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

In a given area, there are 10 famous tourist attractions, of which 8 are for daytime visits and 2 are for nighttime visits. A tour group wants to select 5 from these 10 spots for a two-day tour. The itinerary is arranged with one spot in the morning, one in the afternoon, and one in the evening of the first day, and one spot in the morning and one in the afternoon of the second day.
1. How many different arrangements are there if at least one of the two daytime spots, A and B, must be chosen?
2. How many different arrangements are there if the two daytime spots, A and B, are to be visited on the same day?
3. How many different arrangements are there if the two daytime spots, A and B, are not to be chosen at the same time?

Answer: $2352$


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

In triangle $ABC,$ $M$ is the midpoint of $\overline{BC},$ $AB = 15,$ and $AC = 21.$ Let $E$ be on $\overline{AC},$ and $F$ be on $\overline{AB},$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AM}.$ If $AE = 3AF,$ then find $\frac{EG}{GF}.$

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


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>.

Four elevators in a skyscraper, differing in color (red, blue, green, and yellow), move in different directions at different but constant speeds. An observer timed the events as follows: At the 36th second, the red elevator caught up with the blue one (moving in the same direction). At the 42nd second, the red elevator passed by the green one (moving in opposite directions). At the 48th second, the red elevator passed by the yellow one. At the 51st second, the yellow elevator passed by the blue one. At the 54th second, the yellow elevator caught up with the green one. At what second from the start will the green elevator pass by the blue one, assuming the elevators did not stop or change direction during the observation period?

Answer: $46$


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

Jar C initially contains 6 red buttons and 12 green buttons. Michelle removes the same number of red buttons as green buttons from Jar C and places them into an empty Jar D. After the removal, Jar C is left with $\frac{3}{4}$ of its initial button count. If Michelle were to randomly choose a button from Jar C and a button from Jar D, what is the probability that both chosen buttons are green? Express your answer as a common fraction.

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


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>.

A book of one hundred pages has its pages numbered from 1 to 100. How many pages in this book have the digit 5 in their numbering? (Note: one sheet has two pages.)

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>.

Find the terminating decimal expansion of $\frac{13}{200}$.

Answer: $0.052$


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

If \( A = \frac{0.375 \times 2.6}{2 \frac{1}{2} \times 1 \frac{1}{5}}+\frac{0.625 \times 1.6}{3 \times 1.2 \times 4 \frac{1}{6}}+6 \frac{2}{3} \times 0.12+28+\frac{1 \div 9}{7}+\frac{0.2}{9 \times 22} \), then when \( A \) is expressed as a fraction, the numerator of \( A \) is \( \qquad \) , and the denominator of \( A \) is \( \qquad \) .

Answer: $1901/3360$


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

A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the distance in meters the train has traveled during this period.

Answer: $405$


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

Paint is to be mixed so that the ratio of blue paint to green paint is 5 to 3. If Clara wants to make 45 cans of the mixture and all cans hold the same volume of paint, how many cans of blue paint will she need?

Answer: $28$


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

Mafia is a game where there are two sides: The village and the Mafia. Every night, the Mafia kills a person who is sided with the village. Every day, the village tries to hunt down the Mafia through communication, and at the end of every day, they vote on who they think the mafia are.**p6.** Patrick wants to play a game of mafia with his friends. If he has  $10$  friends that might show up to play, each with probability  $1/2$ , and they need at least  $5$  players and a narrator to play, what is the probability that Patrick can play?**p7.** At least one of Kathy and Alex is always mafia. If there are  $2$  mafia in a game with  $6$  players, what is the probability that both Kathy and Alex are mafia?**p8.** Eric will play as mafia regardless of whether he is randomly selected to be mafia or not, and Euhan will play as the town regardless of what role he is selected as. If there are  $2$  mafia and  $6$  town, what is the expected value of the number of people playing as mafia in a random game with Eric and Euhan?**p9.** Ben is trying to cheat in mafia. As a mafia, he is trying to bribe his friend to help him win the game with his spare change. His friend will only help him if the change he has can be used to form at least  $25$  different values. What is the fewest number of coins he can have to achieve this added to the fewest possible total value of those coins? He can only use pennies, nickels, dimes, and quarters.**p10.** Sammy, being the very poor mafia player he is, randomly shoots another player whenever he plays as the vigilante. What is the probability that the player he shoots is also not shot by the mafia nor saved by the doctor, if they both select randomly in a game with  $8$  people? There are  $2$  mafia, and they cannot select a mafia to be killed, and the doctor can save anyone.

