \frac {a_n}{a_n+2}$ $(n\in\mathbb{N}^*)$, find the value of $a_{10}$.

Answer: $\frac {1}{1023}$


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

(a) A natural number $n$ is less than 120. What is the largest remainder that the number 209 can give when divided by $n$?

(b) A natural number $n$ is less than 90. What is the largest remainder that the number 209 can give when divided by $n$?

Answer: $69$


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

A man buys a house for $20,000 and wants to earn a $6\%$ annual return on his investment. He pays $650 a year in taxes and sets aside $15\%$ of each month's rent for repairs and upkeep. Determine the required monthly rent (in dollars) to meet his financial goals.
A) $165.25$
B) $172.50$
C) $181.38$
D) $190.75$
E) $200.00$

Answer: $181.38$


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>.

What is the remainder when $8x^4 - 18x^3 + 27x^2 - 14x - 30$ is divided by $4x-12$?

Answer: $333$


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

The radian measure of a 60° angle is $$\frac {\pi}{6}$$.

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>.

Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with $a > 0, b > 0$, if the four vertices of square $ABCD$ are on the hyperbola and the midpoints of $AB$ and $CD$ are the two foci of the hyperbola, determine the eccentricity of the hyperbola.

Answer: $\frac{1 + \sqrt{5}}{2}$


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

Given that points $P$ and $Q$ are on the circle $x^2 + (y-6)^2 = 2$ and the ellipse $\frac{x^2}{10} + y^2 = 1$, respectively, what is the maximum distance between $P$ and $Q$?

A) $5\sqrt{2}$
B) $\sqrt{46} + \sqrt{2}$
C) $7 + \sqrt{2}$
D) $6\sqrt{2}$

Answer: $6\sqrt{2}$


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>.

Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.

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


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

Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots $ satisfy the following conditions: $a_{1}=0$, $a_{2}=-|a+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, and so on, then the value of $a_{2022}$ is ____.

Answer: $-1011$


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

Compute $\sqrt{125}\cdot\sqrt{45}\cdot \sqrt{10}$.

Answer: $75\sqrt{10}$


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>.

Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\]

Answer: $\frac{89}{9}$


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>.

In a modified version of a walking game, I play by different rules. On the first move, I stand still, but for each subsequent move $n$ where $2 \le n \le 25$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing all 25 moves, I must return to my original starting point. How many steps is my return journey?

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>.

There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers.

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>.

What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?

Answer: $50$


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

Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\sin A=\sqrt{3}(1-\cos A)$.

$(1)$ Find $A$;

$(2)$ If $a=7$ and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$.

Answer: $10 \sqrt{3}$


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 the smallest composite number that has no prime factors less than 15.

Answer: $289$


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

Given the planar vectors $\overrightarrow{a}, \overrightarrow{b}$, with $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b}=1$, let $\overrightarrow{e}$ be a unit vector in the plane. Find the maximum value of $y=\overrightarrow{a} \cdot \overrightarrow{e} + \overrightarrow{b} \cdot \overrightarrow{e}$.

Answer: $\sqrt{7}$


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

Given the sets $A=\{x|x^2 - mx + m^2 - 19 = 0\}$, $B=\{x|x^2 - 5x + 6 = 0\}$, and $C=\{2, -4\}$. If $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$, find the value of the real number $m$.

Answer: $-2$


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

On the first day, Barry Sotter used his magic to make an object's length increase by $\frac{1}{3}$, so if the original length was $x$, it became $x + \frac{1}{3} x$. On the second day, he increased the new length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day, Barry made the object's length exactly 50 times its original length. Find the value of $n$.

Answer: $147$


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

Given the line $l_{1}: 2x + 5y = 1$ and the circle $C: x^{2} + y^{2} - 2x + 4y = 4$ with center $O_{1}$, let a moving line $l_{2}$ which is parallel to $l_{1}$ intersect the circle $C$ at points $A$ and $B$. Find the maximum value of the area $S_{\triangle ABB_{1}}$.

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>.

Find the largest prime divisor of $36^2 + 49^2$.

Answer: $3697$


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

Determine the area and the circumference of a circle with the center at the point \( R(2, -1) \) and passing through the point \( S(7, 4) \). Express your answer in terms of \( \pi \).

Answer: $10\pi \sqrt{2}$


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>.

Nine points are evenly spaced at intervals of one unit around a $3 \times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{5}$
D) $\frac{1}{6}$
E) $\frac{1}{7}$

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


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

In Mathville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Matt runs around the block on the $500$-foot side of the street, while Mike runs on the opposite side of the street. How many more feet than Matt does Mike run for every lap around the block?

