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

Given that $\overrightarrow{e}$ is a unit vector, $|\overrightarrow{a}|=4$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{e}$ is $\frac{2}{3}π$, find the projection of $\overrightarrow{a}$ onto $\overrightarrow{e}$.

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>.

A certain shopping mall's sales volume (unit: ten thousand pieces) for five consecutive years is as shown in the table below:

| $x$(year) | 1 | 2 | 3 | 4 | 5 |
|-----------|---|---|---|---|---|
| $y$(sales volume) | 5 | 5 | 6 | 7 | 7 |

$(1)$ Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ for the sales volume $y$ and the corresponding year $x$;
$(2)$ If two sets of data are randomly selected from the five sets, find the probability that the two selected sets are exactly the data from two consecutive years.
Note: In the linear regression equation $y=bx+a$, $\hat{b}=\frac{\sum_{i=1}^{n}{{x}_{i}{y}_{i}-n\overline{x}\overline{y}}}{\sum_{i=1}^{n}{{x}_{i}^{2}-n\overline{{x}}^{2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$. Where $\overline{x}$ and $\overline{y}$ are sample averages, and the linear regression equation can also be written as $\hat{y}=\hat{b}x+\hat{a}$.

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


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

In the rectangular coordinate system (xOy), a pole coordinate system is established with O as the pole and the positive semi-axis of x as the polar axis. The shortest distance between a point on the curve C: ρcosθ - ρsinθ = 1 and a point on the curve M: x = -2 + cosφ, y = 1 + sinφ (φ is a parameter) can be calculated.

Answer: $2\sqrt{2}-1$


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

If  $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$  can be written in the form  $a^b$  for positive integers  $a,b$  with  $a$  squarefree, find  $a+b$ .

*Proposed by Deyuan Li and Andrew Milas*

Answer: $2018$


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

A market survey shows that the price $f(t)$ and sales volume $g(t)$ of a particular product in Oriental Department Store over the past month (calculated based on 30 days) approximately satisfy the functions $f(t)=100(1+ \frac {1}{t})$ and $g(t)= \begin{cases} 100+t & 1\leqslant t < 25,t\in N \\ 150-t & 25\leqslant t\leqslant 30,t\in N \end{cases}$, respectively.
(1) Find the daily sales revenue $W(t)$ of the product in terms of time $t (1\leqslant t\leqslant 30,t\in N)$;
(2) Calculate the maximum and minimum daily sales revenue $W(t)$.

Answer: $12100$


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

On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:

| Number of Trees Planted | 5   | 6   | 7   |
|--------------------------|-----|-----|-----|
| Number of Groups         | 3   | 4   | 3   |

The variance of the number of trees planted by these 10 groups is ______.

Answer: $0.6$


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>.

Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes?

Answer: $26$


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

Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle.

Answer: $\frac {1}{10}$


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

In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, the area of mid-to-low-priced houses built each year will increase by 500,000 square meters compared to the previous year. So, by the end of which year:

$(1)$ The cumulative area of mid-to-low-priced houses built in our county over the years (with 2019 as the first cumulative year) will first exceed 22.5 million square meters?

$(2)$ The proportion of the area of mid-to-low-priced houses built in that year to the total area of housing built in that year will first exceed 85%? (Reference data: $1.08^{4}\approx 1.36$, $1.08^{5}\approx 1.47$)

Answer: $2024$


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

The lateral surface area of a circular truncated cone is given by the formula, find the value for the lateral surface area of the cone where the radii of the upper and lower bases are $r=1$ and $R=4$ and the height is $4$.

Answer: $25\pi$


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

Two different numbers are selected from 1, 2, 3, 4, 5 to form a point (x, y). Find the probability that this point lies above the line x+y-5=0.

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


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

Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)?

Answer: $165$


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

In trapezoid \(ABCD\), the side \(AD\) is perpendicular to the bases and is equal to 9. \(CD\) is 12, and the segment \(AO\), where \(O\) is the point of intersection of the diagonals of the trapezoid, is equal to 6. Find the area of triangle \(BOC\).

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


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

Given the sequence 1, $\frac{1}{2}$, $\frac{2}{1}$, $\frac{1}{3}$, $\frac{2}{2}$, $\frac{3}{1}$, $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$, $\frac{4}{1}$, ..., then $\frac{3}{5}$ is the \_\_\_\_\_\_ term of this sequence.

