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

6 books of different types are to be divided into 3 groups, one group containing 4 books and the other two groups containing 1 book each. Calculate the number of different ways this can be done.

Answer: $15$


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

Given $f(x) = 2\sin x (\sin x + \cos x)$, where $x \in \mathbb{R}$.

(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.

(Ⅱ) If $f\left( \frac{a}{2} \right) = 1 + \frac{3\sqrt{2}}{5}$, and $\frac{3\pi}{4} < a < \frac{5\pi}{4}$, find the value of $\cos a$.

Answer: $-\frac{7\sqrt{2}}{10}$


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>.

Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$.

Answer: $140$


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

What is the ratio of the area of the shaded triangle to the area of the square? The square is divided into a 5x5 grid of smaller, equal-sized squares. A triangle is shaded such that it covers half of a square at the center of the grid and three full squares adjacent to this half-covered square. The vertices of the triangle touch the midpoints of the sides of the squares it covers.

Answer: $\frac{7}{50}$


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

In a school event, there are 360 students participating. They need to be arranged in rows such that each row contains the same number of students. The arrangement must have at least 12 rows and at least 18 students in each row. Calculate the sum of all possible numbers of students per row.

Answer: $92$


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

Given the following definition: We call a pair of rational numbers $a$ and $b$ that satisfy the equation $a-b=ab+1$ as "companion rational number pairs," denoted as $\left(a,b\right)$. For example, $3-\frac{1}{2}=3\times \frac{1}{2}+1$, $5-\frac{2}{3}=5\times \frac{2}{3}+1$, so the pairs $(3,\frac{1}{2})$ and $(5,\frac{2}{3})$ are both "companion rational number pairs."

$(1)$ Among the pairs $(-2,\frac{1}{3})$ and $(-\frac{1}{2},-3)$, the "companion rational number pair" is ______;

$(2)$ If $\left(x+1,5\right)$ is a "companion rational number pair," then the value of $x$ is ______;

$(3)$ If $\left(a,b\right)$ is a "companion rational number pair," find the value of $3ab-a+\frac{1}{2}(a+b-5ab)+1$.

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 the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the parabola $y^{2}=8x$ share a common focus $F$, and one of the intersection points of the two curves is $P$. If $|PF|=4$, calculate the eccentricity of the hyperbola.

Answer: $\sqrt{2}+1$


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>.

In the town of Gearville, each bike license plate consists of three letters. The first letter is chosen from the set $\{B, F, G, T, Y\}$, the second from $\{E, U\}$, and the third from $\{K, S, W\}$. Gearville decided to increase the number of possible license plates by adding three new letters. These new letters can be added to one set or distributed among the sets. What is the largest possible number of ADDITIONAL license plates that can be created by optimally placing these three letters?

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>.

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

*Proposed by Andrew Wu*

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 point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\overrightarrow{MA}$ and $\overrightarrow{MB}$.

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


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>.

The Eagles beat the Falcons 3 times and the Falcons won 4 times in their initial meetings. They then played $N$ more times, and the Eagles ended up winning 90% of all the games played, including the additional games, to find the minimum possible value for $N$.

Answer: $33$


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

In an opaque bag, there are $10$ balls each of red, white, and yellow colors, all identical except for the color. At least how many balls must be drawn to ensure that two balls of different colors are drawn? At least how many balls must be drawn to ensure that two yellow balls are drawn?

Answer: $22$


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

There is a pentagon ABCDE. If the vertices A, B, C, D, E are colored with one of three colors: red, yellow, green, such that adjacent vertices are of different colors, then there are a total of     different coloring methods.

Answer: $30$


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

In $\triangle PQR$, we have $PQ = QR = 46$ and $PR = 40$. Point $M$ is the midpoint of $\overline{QR}$. Find the length of segment $PM$.

Answer: $\sqrt{1587}$


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

In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units, what is the distance from $D$ to the midpoint of segment $EF$?

Answer: $7.5$


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

Given that $\binom{23}{5}=33649$, $\binom{23}{6}=42504$, and $\binom{23}{7}=33649$, find $\binom{25}{7}$.

Answer: $152306$


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

Nine hundred forty-three minus eighty-seven equals

Answer: $856$


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

Toner Drum and Celery Hilton are both running for president. A total of  $129$  million people cast their vote in a random order, with exactly  $63$  million and  $66$  million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?

