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

Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$.

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


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

When $0.76\overline{204}$ is expressed as a fraction in the form $\frac{x}{999000}$, what is the value of $x$?

Answer: $761280$


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

$ABCD$  is a rectangle with  $AB = CD = 2$ . A circle centered at  $O$  is tangent to  $BC$ ,  $CD$ , and  $AD$  (and hence has radius  $1$ ). Another circle, centered at  $P$ , is tangent to circle  $O$  at point  $T$  and is also tangent to  $AB$  and  $BC$ . If line  $AT$  is tangent to both circles at  $T$ , find the radius of circle  $P$ .

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


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>.

In recent years, handicrafts made by a certain handicraft village have been very popular overseas. The villagers of the village have established a cooperative for exporting handicrafts. In order to strictly control the quality, the cooperative invites 3 experts to inspect each handicraft made by the villagers. The quality control process is as follows: $(i)$ If all 3 experts consider a handicraft to be of good quality, then the quality of the handicraft is rated as grade $A$; $(ii)$ If only 1 expert considers the quality to be unsatisfactory, then the other 2 experts will conduct a second quality check. If both experts in the second check consider the quality to be good, then the handicraft is rated as grade $B$. If one or both of the experts in the second check consider the quality to be unsatisfactory, then the handicraft is rated as grade $C$; $(iii)$ If 2 or all 3 experts consider the quality to be unsatisfactory, then the handicraft is rated as grade $D$. It is known that the probability of 1 handicraft being considered unsatisfactory by 1 expert in each quality check is $\frac{1}{3}$, and the quality of each handicraft is independent of each other. Find: $(1)$ the probability that 1 handicraft is rated as grade $B$; $(2)$ if handicrafts rated as grade $A$, $B$, and $C$ can be exported with profits of $900$ yuan, $600$ yuan, and $300$ yuan respectively, while grade $D$ cannot be exported and has a profit of $100$ yuan. Find: $①$ the most likely number of handicrafts out of 10 that cannot be exported; $②$ if the profit of 1 handicraft is $X$ yuan, find the distribution and mean of $X$.

Answer: $\frac{13100}{27}$


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>.

Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\).

Answer: $2 - \sqrt{2}$


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

Given \( P = 3659893456789325678 \) and \( 342973489379256 \), the product \( P \) is calculated. The number of digits in \( P \) is:

Answer: $34$


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

A circle $\omega$ is circumscribed around the triangle $ABC$. A circle $\omega_{1}$ is tangent to the line $AB$ at point $A$ and passes through point $C$, and a circle $\omega_{2}$ is tangent to the line $AC$ at point $A$ and passes through point $B$. A tangent to circle $\omega$ at point $A$ intersects circle $\omega_{1}$ again at point $X$ and intersects circle $\omega_{2}$ again at point $Y$. Find the ratio $\frac{AX}{XY}$.

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


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

If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: $30$


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

Given that a basketball player has a 40% chance of scoring with each shot, estimate the probability that the player makes exactly two out of three shots using a random simulation method. The simulation uses a calculator to generate a random integer between 0 and 9, with 1, 2, 3, and 4 representing a scored shot, and 5, 6, 7, 8, 9, 0 representing a missed shot. Three random numbers are grouped to represent the results of three shots. After simulating, the following 20 groups of random numbers were generated:
```
907  966  191  925  271  932  812  458  569  683
431  257  393  027  556  488  730  113  537  989
```
Based on this, estimate the probability that the athlete scores twice out of three shots.

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 solid cube of side length $2$ is removed from each corner of a larger solid cube of side length $4$. Find the number of edges of the remaining solid.

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>.

For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative?

Answer: $12$


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

Let X be a set containing 10 elements, and A, B be two disjoint subsets of X, containing 3 and 4 elements respectively. Calculate the number of subsets of X that contain neither A nor B.