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>.

What is the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}?\]

Answer: $3124$


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

If person A has either a height or weight greater than person B, then person A is considered not inferior to person B. Among 100 young boys, if a person is not inferior to the other 99, he is called an outstanding boy. What is the maximum number of outstanding boys among the 100 boys?

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>.

Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?

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>.

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

Answer: $336$


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

Our school's basketball team has won the national middle school basketball championship multiple times! In one competition, including our school's basketball team, 7 basketball teams need to be randomly divided into two groups (one group with 3 teams and the other with 4 teams) for the group preliminaries. The probability that our school's basketball team and the strongest team among the other 6 teams end up in the same group is ______.

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


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

Given that the radius of the base of a cone is $2$ and the area of the unfolded side of the cone is $8\pi$, find the volume of the inscribed sphere in the cone.

Answer: $\frac{32\sqrt{3}}{27}\pi$


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

Using the numbers $1$, $2$, $3$, $4$ to form a four-digit number without repeating digits, the number of four-digit numbers larger than $2134$ is _____. (Answer in digits)

Answer: $17$


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

Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$.

Answer: $17.28125$


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

What is the smallest positive value of $m$ so that the equation $18x^2 - mx + 252 = 0$ has integral solutions?

Answer: $162$


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

When Lisa squares her favorite  $2$ -digit number, she gets the same result as when she cubes the sum of the digits of her favorite  $2$ -digit number. What is Lisa's favorite  $2$ -digit number?

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>.

The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).

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>.

Isaac repeatedly flips a fair coin. Whenever a particular face appears for the  $2n+1$ th time, for any nonnegative integer  $n$ , he earns a point. The expected number of flips it takes for Isaac to get  $10$  points is  $\tfrac ab$  for coprime positive integers  $a$  and  $b$ . Find  $a + b$ .

*Proposed by Isaac Chen*

Answer: $201$


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

Six chairs are placed in a row. Find the number of ways 3 people can sit randomly in these chairs such that no two people sit next to each other.

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>.

Determine the value of $k$ such that the equation
\[\frac{x + 3}{kx - 2} = x\] has exactly one solution.

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


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

What is the product of the digits in the base 8 representation of $9876_{10}$?

Answer: $96$


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

Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updating Sam's total to two coins. This continues until Bob says, ``$x$ for me, $x$ for you,'' at which he takes $x$ more coins and makes Sam's total $x$ coins in total. After all coins are distributed, Pirate Bob has exactly three times as many coins as Pirate Sam. Find out how many gold coins they have between them?

Answer: $20$


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

Given triangle $\triangle ABC$ with $\cos C = \frac{2}{3}$, $AC = 4$, and $BC = 3$, calculate the value of $\tan B$.

Answer: $4\sqrt{5}$


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

While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments  $AB = 10$  and  $CD = 7$  on a plane. Across all possible configurations of  those two segments, determine the maximum possible area of quadrilateral  $ACBD$ .

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 triangle  $ ABC$ . Point  $ O$  is the center of the excircle touching the side  $ BC$ . Point  $ O_1$  is the reflection of  $ O$  in  $ BC$ . Determine angle  $ A$  if  $ O_1$  lies on the circumcircle of  $ ABC$ .

Answer: $60$


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

Convert the binary number $1101100_{(2)}$ to a decimal number.

Answer: $108$


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

The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15.

Answer: $3.75$


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

Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$.

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>.

Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$.

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>.

Read the following material: The overall idea is a common thinking method in mathematical problem solving: Here is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Regard "$x^{2}+2x$" as a whole, let $x^{2}+2x=y$, then the original expression $=y^{2}+2y+1=\left(y+1\right)^{2}$, and then restore "$y$".

**Question:**
(1) ① The student's factorization result is incorrect, please write down the correct result directly ______;
② According to material $1$, please try to imitate the above method to factorize the polynomial $(x^{2}-6x+8)(x^{2}-6x+10)+1$;
(2) According to material $1$, please try to imitate the above method to calculate:
$(1-2-3-\ldots -2020)\times \left(2+3+\ldots +2021\right)-\left(1-2-3-\ldots -2021\right)\times \left(2+3+\ldots +2020\right)$.

Answer: $2021$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 heights on the two equal sides of an isosceles triangle is equal to the height on the base. Find the sine of the base angle.