*2017 CCA Math Bonanza Individual Round #13*

Answer: $1/43$


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

Given the equation \((7+4 \sqrt{3}) x^{2}+(2+\sqrt{3}) x-2=0\), calculate the difference between the larger root and the smaller root.

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


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 rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, the maintenance cost is 50 yuan per month. To maximize the monthly revenue of the rental company, the monthly rent for each car should be set at ______.

Answer: $4050$


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

A standard domino game consists of 28 tiles. Each tile is made up of two integers ranging from 0 to 6, inclusive. All possible combinations $(a, b)$, where $a \leq b$, are listed exactly once. Note that tile $(4,2)$ is listed as tile $(2,4)$, because $2 \leq 4$. Excluding the piece $(0,0)$, for each of the other 27 tiles $(a, b)$, where $a \leq b$, we write the fraction $\frac{a}{b}$ on a board.

a) How many distinct values are written in the form of fractions on the board? (Note that the fractions $\frac{1}{2}$ and $\frac{2}{4}$ have the same value and should be counted only once.)

b) What is the sum of the distinct values found in the previous item?

Answer: $\frac{13}{2}$


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>.

<u>Set 4</u> **p10.** Eve has nine letter tiles: three  $C$ ’s, three  $M$ ’s, and three  $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of batteries:  $C$  batteries for  $\$ 4 $ per package, and $ D $ batteries for $ \ $7$  per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly  $\$ 2021 $, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?**p12.** A deck of cards consists of $ 30 $ cards labeled with the integers $ 1 $ to $ 30 $, inclusive. The cards numbered $ 1 $ through $ 15 $ are purple, and the cards numbered $ 16 $ through $ 30 $ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $ 32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.

PS. You should use hide for answers.

Answer: $\frac{1}{28}$


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>.

George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal.

Answer: $4.7667$


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

Let f be a function that satisfies the following conditions: $(i)$  If  $x > y$  and  $f(y) - y \geq v \geq f(x) - x$ , then  $f(z) = v + z$ , for some number  $z$  between  $x$  and  $y$ . $(ii)$  The equation  $f(x) = 0$  has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$   $f(0) = 1$ . $(iv)$   $f(1987) \leq 1988$ . $(v)$   $f(x)f(y) = f(xf(y) + yf(x) - xy)$ .

Find  $f(1987)$ .

*Proposed by Australia.*

Answer: $1988$


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

A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$, $S$, and the upper left corner of the square is left unshaded?

[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,4)--(2,0)--(4,2)--(0,4)--cycle,white,linewidth(1));
label("R",(2,0),S);
label("S",(4,2),E);
[/asy]

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 $$\frac {\sin\alpha-2\cos\alpha}{\sin\alpha +2\cos\alpha}=3,$$ calculate:
(1) $$\frac {\sin\alpha+2\cos\alpha}{5\cos\alpha -\sin\alpha};$$
(2) $({\sin\alpha+\cos\alpha})^{2}.$

Answer: $\frac{9}{17}$


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>.

My friend June likes numbers that have an interesting property: they are divisible by 4. How many different pairings of last two digits are possible in numbers that June likes?

Answer: $25$


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

The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transported to the island that day.

Answer: $904$


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

In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establishing a polar coordinate system with the same unit length, the polar coordinate equation of the curve $C$ is $ρ=2\sqrt{2}\sin({θ+\frac{π}{4}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(4,3)$, the intersection points of line $l$ and curve $C$ be $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

Answer: $\frac{{5\sqrt{2}}}{{11}}$


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

The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units and becomes an even function. Let the sequence $\{a_n\}$ be defined by the formula $a_n=f(\frac{n\pi}{6})$. Compute the sum of the first $2018$ terms of $\{a_n\}$.

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>.

A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece?

Answer: $\frac{19}{61}$


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

Regular octagonal pyramid $\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyramid.

Answer: $\frac{250\sqrt{3}(1 + \sqrt{2})}{3}$


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

If I have a $5 \times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn?

Answer: $14400$


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

Boys and girls are standing in a circle (there are both), a total of 20 children. It is known that each boy's neighbor in the clockwise direction is a child in a blue T-shirt, and each girl's neighbor in the counterclockwise direction is a child in a red T-shirt. Can you uniquely determine how many boys are in the circle?

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>.

In a rectangular grid comprising 5 rows and 4 columns of squares, how many different rectangles can be traced using the lines in the grid?

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>.

How many positive integers have cube roots that are less than 20?

Answer: $7999$


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

Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there?