PS. You should use hide for answers.

Answer: $319/512$


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

Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB < BC$. There is a point $E$ on line segment $CD$ such that $\triangle ABC \sim \triangle CED$ and the area of $\triangle AED$ is $9$ times the area of $\triangle CED$. What is $\tfrac{BC}{AB}$?

Answer: $10.25$


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 that a set of $n$ people participate in an online video soccer tournament, the statistics from the tournament reveal: The average number of complete teams wholly contained within randomly chosen subsets of $10$ members equals twice the average number of complete teams found within randomly chosen subsets of $7$ members. Find out how many possible values for $n$, where $10\leq n\leq 2017$, satisfy this condition.

Answer: $450$


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

A nut has the shape of a regular hexagonal prism. Each lateral face of the nut is painted in one of three colors: white, red, or blue, with adjacent faces painted in different colors. How many different nut paintings are possible? (It is not necessary to use all three colors in the painting.)

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>.

It is easy to place the complete set of ships for the game "Battleship" on a $10 \times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.)

Answer: $7 \times 7$


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

Define a function \( f \) on the set of positive integers \( N \) as follows:
(i) \( f(1) = 1 \), \( f(3) = 3 \);
(ii) For \( n \in N \), the function satisfies
\[
\begin{aligned}
&f(2n) = f(n), \\
&f(4n+1) = 2f(2n+1) - f(n), \\
&f(4n+3) = 3f(2n+1) - 2f(n).
\end{aligned}
\]

Find all \( n \) such that \( n \leqslant 1988 \) and \( f(n) = n \).

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 12 points in a diagram, calculate the number of groups of 3 points that can be formed to create a triangle.

Answer: $200$


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

Find the smallest real number $a$ such that for any non-negative real numbers $x, y, z$ whose sum is 1, the inequality $a\left(x^2 + y^2 + z^2\right) + xyz \geq \frac{9}{3} + \frac{1}{27}$ holds.

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


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

There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person's number instead. The sum of all the numbers reported by the 37 people is 2011. Which person made the mistake?

Answer: $34$


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

Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning.

Answer: $258$


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

Given that $F$ is the right focus of the hyperbola $C:x^{2}-\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6\sqrt{6})$, the minimum perimeter of $\triangle APF$ is $\_\_\_\_\_\_$.

Answer: $32$


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

If $\frac{1}{8}$ of $2^{32}$ equals $8^y$, what is the value of $y$?

Answer: $9.67$


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>.

There are 6 male doctors and 4 female doctors. From them, 3 male doctors and 2 female doctors are selected to provide medical services in 5 different regions. It is specified that male doctor A cannot go to region A. How many different assignment plans are there in total?

Answer: $12960$


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

A store has three types of boxes containing marbles in large, medium, and small sizes, respectively holding 13, 11, and 7 marbles. If someone wants to buy 20 marbles, it can be done without opening the boxes (1 large box plus 1 small box). However, if someone wants to buy 23 marbles, a box must be opened. Find the smallest number such that any purchase of marbles exceeding this number can always be done without opening any boxes. What is this smallest number?

Answer: $30$


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

Find the pattern and fill in the blanks:

1. 12, 16, 20, \_\_\_\_\_\_, \_\_\_\_\_\_
2. 2, 4, 8, \_\_\_\_\_\_, \_\_\_\_\_\_

Answer: $32$


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

$A_1, A_2, ..., A_n$  are the subsets of  $|S|=2019$  such that union of any three of them gives  $S$  but if we combine two of subsets it doesn't give us  $S$ . Find the maximum value of  $n$ .

Answer: $64$


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

Pedro wrote a list of all positive integers less than 10000 in which each of the digits 1 and 2 appear exactly once. For example, 1234, 231, and 102 were written on the list, but 1102 and 235 are not on the list. How many numbers are there on Pedro's list?