Answer: $\frac{\sqrt{15}}{4}$


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>.

Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that results in 24 rows, with the rows alternating between 7 and 8 pipes.

After the crates have been packed with an equal number of 200 pipes each, what is the positive difference in the total heights (in cm) of the two packings?

Answer: $200 - 96\sqrt{3}$


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 three-digit natural number $abc$ is termed a "convex number" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\{5, 6, 7, 8, 9\}$ and are distinct from one another, find the probability that a randomly chosen three-digit number $abc$ is a "convex number".

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>.

A rectangular garden needs to be enclosed on three sides using a 70-meter rock wall as one of the sides. Fence posts are placed every 10 meters along the fence, including at the ends where the fence meets the rock wall. If the area of the garden is 2100 square meters, calculate the fewest number of posts required.

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>.

Which of the following is equal to $2017 - \frac{1}{2017}$?

Answer: $\frac{2018 \times 2016}{2017}$


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

In how many ways can the number 5 be expressed as the sum of one or more positive integers?

Answer: $16$


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

A line passes through the distinct vectors $\mathbf{u}$ and $\mathbf{v}.$ For some value of $k$, the vector
\[k \mathbf{u} + \frac{5}{8} \mathbf{v}\] must also lie on the line. Find $k$.

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


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

Given the random variable $X$ follows a normal distribution $N(-1, \sigma^2)$, and $P(-3 \leq X \leq -1) = 0.4$, calculate the probability of $X$ being greater than or equal to $1$.

Answer: $0.1$


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>.

Using the digits $1$, $2$, $3$, $5$, and $6$ exactly once, the five-digit positive integers are formed and arranged in ascending order. What is the $60^{\text{th}}$ integer in this list?

Answer: $32651$


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

Given the set $\{a,b,c\}=\{1,2,3\}$, and three relations: $①a\neq 3$, $②b=3$, $③c\neq 1$. Only one of these relations is correct. Find the value of $100a+10b+c=\_\_\_\_\_\_.$

Answer: $312$


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

Given that the right focus of ellipse $I$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.
(1) Find the equation of ellipse $I$;
(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $A$ and $B$. Construct an isosceles triangle $PAB$ with base $AB$ and vertex $P$ at coordinates $(-3,2)$. Find the area of $∆PAB$.

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>.

A sphere is inside a cube with an edge length of $3$, and it touches all $12$ edges of the cube. Find the volume of the sphere.

Answer: $9\sqrt{2}\pi$


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

What is the total area of two overlapping circles where circle A has center at point $A(2, -1)$ and passes through point $B(5, 4)$, and circle B has center at point $C(3, 3)$ and passes through point $D(5, 8)$? Express your answer in terms of $\pi$.

Answer: $63\pi$


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

The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$.

Answer: $16384$


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

Given a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. One of its asymptotes is $x+\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\_1 \perp x$-axis. If $F\_2$ is also a focus of the parabola $y^{2}=12x$, find the distance from $F\_1$ to line $F\_2M$.

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


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>.

The congruent sides of an isosceles triangle are each 8 cm long, and the perimeter is 26 cm. In centimeters, what is the length of the base? Also, find the area of the triangle.

Answer: $5\sqrt{39}$


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

Given the geometric sequence with 8 inserted numbers between 1 and 3, find the product of these 8 inserted numbers.

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>.

During a journey, the distance read on the odometer was 450 miles. On the return trip, using snow tires for the same distance, the reading was 440 miles. If the original wheel radius was 15 inches, find the increase in the wheel radius, correct to the nearest hundredth of an inch.

Answer: $0.34$


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

A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed?

Answer: $40$


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

The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team.

Answer: $360$


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

Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. In the diagram, $AB = 34$, $BC = 35$, and $AC = 39$. Let the coordinates be:
- $A = (-17,0)$,
- $B = (17,0)$,
- $C = (0,30)$.
Express your answer as a common fraction.

Answer: $\frac{35}{39}$


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

A triangular pyramid stack of apples is formed with a base consisting of 6 apples on each side. Each higher layer is formed by 3 apples per row, with the top layer consisting of a single apple. Calculate the total number of apples used in the entire stack.