Answer: $840$


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

My five friends and I play doubles badminton every weekend. Each weekend, two of us play as a team against another two, while the remaining two rest. How many different ways are there for us to choose the two teams and the resting pair?

Answer: $45$


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

What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003}+1)^{11/7}$?

Answer: $571$


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

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=-\sqrt{2}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: $\dfrac{3\pi}{4}$


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 deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.)

Answer: $210$


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

Given the ellipse $C:\dfrac{x^2}{m^2}+y^2=1$ (where $m > 1$ is a constant), $P$ is a moving point on curve $C$, and $M$ is the right vertex of curve $C$. The fixed point $A$ has coordinates $(2,0)$.

$(1)$ If $M$ coincides with $A$, find the coordinates of the foci of curve $C$;

$(2)$ If $m=3$, find the maximum and minimum values of $|PA|$.

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


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

Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads?

Answer: $\frac{9}{16}$


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>.

Consider all angles measured in degrees. Calculate the product $\prod_{k=1}^{22} \csc^2(4k-3)^\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find the value of $m+n$.

Answer: $23$


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

Given a point P on the curve $y = x^2 - \ln x$, find the minimum distance from point P to the line $y = x + 2$.

Answer: $\sqrt{2}$


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

For Beatrix's latest art installation, she has fixed a $2 \times 2$ square sheet of steel to a wall. She has two $1 \times 2$ magnetic tiles, both of which she attaches to the steel sheet, in any orientation, so that none of the sheet is visible and the line separating the two tiles cannot be seen. One tile has one black cell and one grey cell; the other tile has one black cell and one spotted cell. How many different looking $2 \times 2$ installations can Beatrix obtain?

A 4
B 8
C 12
D 14
E 24

Answer: $12$


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

Given that the random variable $X$ follows a two-point distribution with $E(X) = 0.7$, determine its success probability.

A) $0$
B) $1$
C) $0.3$
D) $0.7$

Answer: $0.7$


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>.

How many $3$-digit positive integers have digits whose product equals $36$?

Answer: $21$


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

Let  $x \neq y$  be positive reals satisfying  $x^3+2013y=y^3+2013x$ , and let  $M = \left( \sqrt{3}+1 \right)x + 2y$ .  Determine the maximum possible value of  $M^2$ .

*Proposed by Varun Mohan*

Answer: $16104$


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

What is the number of radians in the smaller angle formed by the hour and minute hands of a clock at 3:40? Express your answer as a decimal rounded to three decimal places.

Answer: $2.278$


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

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 3 and 9, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Calculate the length of the chord expressed in the form $\frac{m\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, and provide $m+n+p.$

Answer: $22$


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

In a square with a side length of 12 cm, the midpoints of its adjacent sides are connected to each other and to the opposite side of the square. Find the radius of the circle inscribed in the resulting triangle.

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


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

A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:

- If an adventurer has diamonds, then they have either emeralds or sapphires (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or diamonds (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?

Answer: $16$


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

A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:

10.12, 9.97, 10.01, 9.95, 10.02, 9.98, 9.21, 10.03, 10.04, 9.99, 9.98, 9.97, 10.01, 9.97, 10.03, 10.11

The mean ($\bar{x}$) and standard deviation ($s$) are calculated as follows:

$\bar{x} \approx 9.96$, $s \approx 0.20$

(I) If there is a component with a dimension outside the range of ($\bar{x} - 3s$, $\bar{x} + 3s$), it is considered that an abnormal situation has occurred in the production process of that day, and the production process of that day needs to be inspected. Based on the inspection results of that day, is it necessary to inspect the production process of that day? Please explain the reason.

(II) Among the 16 different components inspected that day, two components are randomly selected from those with dimensions in the range of (10, 10.1). Calculate the probability that the dimensions of both components are greater than 10.02.

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


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 product of two consecutive integers is $20{,}412$. What is the sum of these two integers?

Answer: $287$


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

A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$.