Answer: $336$


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

How many non-empty subsets $S$ of $\{1,2,3,\ldots,20\}$ have the following three properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$(3)$ $S$ contains at least one number greater than $10$.

Answer: $2526$


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

How many four-digit numbers greater than 3999 can be formed such that the product of the middle two digits exceeds 10?

Answer: $3480$


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

Fill the numbers 1 to 16 into a $4 \times 4$ grid such that each number in a row is larger than the number to its left and each number in a column is larger than the number above it. Given that the numbers 4 and 13 are already placed in the grid, determine the number of different ways to fill the remaining 14 numbers.

Answer: $1120$


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

Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahedron?

Answer: $15$


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

Given the function $f(x)=2\cos x(\sin x-\cos x)$, where $x\in R$.
(I) Find the symmetry center of the graph of the function $f(x)$;
(II) Find the minimum and maximum values of the function $f(x)$ on the interval $[\frac{\pi}{8}, \frac{3\pi}{4}]$.

Answer: $-2$


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

A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken? $\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$

Answer: $245$


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

In the diagram, $\triangle ABC$ is right-angled at $A$ with $AB = 3$ and $AC = 3\sqrt{3}$. Altitude $AD$ intersects median $BE$ at point $G$. Find the length of $AG$. Assume the diagram has this configuration:

[Contextual ASY Diagram]
```
draw((0,0)--(9,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(4.5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49));
label("$A$",(0,0),SW); label("$E$",(4.5,0),S); label("$B$",(9,0),SE);
label("$D$",(7.5,4.33),NE); label("$C$",(0,10*sqrt(3)),N); label("$G$",(4.29,2.47),NW);
```

Answer: $-\frac{1.5\sqrt{3}}{3}$


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>.

From the numbers 1, 2, 3, 4, 5, and 6, two distinct numbers are taken out each time and denoted as $a$ and $b$. The total number of distinct values obtained for $3^{\frac{a}{b}}$ is ______.

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>.

Compute the square of 1023 without a calculator.

Answer: $1046529$


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

How many ways can you tile the white squares of the following \(2 \times 24\) grid with dominoes? (A domino covers two adjacent squares, and a tiling is a non-overlapping arrangement of dominoes that covers every white square and does not intersect any black square.)

Answer: $27$


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

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

Answer: $20\sqrt{5}$


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

The polynomial $x^3 - ax^2 + bx - 2310$ has three positive integer roots. What is the smallest possible value of $a$?

Answer: $88$


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

The number of elderly employees in the sampled group is what fraction of the number of young employees in the sampled group if the sampled group consists of the employees of a certain unit, which has a total of 430 employees, among which there are 160 young employees, and the number of middle-aged employees is twice the number of elderly employees?

Answer: $18$


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

A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters from the well's top for the second time. Determine the total time the frog spends climbing from the bottom to the top of the well in minutes.

Answer: $22$


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

In the equation on the right, each Chinese character represents one of the ten digits from 0 to 9. The same character represents the same digit, and different characters represent different digits. What is the four-digit number represented by "数学竞赛"?

Answer: $1962$


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

Given a triangle \(PQR\). Point \(T\) is the center of the inscribed circle.

The rays \(PT\) and \(QT\) intersect side \(PQ\) at points \(E\) and \(F\) respectively. It is known that the areas of triangles \(PQR\) and \(TFE\) are equal. What part of side \(PQ\) constitutes from the perimeter of triangle \(PQR\)?

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


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

A fair standard four-sided die is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one "3" is rolled?
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac{3}{10}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$

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


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

In $\triangle ABC$, it is known that $\sin A : \sin B : \sin C = 3 : 5 : 7$. The largest interior angle of this triangle is equal to ______.

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


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

The diagram shows a karting track circuit. The start and finish are at point $A$, and the kart driver can return to point $A$ and continue on the circuit as many times as desired.

The time taken to travel from $A$ to $B$ or from $B$ to $A$ is one minute. The time taken to travel around the loop is also one minute. The direction of travel on the loop is counterclockwise (as indicated by the arrows). The kart driver does not turn back halfway or stop. The duration of the race is 10 minutes. Find the number of possible distinct routes (sequences of section traversals).