Answer: $112$


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

Given the ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$: $x^{2}- \frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\_2$ intersects with the circle having the major axis of $C\_1$ as its diameter at points $A$ and $B$. If $C\_1$ precisely trisects the line segment $AB$, then the length of the minor axis of the ellipse $C\_1$ is _____.

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>.

Given $\triangle ABC$ with $AC=1$, $\angle ABC= \frac{2\pi}{3}$, $\angle BAC=x$, let $f(x)= \overrightarrow{AB} \cdot \overrightarrow{BC}$.

$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;

$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of $g(x)$ is $\left[- \frac{3}{2},1\right)$, find the value of the real number $m$.

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>.

Define the operation "" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \lg|x + 2|$ (where $x \neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$.

Answer: $-8$


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>.

What is the least positive integer $n$ such that $6375$ is a factor of $n!$?

Answer: $17$


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

Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \in \mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$.

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>.

Let $f(x) = 4\cos(wx+\frac{\pi}{6})\sin(wx) - \cos(2wx) + 1$, where $0 < w < 2$.
1. If $x = \frac{\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.
2. If the function $f(x)$ is increasing on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$, find the maximum value of $w$.

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>.

Evaluate the expression $2000 \times 1995 \times 0.1995 - 10$.

Answer: $0.2 \times 1995^2 - 10$


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 the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin.

Answer: $3(\sqrt{3} - 1)$


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

Given a geometric sequence $\{a_n\}$ satisfies $a_2a_5=2a_3$, and $a_4$, $\frac{5}{4}$, $2a_7$ form an arithmetic sequence, the maximum value of $a_1a_2a_3…a_n$ is \_\_\_\_\_\_.

Answer: $1024$


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

During a space experiment conducted by astronauts, they must implement a sequence of 6 procedures. Among them, Procedure A can only occur as the first or the last step, and Procedures B and C must be adjacent when conducted. How many different sequences are there to arrange the experiment procedures?

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>.

Every 1 kilogram of soybeans can produce 0.8 kilograms of soybean oil. With 20 kilograms of soybeans, you can produce \_\_\_\_\_\_ kilograms of soybean oil. To obtain 20 kilograms of soybean oil, you need \_\_\_\_\_\_ kilograms of soybeans.

Answer: $25$


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

Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6).

Answer: $-3\sqrt{3}$


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

What is the greatest number of consecutive integers whose sum is $36$?

Answer: $72$


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

The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $\frac{18\sqrt{41} + 40}{\sqrt{41}}$
B) $\frac{360}{17}$
C) $\frac{107}{5}$
D) $\frac{43}{2}$
E) $\frac{281}{13}$

Answer: $\frac{18\sqrt{41} + 40}{\sqrt{41}}$


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

Let \( F_1 = (0,2) \) and \( F_2 = (6,2) \). Find the set of points \( P \) such that
\[
PF_1 + PF_2 = 10
\]
forms an ellipse. The equation of this ellipse can be written as
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
\]
Find \( h + k + a + b \).

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>.

Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play?

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>.

In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\cos F$.

Answer: $\frac{8}{17}$


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>.

It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have?

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>.

Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-1$, $b=2$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$.

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>.

Let \( Q_1 \) be a regular \( t \)-gon and \( Q_2 \) be a regular \( u \)-gon \((t \geq u \geq 3)\) such that each interior angle of \( Q_1 \) is \( \frac{60}{59} \) as large as each interior angle of \( Q_2 \). What is the largest possible value of \( u \)?

Answer: $119$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 on a plane, the parametric equations of curve $C$ are given by $\begin{cases} x = 2\cos θ \\ y = \sqrt{3}\sin θ \end{cases}$ ($θ$ is the parameter). A polar coordinate system is established with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The line $l$ passes through two points $A(\sqrt{2}, \frac{π}{4})$ and $B(3, \frac{π}{2})$ in the polar coordinate system.

(I) Write the general equation of curve $C$ and find the slope of line $l$.
(II) Suppose line $l$ intersects curve $C$ at points $P$ and $Q$. Compute $|BP| \cdot |BQ|$.

Answer: $\frac{120}{19}$


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>.

Given the functions f(x) = x and g(x) = ax^2 - x, where a > 0. If for all x1 in the interval [1, 2], there exists an x2 in the interval [1, 2] such that f(x1) * f(x2) = g(x1) * g(x2) holds true, then find the value of a.

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>.