Answer: $240$


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

Given $\sqrt{2 + \frac{2}{3}} = 2\sqrt{\frac{2}{3}}, \sqrt{3 + \frac{3}{8}} = 3\sqrt{\frac{3}{8}}, \sqrt{4 + \frac{4}{15}} = 4\sqrt{\frac{4}{15}}\ldots$, if $\sqrt{6 + \frac{a}{b}} = 6\sqrt{\frac{a}{b}}$ (where $a,b$ are real numbers), please deduce $a = \_\_\_\_$, $b = \_\_\_\_$.

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 a "double-single" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000.

Answer: $81$


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

In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$.

Answer: $145$


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

Let $x,$ $y,$ and $z$ be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of $x^2 + y^2 + z^2.$

Answer: $\frac{40}{7}$


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>.

On a table there are  $100$  red and  $k$  white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of  $k$ .

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>.

On a table near the sea, there are  $N$  glass boxes where  $N<2021$ , each containing exactly  $2021$  balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let  $S$  be the sum of all values of  $N$  for which Sowdha has a winning strategy and let  $R$  be the sum of all values of  $N$  for which Rafi has a winning strategy. What is the value of  $\frac{R-S}{10}$ ?

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>.

Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$.

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


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>.

Read the following problem-solving process:
The first equation: $\sqrt{1-\frac{3}{4}}=\sqrt{\frac{1}{4}}=\sqrt{(\frac{1}{2})^2}=\frac{1}{2}$.
The second equation: $\sqrt{1-\frac{5}{9}}=\sqrt{\frac{4}{9}}=\sqrt{(\frac{2}{3})^2}=\frac{2}{3}$;
The third equation: $\sqrt{1-\frac{7}{16}}=\sqrt{\frac{9}{16}}=\sqrt{(\frac{3}{4})^2}=\frac{3}{4}$;
$\ldots $
$(1)$ According to the pattern you discovered, please write down the fourth equation: ______
$(2)$ According to the pattern you discovered, please write down the nth equation (n is a positive integer): ______;
$(3)$ Using this pattern, calculate: $\sqrt{(1-\frac{3}{4})×(1-\frac{5}{9})×(1-\frac{7}{16})×⋯×(1-\frac{21}{121})}$;

Answer: $\frac{1}{11}$


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

According to the Shannon formula $C=W\log_{2}(1+\frac{S}{N})$, if the bandwidth $W$ is not changed, but the signal-to-noise ratio $\frac{S}{N}$ is increased from $1000$ to $12000$, then find the approximate percentage increase in the value of $C$.

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>.

Expand $-(4-c)(c+2(4-c) + c^2)$. What is the sum of the coefficients of the expanded form?

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>.

Compute: \(103 \times 97\).

Answer: $9991$


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

A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated two independent times, obtaining two real numbers $x$ and $y$. What is the probability that $|x-y| > \frac{1}{2}$?

Answer: $\frac{7}{16}$


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 $cos({α-\frac{π}{6}})=\frac{1}{3}$, find $sin({2α+\frac{π}{6}})$.

Answer: $-\frac{7}{9}$


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 Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics books together.

Answer: $25920$


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

The number \(abcde\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \(edcba\), which is the reverse of \(abcde\). What is the value of \(a + b + c + d + e\)?

Answer: $27$


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

Given a plane $\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:
(1) If $m \parallel \alpha$ and $n \subseteq \alpha$, then $m \parallel n$.
(2) If $m \parallel \alpha$ and $n \parallel \alpha$, then $m \parallel n$.
(3) If $m \parallel n$ and $n \subseteq \alpha$, then $m \parallel \alpha$.
(4) If $m \parallel n$ and $m \parallel \alpha$, then $n \parallel \alpha$ or $n \subseteq \alpha$.
Identify which of the above propositions are correct (write the number).

Answer: $(4)$


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

Given the set $M=\{m\in \mathbb{Z} | x^2+mx-36=0 \text{ has integer solutions}\}$, a non-empty set $A$ satisfies the conditions:
(1) $A \subseteq M$,
(2) If $a \in A$, then $-a \in A$, the number of all such sets $A$ is.

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>.

Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).

Given points:
\( A_{1}(1, -1, 1) \)

\( A_{2}(-2, 0, 3) \)

\( A_{3}(2, 1, -1) \)

\( A_{4}(2, -2, -4) \)

Answer: $\frac{33}{\sqrt{101}}$


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>.

Given the function $f(x)=\vec{a}\cdot \vec{b}$, where $\vec{a}=(2\cos x,\sqrt{3}\sin 2x)$, $\vec{b}=(\cos x,1)$, and $x\in \mathbb{R}$.