Answer: $56$


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

Given the vectors $\overrightarrow{a}=(\cos 40^{\circ},\sin 40^{\circ})$, $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, and $\overrightarrow{u}= \sqrt {3} \overrightarrow{a}+λ \overrightarrow{b}$ (where $λ∈R$), find the minimum value of $|\overrightarrow{u}|$.

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


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

Adnan is trying to remember his four-digit PIN. He is sure it contains the digits 5, 3, 7, and 0 but can't recall the order in which they appear. How many different arrangements are possible for his PIN?

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>.

What is the largest five-digit number whose digits add up to 20?

Answer: $99200$


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

The angle that has the same terminal side as $- \frac{\pi}{3}$ is $\frac{\pi}{3}$.

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


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

A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum?

Answer: $13$


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

Given a sequence $\{a_n\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \frac{S_1 + S_2 + \dots + S_n}{n}$ as the "ideal number" of the sequence $a_1, a_2, \dots, a_n$. If the "ideal number" of the sequence $a_1, a_2, \dots, a_{502}$ is $2012$, calculate the "ideal number" of the sequence $2, a_1, a_2, \dots, a_{502}$.

Answer: $2010$


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

Given two vectors in the plane, $\mathbf{a} = (2m+1, 3)$ and $\mathbf{b} = (2, m)$, and $\mathbf{a}$ is in the opposite direction to $\mathbf{b}$, calculate the magnitude of $\mathbf{a} + \mathbf{b}$.

Answer: $\sqrt{2}$


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

All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?

Answer: $156$


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

Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______.

Answer: $84$


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

What time is it 2017 minutes after 20:17?

Answer: $05:54$


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

$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$  is a function that satisfies   $$ g(x)+g(y)=g(x+y)-xy. $$   If  $g(23)=0$ , what is the sum of all possible values of  $g(35)$ ?

Answer: $210$


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

A note contains three two-digit numbers that are said to form a sequence with a fourth number under a cryptic condition. The numbers provided are 46, 19, and 63, but the fourth number is unreadable. You know that the sum of the digits of all four numbers is $\frac{1}{4}$ of the total sum of these four numbers. What is the fourth number?

Answer: $28$


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

Given $f(\alpha)=\frac{2\sin(2\pi-\alpha)\cos(2\pi+\alpha)-\cos(-\alpha)}{1+\sin^{2}\alpha+\sin(2\pi+\alpha)-\cos^{2}(4\pi-\alpha)}$, find the value of $f\left(-\frac{23}{6}\pi \right)$.

Answer: $-\sqrt{3}$


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

Let $P$ be the centroid of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. The area of triangle $ABC$ is 24. Find the area of triangle $G_1 G_2 G_3$.

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


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

Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:
- Less than 1 year: 3 marks
- 1 to less than 2 years: 6 marks
- 2 to less than 3 years: 5 marks
- 3 to less than 4 years: 4 marks
- 4 to less than 5 years: 2 marks
- 5 to less than 6 years: 2 marks
- 6 to less than 7 years: 3 marks
- 7 to less than 8 years: 2 marks
- 8 to less than 9 years: 1 mark
- 9 to less than 10 years: 1 mark
Determine what percent of the employees have worked there for $6$ years or more.

Answer: $24.14\%$


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

Let  $ABC$  be a right triangle with a right angle at  $C.$  Two lines, one parallel to  $AC$  and the other parallel to  $BC,$  intersect on the hypotenuse  $AB.$  The lines split the triangle into two triangles and a rectangle. The two triangles have areas  $512$  and  $32.$  What is the area of the rectangle?

*Author: Ray Li*

Answer: $256$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $\left\{a_{n}\right\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then $S_{40}$ is equal to:

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>.

John has saved up $5235_9$ dollars for a trip to Japan. A round-trip airline ticket costs $1250_8$ dollars. In base ten, how many dollars will he have left for lodging and food?

Answer: $3159$


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

What is the product of all real numbers that are tripled when added to their reciprocals?

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>.

Calculate: $|\sqrt{8}-2|+(\pi -2023)^{0}+(-\frac{1}{2})^{-2}-2\cos 60^{\circ}$.

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


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

Given the function $y=\sin (3x+ \frac {\pi}{3})\cos (x- \frac {\pi}{6})+\cos (3x+ \frac {\pi}{3})\sin (x- \frac {\pi}{6})$, find the equation of one of the axes of symmetry.

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>.

Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______.