A certain store sells a type of handbag, and it is known that the cost price of this handbag is $50$ yuan per unit. Market research shows that the daily sales quantity $y$ (unit: units) of this handbag is related to the selling price $x$ (unit: yuan) as follows: $y=-x+80$ $(50 < x < 80)$. Let $w$ represent the daily sales profit of this handbag.
$(1)$ At what price should the selling price of this handbag be set for the store to maximize its daily sales profit? What is the maximum profit per day?
$(2)$ If the price department stipulates that the selling price of this handbag should not exceed $68$ yuan, and the store needs to make a daily sales profit of $200$ yuan from selling this handbag, what should be the selling price per unit?

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>.

In $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. It is known that $a$, $b$, $c$ are in geometric progression, and $a^{2}-c^{2}=ac-bc$,
(1) Find the measure of $\angle A$;
(2) Find the value of $\frac{b\sin B}{c}$.

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>.

Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:
(1) Person A and Person B must stand at the two ends;
(2) All boys must stand together;
(3) No two boys stand next to each other;
(4) Exactly one person stands between Person A and Person B.

Answer: $1200$


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

A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid?

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


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

Simplify the product \[\frac{9}{5}\cdot\frac{14}{9}\cdot\frac{19}{14} \dotsm \frac{5n+4}{5n-1} \dotsm \frac{1009}{1004}.\]

Answer: $\frac{1009}{5}$


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>.

To welcome the 2008 Olympic Games, a craft factory plans to produce the Olympic logo "China Seal" and the Olympic mascot "Fuwa". The factory mainly uses two types of materials, A and B. It is known that producing a set of the Olympic logo requires 4 boxes of material A and 3 boxes of material B, and producing a set of the Olympic mascot requires 5 boxes of material A and 10 boxes of material B. The factory has purchased 20,000 boxes of material A and 30,000 boxes of material B. If all the purchased materials are used up, how many sets of the Olympic logo and Olympic mascots can the factory produce?

Answer: $2400$


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

Given functions $y_1=\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\neq 0$).
$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.
$(2)$ If point $C(-1,n)$ is translated $6$ units upwards and falls exactly on function $y_{1}$, and point $C(-1,n)$ is translated $2$ units to the right and falls exactly on function $y_{2}$, and $k_{1}+k_{2}=0$, find the value of $b$.

Answer: $-6$


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

In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $a=2$, $b=3$, $\cos C=\frac{1}{3}$. The radius of the circumcircle is ______.

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


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

How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits?

Answer: $90$


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

In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\frac{MO}{MF}$ is _______.

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>.

Given the graph of the power function $y=f(x)$ passes through the point $\left(\frac{1}{3},\frac{\sqrt{3}}{3}\right)$, then the value of $\log_{2}f(2)$ 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>.

What is the least positive integer  $m$  such that the following is true?

*Given  $\it m$  integers between  $\it1$  and  $\it{2023},$  inclusive, there must exist two of them  $\it a, b$  such that  $1 < \frac ab \le 2.$*

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 every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0.

Answer: $\frac{1}{72}$


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

After shifting the graph of the function $y=\sin^2x-\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\sin x\cos x$ ($k>0$) with respect to the point $\left( \frac{\pi}{3}, 0 \right)$. Find the minimum positive value of $k+m$.

Answer: $2+ \frac{5\pi}{12}$


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>.

What is the probability that the same number will be facing up on each of five standard six-sided dice that are tossed simultaneously? Express your answer as a simplified common fraction.

Answer: $\frac{1}{1296}$


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

Let $f(x)$ be an odd function on $R$, $f(x+2)=-f(x)$. When $0\leqslant x\leqslant 1$, $f(x)=x$. Find $f(5.5)$.

Answer: $0.5$


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>.

A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet?

Answer: $18\sqrt{5}$


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>.

Simplify first, then evaluate: $\left(2m+n\right)\left(2m-n\right)-\left(2m-n\right)^{2}+2n\left(m+n\right)$, where $m=2$, $n=-1^{2023}$.

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>.

The total in-store price for a laptop is $299.99. A radio advertisement offers the same laptop for five easy payments of $55.98 and a one-time shipping and handling charge of $12.99. Calculate the amount of money saved by purchasing the laptop from the radio advertiser.

Answer: $710$


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

Given a fixed integer \( n \) where \( n \geq 2 \):

a) Determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4\) holds for all nonnegative real numbers \( x_1, x_2, \ldots, x_n \geq 0 \).

b) For this constant \( c \), determine the necessary and sufficient conditions for equality to hold.

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


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

Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. Determine the area of their overlapping interiors. Express your answer in expanded form in terms of $\pi$.