Answer: $34$


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

The length of a circular race track is $1 \mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ on the track, and then for the second time at point $A$. How far did the first motorcyclist travel from the starting point to point $B$?

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


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

As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot Y moves clockwise, and robots J and B move counterclockwise. After 12 seconds, robot J reaches point B. After another 9 seconds, when robot J first catches up with robot B, it also meets robot Y for the first time. How many seconds after B first reaches point A will Y first reach point B?

Answer: $56$


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

For \(50 \le n \le 150\), how many integers \(n\) are there such that \(\frac{n}{n+1}\) is a repeating decimal and \(n+1\) is not divisible by 3?

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>.

Let $z$ be a complex number such that $|z| = 3.$ Find the largest possible distance between $(1 + 2i)z^3$ and $z^4$ when plotted in the complex plane.

Answer: $216$


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

In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\).

Answer: $57$


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

Given that $\sec x - \tan x = \frac{5}{4},$ find all possible values of $\sin x.$

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


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

A right triangle when rotating around a large leg forms a cone with a volume of  $100\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of  $180^o$  around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to  $17$ .

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>.

The probability that the blue ball is tossed into a higher-numbered bin than the yellow ball.

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


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

Given the set $\{-25, -4, -1, 3, 5, 9\}$, find the largest quotient and the smallest product that can be formed using two numbers chosen from the set.

Answer: $-225$


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

A math test starts at 12:35 PM and lasts for $4 \frac{5}{6}$ hours. At what time does the test end?

Answer: $17:25$


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>.

Points \(O\) and \(I\) are the centers of the circumscribed and inscribed circles of triangle \(ABC\), \(M\) is the midpoint of the arc \(AC\) of the circumscribed circle (which does not contain \(B\)). It is known that \(AB = 15\), \(BC = 7\), and \(MI = MO\). Find \(AC\).

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>.

What is the largest four-digit number that is divisible by 6?

Answer: $9996$


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

In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 participants, compete against each other. During the tournament, each schoolchild must play with each student exactly once, with the condition that everyone can play at most once per day. Different numbers of games could be played on different days.

At some point in the tournament, the organizer noticed that there is exactly one way to schedule the next day with 15 games and $N$ ways to schedule the next day with just 1 game (the order of games in the schedule does not matter, only who plays with whom matters). Find the maximum possible value of $N$.

Answer: $120$


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

What is $2453_6 + 16432_6$? Express your answer in both base 6 and base 10.

Answer: $3881$


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

The Seattle weather forecast suggests a 60 percent chance of rain each day of a five-day holiday. If it does not rain, then the weather will be sunny. Stella wants exactly two days to be sunny during the holidays for a gardening project. What is the probability that Stella gets the weather she desires? Give your answer as a fraction.

Answer: $\frac{4320}{15625}$


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

Find  the largest real number  $c$  such that   $$ \sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i $$  for  any positive integer  $n $  and  any real numbers  $x_1,x_2,\dots,x_n.$

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 vectors $\overrightarrow{a} = (1, -1)$ and $\overrightarrow{b} = (6, -4)$, if $\overrightarrow{a}$ is perpendicular to $(t\overrightarrow{a} + \overrightarrow{b})$, find the value of the real number $t$.

Answer: $-1$


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

The product   $$ \left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right) $$   can be written as  $\frac{r}{s2^t}$  where  $r$ ,  $s$ , and  $t$  are positive integers and  $r$  and  $s$  are odd and relatively prime. Find  $r+s+t$ .

Answer: $3769$


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

Express $3.\overline{36}$ as a common fraction in lowest terms.

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


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>.

Two-thirds of a number plus two times another number equals 20. If one-fourth of the first number minus the second number equals 2, find these two numbers.

Answer: $\frac{22}{7}$


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

If both $a$ and $b$ in the fraction $\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction.

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>.

Twelve delegates, four each from three different countries, randomly select chairs at a round table that seats twelve people. Calculate the probability that each delegate sits next to at least one delegate from another country, and express this probability as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

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>.

An abstract animal lives in groups of two and three.

In a forest, there is one group of two and one group of three. Each day, a new animal arrives in the forest and randomly chooses one of the inhabitants. If the chosen animal belongs to a group of three, that group splits into two groups of two; if the chosen animal belongs to a group of two, they form a group of three. What is the probability that the $n$-th arriving animal will join a group of two?