Answer: $5 - 2\sqrt{3}$


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

Find the square root of $\dfrac{10!}{210}$.

Answer: $24\sqrt{30}$


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

In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ).

Answer: $19$


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

In the multiplication shown, $P, Q,$ and $R$ are all different digits such that

$$
\begin{array}{r}
P P Q \\
\times \quad Q \\
\hline R Q 5 Q
\end{array}
$$

What is the value of $P + Q + R$?

Answer: $17$


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

Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place?

Answer: $448$


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

Encrypt integers by the following method: the digit of each number becomes the units digit of its product with 7, then replace each digit _a_ with $10 - _a_$. If a number is encrypted by the above method and becomes 473392, then the original number is ______.

Answer: $891134$


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

A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?

Answer: $115$


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

A rhombus  $ABCD$  is given with  $\angle BAD = 60^o$  . Point  $P$  lies inside the rhombus such that  $BP = 1$ ,  $DP = 2$ ,  $CP = 3$ . Determine the length of the segment  $AP$ .

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>.

Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and at the segments, the products of the numbers at the endpoints were placed. Then, the sum of the numbers on the red segments was subtracted by the sum of the numbers on the blue segments. What is the largest possible value that could be obtained?

I. Bogdanov

Answer: $1/2$


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

Six cubes, each an inch on an edge, are fastened together, as shown.  Find the total surface area in square inches.  Include the top, bottom and sides.

[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy]

Answer: $26$


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

Given the function f(x) = $\sqrt {2}$sin $\frac {x}{2}$cos $\frac {x}{2}$ - $\sqrt {2}$sin2 $\frac {x}{2}$,
(1) Find the smallest positive period of f(x);
(2) Find the minimum value of f(x) in the interval [-π, 0].

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


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>.

Given the function $y=4\cos (2x+\frac{\pi}{4})$, determine the direction and magnitude of horizontal shift required to obtain the graph of the function $y=4\cos 2x$.

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


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>.

Evaluate the value of $3^2 \times 4 \times 6^3 \times 7!$.

Answer: $39191040$


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

Given that $\tan α$ and $\tan β$ are the roots of the equation $x^{2}+3 \sqrt {3}x+4=0$, and $\(- \frac {π}{2} < α < \frac {π}{2}\)$, $\(- \frac {π}{2} < β < \frac {π}{2}\)$, find $α+β$.

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


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

In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\frac{|MO|}{|MF|}$.

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


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

If the line passing through the point $P(2,1)$ intersects the graph of the function $f(x)= \frac{2x+3}{2x-4}$ at points $A$ and $B$, and $O$ is the origin, calculate the value of $( \overrightarrow{OA}+ \overrightarrow{OB})\cdot \overrightarrow{OP}$.

Answer: $10$


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

Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$

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


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 Square $ABCD$ has side length $5$, point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$, calculate the degree measure of $\angle AMD$.

Answer: $45$


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

Plane M is parallel to plane N. There are 3 different points on plane M and 4 different points on plane N. The maximum number of tetrahedrons with different volumes that can be determined by these 7 points is ____.

Answer: $34$


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

Let  $ABC$  be an acute triangle. Let $H$  and  $D$  be points on  $[AC]$  and  $[BC]$ , respectively, such that  $BH \perp AC$  and  $HD \perp BC$ . Let  $O_1$  be the circumcenter of  $\triangle ABH$ , and  $O_2$  be the circumcenter of  $\triangle BHD$ , and  $O_3$  be the circumcenter of  $\triangle HDC$ . Find the ratio of area of  $\triangle O_1O_2O_3$  and  $\triangle ABH$ .

Answer: $1/4$


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>.

The volume of the box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$?

Answer: $12$


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

Given that \\(AB\\) is a chord passing through the focus of the parabola \\(y^{2} = 4\sqrt{3}x\\), and the midpoint \\(M\\) of \\(AB\\) has an x-coordinate of \\(2\\), calculate the length of \\(AB\\.

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


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>.

In triangle $ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$.

Answer: $\frac{50}{3}$


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 hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$.

Answer: $36$


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

A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)?

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>.

Given that the leftmost position can be occupied by student A or B, and the rightmost position cannot be occupied by student A, find the number of different arrangements of the six high-performing students from Class 1, Grade 12.