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


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

Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_.

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 $f(\cos x) = -f'(\frac{1}{2})\cos x + \sqrt{3}\sin^2 x$, find the value of $f(\frac{1}{2})$.

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>.

Given that $α$ is an angle in the second quadrant and $\sin α= \frac {3}{5}$, find $\sin 2α$.

Answer: $- \frac{24}{25}$


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

In a trapezoid, the two non parallel sides and a base have length  $1$ , while the other base and both the diagonals have length  $a$ . Find the value of  $a$ .

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


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 $\sin \left(x+ \frac {\pi}{3}\right)= \frac {1}{3}$, then the value of $\sin \left( \frac {5\pi}{3}-x\right)-\cos \left(2x- \frac {\pi}{3}\right)$ is \_\_\_\_\_\_.

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


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

Express $0.000 000 04$ in scientific notation.

Answer: $4 \times 10^{-8}$


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

A taxi driver passes through six traffic checkpoints on the way from the restaurant to the train station. Assuming that the events of encountering a red light at each checkpoint are independent of each other and the probability is $\frac{1}{3}$ at each checkpoint, calculate the probability that the driver has passed two checkpoints before encountering a red light.

Answer: $\frac{4}{27}$


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

Given an arithmetic sequence $\{a_n\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \ldots + a_{99}$.

Answer: $-82$


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

Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \in \mathbb{N^*}$ and $m < n$.

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>.

Given the function $f(x)=\cos(\frac{1}{2}x-\frac{π}{3})$, the graph is shifted to the right by $φ(0＜φ＜\frac{π}{2})$ units to obtain the graph of the function $g(x)$, and $g(x)+g(-x)=0$. Determine the value of $g(2φ+\frac{π}{6})$.

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


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

Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$.

Answer: $111$


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

If $\sec y + \tan y = 3,$ then find $\sec y - \tan y.$

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>.

How many times does the digit 9 appear in the list of all integers from 1 to 1000?

Answer: $300$


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

Given the function $f(x)= \sqrt {2}\cos (x+ \frac {\pi}{4})$, after translating the graph of $f(x)$ by the vector $\overrightarrow{v}=(m,0)(m > 0)$, the resulting graph exactly matches the function $y=f′(x)$. The minimum value of $m$ is \_\_\_\_\_\_.

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


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>.

Circle  $\Gamma$  has diameter  $\overline{AB}$  with  $AB = 6$ .  Point  $C$  is constructed on line  $AB$  so that  $AB = BC$  and  $A \neq C$ . Let  $D$  be on  $\Gamma$  so that  $\overleftrightarrow{CD}$  is tangent to  $\Gamma$ . Compute the distance from line  $\overleftrightarrow{AD}$  to the circumcenter of  $\triangle ADC$ .

*Proposed by Justin Hsieh*

Answer: $4\sqrt{3}$


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

Let  $ a,b,c,d$  be rational numbers with  $ a>0$ . If for every integer  $ n\ge 0$ , the number  $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$  is also integer, then the minimal value of  $ a$  will be

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


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

Three cars start simultaneously from City A, heading towards City B along the same highway. The second car travels 4 kilometers per hour less than the first car and 6 kilometers per hour more than the third car. The second car arrives at City B 3 minutes later than the first car and 5 minutes earlier than the third car. Assuming they do not stop on the way and their speeds are constant, the distance from City A to City B is     kilometers, and the speed of the second car is     kilometers per hour.

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>.

An angle can be represented by two uppercase letters on its sides and the vertex letter, such as $\angle A O B$ (where “ $\angle$ " represents an angle), or by $\angle O$ if the vertex has only one angle. In the triangle $\mathrm{ABC}$ shown below, $\angle B A O=\angle C A O, \angle C B O=\angle A B O$, $\angle A C O=\angle B C O$, and $\angle A O C=110^{\circ}$, find $\angle C B O=$ $\qquad$ .

Answer: $20$


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

Given the sequence $$1, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2}, \frac{3}{1}, \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}, \ldots$$, find the position of $$\frac{8}{9}$$ in this sequence.

Answer: $128$


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

Determine the coefficient of the term containing $x^3$ in the expansion of ${(1+2x)}^{5}$. (The result should be represented as a number.)

Answer: $80$


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

Given the function $f(x)=e^{x}\cos x-x$.

(Ⅰ) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$;

(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$.

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>.