Answer: $4/7$


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

Let  $ABC$  be a non-equilateral triangle with integer sides. Let  $D$  and  $E$  be respectively the mid-points of  $BC$  and  $CA$  ; let  $G$  be the centroid of  $\Delta{ABC}$ . Suppose,  $D$ ,  $C$ ,  $E$ ,  $G$  are concyclic. Find the least possible perimeter of  $\Delta{ABC}$ .

Answer: $37$


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

Point  $P$  lies outside a circle, and two rays are drawn from  $P$  that intersect the circle as shown. One ray intersects the circle at points  $A$  and  $B$  while the other ray intersects the circle at  $M$  and  $N$ .  $AN$  and  $MB$  intersect at  $X$ . Given that  $\angle AXB$  measures  $127^{\circ}$  and the minor arc  $AM$  measures  $14^{\circ}$ , compute the measure of the angle at  $P$ .

[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label(" $A$ ",A,plain.NE);label(" $B$ ",B,plain.NW);label(" $M$ ",M,SE);
label(" $P$ ",P,E);label(" $N$ ",N,dir(250));label(" $X$ ",X,plain.N);[/asy]

Answer: $39$


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

Consider the function:
\[ f(x) = \left\{
\begin{aligned}
3x + 1 & \quad \text{ if } x \leq 2 \\
x^2 & \quad \text{ if } x > 2
\end{aligned}
\right.\]
The function has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(0) + \dots + f^{-1}(8) + f^{-1}(9)$.

Answer: $22 + 2\sqrt{2}$


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>.

Calculate the double integrals and change the order of integration.

$$
\text{a) } \int_{0}^{2} dx \int_{0}^{3}\left(x^{2} + 2xy\right) dy
$$

b) $\int_{-2}^{0} dy \int_{0}^{y^{2}}(x+2y) dx$

c) $\int_{0}^{5} dx \int_{0}^{5-x} \sqrt{4 + x + y} dy$

Answer: $\frac{506}{3}$


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

In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)?

Answer: $108$


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

Masha wrote the numbers $4, 5, 6, \ldots, 16$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value that the sum of the remaining numbers on the board can have?

Answer: $121$


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

18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of "circular triples" (where each of the three participants wins against one and loses to another) in the tournament.

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>.

What is the smallest square number whose first five digits are 4 and the sixth digit is 5?

Answer: $666667$


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

Calculate the definite integral:

$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{3 x+2}) d x}{(\sqrt{3 x+2}+4 \sqrt{2-x})(3 x+2)^{2}}
$$

Answer: $\frac{1}{32} \ln 5$


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>.

If \( x, \) \( y, \) and \( k \) are positive real numbers such that
\[
4 = k^2\left(\frac{x^2}{y^2}+\frac{y^2}{x^2}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right),
\]
find the maximum possible value of \( k \).

Answer: $\frac{-1+\sqrt{17}}{4}$


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

In the right triangle \(ABC\) with the right angle at \(A\), an altitude \(AH\) is drawn. The circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\), given that \(AX = 5\), \(AY = 6\), and \(AB = 9\).

Answer: $13.5$


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 $a_{1}, a_{2}, \cdots, a_{10}$ are ten different positive integers satisfying  the equation $\left|a_{i+1}-a_{i}\right|=2 \text { or } 3$, where $i=1,2, \cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1}, a_{2}, \cdots, a_{10}$ and $m$ is the minimum number among $a_{1}, a_{2}, \cdots, a_{10}$.

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>.

The number $2027$ is prime. Let $T = \sum \limits_{k=0}^{72} \binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$?

Answer: $1369$


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

Given the decimal representation of $\frac{1}{30^{30}}$, determine how many zeros immediately follow the decimal point.

Answer: $44$


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

A uniform cubic die with faces numbered $1, 2, 3, 4, 5, 6$ is rolled three times independently, resulting in outcomes $a_1, a_2, a_3$. Find the probability of the event "$|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_1| = 6$".

Answer: $1/4$


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

Define a  $\textit{tasty residue}$  of  $n$  to be an integer  $1<a<n$  such that there exists an integer  $m>1$  satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of  $2016$ .