Answer: $216$


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

A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys.
A) $20$ inches
B) $10\sqrt{10}$ inches
C) $40$ inches
D) $50$ inches

Answer: $10\sqrt{10}$


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>.

Given that p, q, r, and s are integers in the set {0, 1, 2, 3, 4}, calculate the number of ordered quadruples (p, q, r, s) such that p·s + q·r is odd.

Answer: $168$


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

A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder.

Answer: $16\pi$


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

Two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are given, and $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$. The cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________.

Answer: $\frac{\sqrt{10}}{5}$


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 number of 0-1 binary sequences formed by six 0's and six 1's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not.

Answer: $357$


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

(1) Find the value of $\cos\frac{5\pi}{3}$.

(2) Given that $\frac{\sin\alpha + 2\cos\alpha}{5\cos\alpha - \sin\alpha} = \frac{5}{16}$, find the value of $\tan\alpha$.

(3) Given that $\sin\theta = \frac{1}{3}$ and $\theta \in (0, \frac{\pi}{2})$, find the value of $\tan 2\theta$.

(4) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Moreover, $a\sin A\cos C + c\sin A\cos A = \frac{1}{3}c$, $D$ is the midpoint of $AC$, and $\cos B = \frac{2\sqrt{5}}{5}$, $BD = \sqrt{26}$. Find the length of the shortest side of triangle $ABC$.

Answer: $2\sqrt{2}$


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

Let  $AB$  be a diameter of a circle and let  $C$  be a point on the segement  $AB$  such that  $AC : CB = 6 : 7$ . Let  $D$  be a point on the circle such that  $DC$   is perpendicular to  $AB$ . Let  $DE$  be the diameter through  $D$ . If  $[XYZ]$  denotes the area of the triangle  $XYZ$ , find  $[ABD]/[CDE]$  to the nearest integer.

Answer: $13$


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

Let  $ABC$  be a triangle (right in  $B$ ) inscribed in a semi-circumference of diameter  $AC=10$ . Determine the distance of the vertice  $B$  to the side  $AC$  if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.

Answer: $5/2$


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>.

The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$.

Answer: $180\pi$


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

If physical education is not the first class, and Chinese class is not adjacent to physics class, calculate the total number of different scheduling arrangements for five subjects - mathematics, physics, history, Chinese, and physical education - on Tuesday morning.

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>.

Suppose  $xy-5x+2y=30$ , where  $x$  and  $y$  are positive integers. Find the sum of all possible values of  $x$

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>.

A cinema has 21 rows of seats, with 26 seats in each row. How many seats are there in total in this cinema?

Answer: $546$


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

Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$.

Answer: $14 - 2\sqrt{13}$


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>.

Convert the following expressions between different bases: $110011_{(2)} = \_{(10)} = \_{(5)}$

Answer: $51_{(10)} = 201_{(5)}$


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>.

Determine how many non-similar regular 500-pointed stars exist, given that a regular $n$-pointed star adheres to the rules set in the original problem description.

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>.

The line \(y = -\frac{1}{2}x + 8\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Point \(T(r, s)\) is on line segment \(PQ\). If the area of \(\triangle POQ\) is twice the area of \(\triangle TOP\), then what is the value of \(r+s\)?

Answer: $12$


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

Given that $(a+1)x - 1 - \ln x \leqslant 0$ holds for any $x \in [\frac{1}{2}, 2]$, find the maximum value of $a$.

Answer: $1 - 2\ln 2$


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

Given the ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$, let ${F}_{1}$ and ${F}_{2}$ be its left and right foci, respectively. A line $l$ passing through point ${F}_{2}$ with a slope of $1$ intersects ellipse $C$ at two distinct points $M$ and $N$. Calculate the area of triangle $MN{F}_{1}$.

Answer: $\frac{12\sqrt{2}}{7}$


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

If the complex number $z=(2-i)(a-i)$ (where $i$ is the imaginary unit) is a pure imaginary number, then the value of the real number $a$ is \_\_\_\_\_\_.

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


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

In the geometric sequence $\{a_n\}$, if $a_n > a_{n+1}$, and $a_7 \cdot a_{14} = 6, a_4 + a_{17} = 5$, calculate $\frac{a_5}{a_{18}}$.

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


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

Given the function $f(x)= \begin{cases} a+\ln x,x > 0 \\ g(x)-x,x < 0\\ \end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$.

Answer: $-1-e$