For any interval  $\mathcal{A}$  in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form  $\frac 1x$  where  $x$  is an element in  $\mathcal{A}$ .  Compute the number of ordered pairs of positive integers  $(m,n)$  with  $m< n$  such that the length of the interval  $[m,n]$  is  $10^{10}$  times the length of its reciprocal.

*Proposed by David Altizio*

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>.

Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C.
(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there?
(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, how many different arrangements are there?

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>.

Find the sum of all values of  $a + b$ , where  $(a, b)$  is an ordered pair of positive integers and  $a^2+\sqrt{2017-b^2}$  is a perfect square.

Answer: $67$


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

Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$.

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>.

The ancient Chinese mathematical classic "The Nine Chapters on the Mathematical Art" contains a problem called "Rice and Grain Separation". During the collection of grain in a granary, 1524 "shi" (a unit of weight) of rice was received, but it was found to contain grains of another type mixed in. A sample of rice was taken and it was found that out of 254 grains, 28 were not rice. Approximately how much of this batch of rice is not rice?

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>.

Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[x^2 + y^2 + \frac{4}{(x + y)^2}.\]

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>.

A $20$-quart radiator initially contains a mixture of $18$ quarts of water and $2$ quarts of antifreeze. Six quarts of the mixture are removed and replaced with pure antifreeze liquid. This process is repeated three more times. Calculate the fractional part of the final mixture that is water.
**A)** $\frac{10.512}{20}$
**B)** $\frac{1}{3}$
**C)** $\frac{7.42}{20}$
**D)** $\frac{4.322}{20}$
**E)** $\frac{10}{20}$

Answer: $\frac{4.322}{20}$


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

Given the function $f(x)=e^{x}-ax-1$ ($a$ is a real number), and $g(x)=\ln x-x$.

(I) Discuss the monotonic intervals of the function $f(x)$.

(II) Find the extreme values of the function $g(x)$.

Answer: $-1$


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

Given the parabola $y^2 = 4x$ whose directrix intersects the x-axis at point $P$, draw line $l$ through point $P$ with the slope $k (k > 0)$, intersecting the parabola at points $A$ and $B$. Let $F$ be the focus of the parabola. If $|FB| = 2|FA|$, then calculate the length of segment $AB$.

Answer: $\frac{\sqrt{17}}{2}$


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 \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) be a function that satisfies the following conditions:

1. \( f(1)=1 \)
2. \( f(2n)=f(n) \)
3. \( f(2n+1)=f(n)+1 \)

What is the greatest value of \( f(n) \) for \( 1 \leqslant n \leqslant 2018 \) ?

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>.

$(1)$ Calculate: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation.

Answer: $-4$


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

Given that the focus of the parabola $C: y^{2}=4x$ is $F$, two lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $C$ at points $M$ and $N$. If the product of the slopes of $l_{1}$ and $l_{2}$ is $-1$, calculate the minimum value of $|AB|+|MN|$.

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>.

If $\theta \in (0^\circ, 360^\circ)$ and the terminal side of angle $\theta$ is symmetric to the terminal side of the $660^\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\theta$ (not the origin), find the value of $$\frac {xy}{x^{2}+y^{2}}.$$

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


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

Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction.

Answer: $\frac{7}{15}$


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

In $\triangle ABC$, $\sin (C-A)=1$, $\sin B= \frac{1}{3}$.
(I) Find the value of $\sin A$;
(II) Given $b= \sqrt{6}$, find the area of $\triangle ABC$.

Answer: $3\sqrt{2}$


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

Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\cos ^{2} \frac {A}{2}+\cos A=0$.
(1) Find the value of angle $A$;
(2) If $a=2 \sqrt {3},b+c=4$, find the area of $\triangle ABC$.

Answer: $\sqrt {3}$


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

Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$".

Answer: $\sqrt[9]{81} + 2$


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

On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?

Answer: $25$


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

Given \(0 \le x_0 < 1\), let
\[x_n = \left\{ \begin{array}{ll}
3x_{n-1} & \text{if } 3x_{n-1} < 1 \\
3x_{n-1} - 1 & \text{if } 1 \le 3x_{n-1} < 2 \\
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 2
\end{array}\right.\]
for all integers \(n > 0\), determine the number of values of \(x_0\) for which \(x_0 = x_6\).

Answer: $729$


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

Let \( a, b, c \) be real numbers such that \( 9a^2 + 4b^2 + 25c^2 = 1 \). Find the maximum value of
\[ 3a + 4b + 5c. \]

Answer: $\sqrt{6}$