(Ⅰ) Find the period and the intervals of monotonic increase for the function $y=f(x)$;

(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $f(A)=2$, $a=\sqrt{7}$, and $\sin B=2\sin C$. Calculate the area of $\triangle ABC$.

Answer: $\frac{7\sqrt{3}}{6}$


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

Calculate the value of $$1+\cfrac{2}{3+\cfrac{6}{7}}$$ and express it as a simplified fraction.

Answer: $\frac{41}{27}$


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

Given a geometric sequence $a$, $a(a-1)$, $a(a-1)^{2}$, $...$, where $S_{n}$ represents the sum of its first $n$ terms.
1) Find the range of real values for $a$ and the expression for $S_{n}$;
2) Does there exist a real value of $a$ such that $S_{1}$, $S_{3}$, $S_{2}$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain the reason.

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>.

From the $10$ numbers $0-9$, select $3$ numbers. Find:

$(1)$ How many unique three-digit numbers can be formed without repeating any digits?

$(2)$ How many unique three-digit odd numbers can be formed without repeating any digits?

Answer: $320$


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

Determine the number of real number  $a$ , such that for every  $a$ , equation  $x^3=ax+a+1$  has a root  $x_0$  satisfying following conditions:
(a)  $x_0$  is an even integer;
(b)  $|x_0|<1000$ .

Answer: $999$


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

Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\).

Answer: $18$


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

What is the minimum number of equilateral triangles, each of side length 1 unit, needed to completely cover an equilateral triangle of side length 15 units?

Answer: $225$


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

In the expression $(1+x)^{56}$, the parentheses are expanded and like terms are combined. Find the coefficients of $x^8$ and $x^{48}$.

Answer: $\binom{56}{8}$


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

A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and at least one open-ended question are selected is __________.

Answer: $864$


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

Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B.

Answer: $24$


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

Given \\(a > 0\\), \\(b > 0\\), and \\(a+4b={{(ab)}^{\\frac{3}{2}}}\\).

\\((\\)I\\()\\) Find the minimum value of \\(a^{2}+16b^{2}\\);

\\((\\)II\\()\\) Determine whether there exist \\(a\\) and \\(b\\) such that \\(a+3b=6\\), and explain the reason.

Answer: $32$


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

Given that $||\overrightarrow{OA}||=||\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\overrightarrow{OC}||$ is $1$, find the minimum value of $||\overrightarrow{OA}-t\overrightarrow{OB}||$.

Answer: $\sqrt{3}$


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

In a two-day problem-solving tournament, Alpha and Gamma both attempted questions worth a total of 600 points. Alpha scored 210 points out of 350 points on the first day, and 150 points out of 250 points on the second day. Gamma, who did not attempt 350 points on the first day, had a positive integer score on each of the two days, and Gamma's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{360}{600} = 3/5$.

Find the largest possible two-day success ratio that Gamma could have achieved.

Answer: $\frac{359}{600}$


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

$ABCDEFGH$ is a cube. Find $\sin \angle HAD$.

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>.

In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$, calculate the spherical distance between points $A$ and $C$.

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


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

What is the coefficient of $x^5$ when $$2x^5 - 4x^4 + 3x^3 - x^2 + 2x - 1$$ is multiplied by $$x^3 + 3x^2 - 2x + 4$$ and the like terms are combined?

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>.

Let \( m \in \mathbb{N} \), and let the integer part of \( \log_2 m \) be denoted as \( f(m) \). Calculate the value of \( f(1) + f(2) + \cdots + f(1024) \).

Answer: $8204$


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

Given the function $f(x)=2\sin (\\omega x)$, where $\\omega > 0$.

(1) When $ \\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{2})$ and explain the reason.

(2) When $ \\omega =2$, the graph of the function $y=f(x)$ is translated to the left by $ \\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{6}$ unit, and then translated upward by 1 unit to obtain the graph of the function $y=g(x)$. Find all possible values of the number of zeros of $y=g(x)$ in the interval $[a,a+10π]$ for any $a∈R$.

Answer: $20$


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

Find the number of digit of  $\sum_{n=0}^{99} 3^n$ .
You may use  $\log_{10} 3=0.4771$ .

2012 Tokyo Institute of Technology entrance exam, problem 2-A

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>.

On the same road, two trucks are driving in the same direction. Initially, Truck A is 4 kilometers ahead of Truck B. The speed of Truck A is 45 kilometers per hour, and the speed of Truck B is 60 kilometers per hour. How far apart are the two trucks 1 minute before Truck B catches up to Truck A, in meters?

Answer: $250$