Answer: $831$


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

The graphs of the lines $y = \frac{3}{2}x$ and $y=2x - 1$ intersect and form an acute angle. Find the slope $k$ of the angle bisector of this acute angle that is oriented positively with respect to the x-axis.

Answer: $\frac{7 - \sqrt{29}}{4}$


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

In quadrilateral  $ABCD$ , we have  $AB = 5$ ,  $BC = 6$ ,  $CD = 5$ ,  $DA = 4$ , and  $\angle ABC = 90^\circ$ . Let  $AC$  and  $BD$  meet at  $E$ . Compute  $\dfrac{BE}{ED}$ .

Answer: $5/4$


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>.

For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)?

Answer: $2048$


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

Given that \( 990 \times 991 \times 992 \times 993 = \overline{966428 A 91 B 40} \), find the values of \( \overline{A B} \).

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>.

On the shore of a circular island (viewed from above) are the cities $A$, $B$, $C$, and $D$. The straight asphalt road $AC$ divides the island into two equal halves. The straight asphalt road $BD$ is shorter than the road $AC$ and intersects it. The cyclist's speed on any asphalt road is 15 km/h. The island also has straight dirt roads $AB$, $BC$, $CD$, and $AD$, where the cyclist's speed on any dirt road is the same. The cyclist travels from point $B$ to each of the points $A$, $C$, and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$.

Answer: $450$


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

Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let \( p \) denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let \( q \) denote the probability that the second shot is also a blank if the cylinder is not spun after the first shot. Find the smallest integer \( N \) such that
\[
N \geq \frac{100 p}{q} .
\]

Answer: $89$


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

Three natural numbers are written on the board: two ten-digit numbers \( a \) and \( b \), and their sum \( a + b \). What is the maximum number of odd digits that could be written on the board?

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>.

Children get in for half the price of adults. The price for $8$ adult tickets and $7$ child tickets is $42$. If a group buys more than $10$ tickets, they get an additional $10\%$ discount on the total price. Calculate the cost of $10$ adult tickets and $8$ child tickets.

Answer: $46$


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

The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. After a 15-minute stop, it heads to the provincial capital, arriving by 11:00 AM. Another bus departs from the provincial capital directly to Lishan Town at 9:00 AM on the same day, traveling at 60 kilometers per hour. When the two buses meet, the one traveling from the provincial capital to Lishan Town has been traveling for how many minutes?

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>.

Two right triangles share a side such that the common side AB has a length of 8 units, and both triangles ABC and ABD have respective heights from A of 8 units each. Calculate the area of triangle ABE where E is the midpoint of side CD and CD is parallel to AB. Assume that side AC = side BC.

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>.

Let  $f(t)$  be the cubic polynomial for  $t$  such that  $\cos 3x=f(\cos x)$  holds for all real number  $x$ .
Evaluate

\[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

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


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

On a 10-ring target, the probabilities of hitting scores 10, 9, 8, 7, and 6 are $\frac{1}{5}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8},$ and $\frac{1}{10}$ respectively. The probability of hitting any other score (from 5 to 1) is $\frac{1}{12}$. $A$ pays $B$ the score amount in forints for any hit that is at least 6, and 1.7 forints for any other hit. How much should $B$ pay in case of a miss so that the bet is fair?

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>.

The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle \(X\) is 9, what is the number in circle \(Y\)?

Answer: $11$


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

Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance.

Answer: $44$


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

Determine the absolute value of the difference between single-digit integers $C$ and $D$ such that:
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
 & & & D& D & C_6\\
 & & & \mathbf{5} & \mathbf{2} & D_6\\
 & & + & C & \mathbf{2} & \mathbf{4_6}\\
 \cline{2-6}
 & & D & \mathbf{2} & \mathbf{0} & \mathbf{3_6} \\
\end{array} $$
Express your answer in base $6$.

Answer: $1_6$


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>.

Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \).

Answer: $\frac{\sqrt{31}}{6}$


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>.

If \(\tan \gamma = 5\) and \(\tan \beta = 3,\) find \(\tan(\gamma + \beta)\) and \(\tan(\gamma - \beta)\).

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>.

6 boys and 4 girls are each assigned as attendants to 5 different buses, with 2 attendants per bus. Assuming that boys and girls are separated, and the buses are distinguishable, how many ways can the assignments be made?

Answer: $5400$


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

The expression $\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$?

Answer: $2001$


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

The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately he will provide the value of the expression \( P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right) \). Based on this information, the children must find the polynomial that the teacher might have in mind. What is the smallest possible \( k \) for which the teacher can compose a problem of this type such that the polynomial found by the children will necessarily match the intended one?

Answer: $2017$


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

There are four cards, each with a number on both sides. The first card has 0 and 1, the other three cards have 2 and 3, 4 and 5, and 7 and 8 respectively. If any three cards are selected and arranged in a row, how many different three-digit numbers can be formed?

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>.

Given vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) such that
\[
|a|=|b|=3, |c|=4, \boldsymbol{a} \cdot \boldsymbol{b}=-\frac{7}{2},
\boldsymbol{a} \perp \boldsymbol{c}, \boldsymbol{b} \perp \boldsymbol{c}
\]

Find the minimum value of the expression
\[
|x \boldsymbol{a} + y \boldsymbol{b} + (1-x-y) \boldsymbol{c}|
\]
for real numbers \(x\) and \(y\).

Answer: $\frac{4 \sqrt{33}}{15}$


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>.

Calculate the definite integral:

$$
\int_{1}^{8} \frac{5 \sqrt{x+24}}{(x+24)^{2} \cdot \sqrt{x}} \, dx
$$

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>.

Buses. At the stop near Absent-Minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other and their schedules do not depend on each other. At a completely random moment, the Absent-Minded Scientist arrives at the stop and boards the first bus that arrives to get to the metro. What is the probability that the Scientist will board bus No. 251?

Answer: $5/14$


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 triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\frac{S}{b^{2}+2c^{2}}$.

Answer: $\frac{\sqrt{14}}{24}$


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

The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many negative integer values of $x$ will the value of $12 \star x$ be a positive integer?

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>.

There are three two-digit numbers $A$, $B$, and $C$.

- $A$ is a perfect square, and each of its digits is also a perfect square.
- $B$ is a prime number, and each of its digits is also a prime number, and their sum is also a prime number.
- $C$ is a composite number, and each of its digits is also a composite number, the difference between its two digits is also a composite number. Furthermore, $C$ is between $A$ and $B$.

What is the sum of these three numbers $A$, $B$, and $C$?

Answer: $120$


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

Solve the following equation and provide its root. If the equation has multiple roots, provide their product.

\[ \sqrt{2 x^{2} + 8 x + 1} - x = 3 \]

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>.

Given the function $f(x)=2\sin x\cos (x+ \frac {π}{3})+ \frac { \sqrt {3}}{2}$.
(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac {π}{2}]$.

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>.

Given that \( f(x) \) is an odd function with a period of 4, and for \( x \in (0,2) \), \( f(x) = x^2 - 16x + 60 \). Find the value of \( f(2 \sqrt{10}) \).

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>.

In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon.

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>.

Find the intercept on the $x$-axis of the line that is perpendicular to the line $3x-4y-7=0$ and forms a triangle with both coordinate axes having an area of $6$.

Answer: $-3$


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

A certain school sends two students, A and B, to form a "youth team" to participate in a shooting competition. In each round of the competition, A and B each shoot once. It is known that the probability of A hitting the target in each round is $\frac{1}{2}$, and the probability of B hitting the target is $\frac{2}{3}$. In each round of the competition, whether A and B hit the target or not does not affect each other, and the results of each round of the competition do not affect each other.
$(1)$ Find the probability that the "youth team" hits exactly $1$ time in one round of the competition.
$(2)$ Find the probability that the "youth team" hits exactly $3$ times in three rounds of the competition.

Answer: $\frac{7}{24}$


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

A circle with its center on the line \( y = b \) intersects the parabola \( y = \frac{3}{4} x^{2} \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{3}{4} x + b \). Find all values of \( b \) for which the described configuration is possible.

Answer: $\frac{25}{12}$


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>.

Determine the smallest integer $B$ such that there exists a sequence of consecutive integers, which must include at least one odd and one even number, adding up to 2023.

Answer: $1011$