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


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

The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following rule: $$\frac {1}{2}, \frac {1}{3}, \frac {2}{3}, \frac {1}{4}, \frac {2}{4}, \frac {3}{4}, \frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \ldots, \frac {1}{n}, \frac {2}{n}, \ldots, \frac {n-1}{n}, \ldots$$ If there exists a positive integer k such that $S_{k-1} < 10$ and $S_k > 10$, then $a_k = \_\_\_\_\_\_$.

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


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>.

The value of \( a \) is chosen such that the number of roots of the first equation \( 4^{x} - 4^{-x} = 2 \cos a x \) is 2007. How many roots does the second equation \( 4^{x} + 4^{-x} = 2 \cos a x + 4 \) have for the same \( a \)?

Answer: $4014$


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

The horse walks a distance that is half of the previous day's distance each day, and after walking for 7 days, it has covered a total distance of 700 li. Calculate the total distance it will cover from the 8th day to the 14th day.

Answer: $\frac {175}{32}$


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>.

The complex number $2+i$ and the complex number $\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, calculate the angle $\angle AOB$.

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


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>.

Consider numbers of the form  $1a1$ , where  $a$  is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?

*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples:  $353$ ,  $91719$ .*

Answer: $55$


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

How many natural numbers between 200 and 400 are divisible by 8?

Answer: $24$


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

Given the function $f(x) = \sqrt{ax^2 + bx + c}$ $(a < 0)$, and all points $\left(\begin{matrix}s, f(t) \end{matrix}\right)$ (where $s, t \in D$) form a square region, determine the value of $a$.

Answer: $-4$


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

The coefficient of \\(x^4\\) in the expansion of \\((1+x+x^2)(1-x)^{10}\\).

Answer: $135$


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

How many kings can be placed on an $8 \times 8$ chessboard without any of them being in check?

Answer: $16$


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

A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\bigtriangleup ADF$. Express your answer in simplest radical form.

Answer: $4\sqrt{3}$


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

Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$.

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>.

Given the sequence $\{a\_n\}$ that satisfies the condition: when $n \geqslant 2$ and $n \in \mathbb{N}^+$, we have $a\_n + a\_{n-1} = (-1)^n \times 3$. Calculate the sum of the first 200 terms of the sequence $\{a\_n\}$.

Answer: $300$


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

If Greg rolls six fair six-sided dice, what is the probability that he rolls more 2's than 5's?

Answer: $\dfrac{16710}{46656}$


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

In 2000, there were 60,000 cases of a disease reported in a country. By 2020, there were only 300 cases reported. Assume the number of cases decreased exponentially rather than linearly. Determine how many cases would have been reported in 2010.

Answer: $4243$


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

The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers?

Answer: $\frac{53}{3}$


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

Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one tie such that the shirt and hat are never of the same color, and the tie must match the color of the hat?

Answer: $18900$


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

Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: $0.472$


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

(1) Given that $\tan \alpha = -2$, calculate the value of $\dfrac {3\sin \alpha + 2\cos \alpha}{5\cos \alpha - \sin \alpha}$.
(2) Given that $\sin \alpha = \dfrac {2\sqrt{5}}{5}$, calculate the value of $\tan (\alpha + \pi) + \dfrac {\sin \left( \dfrac {5\pi}{2} + \alpha \right)}{\cos \left( \dfrac {5\pi}{2} - \alpha \right)}$.

Answer: $-\dfrac{5}{2}$


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>.

Anton makes custom watches for a jewelry store. Each watch consists of a bracelet, a gemstone, and a clasp.

Bracelets are available in silver, gold, and steel. Anton has precious stones: cubic zirconia, emerald, quartz, diamond, and agate - and clasps: classic, butterfly, buckle. Anton is happy only when three watches are displayed in a row from left to right according to the following rules:

- There must be a steel watch with a classic clasp and cubic zirconia stone.
- Watches with a classic clasp must be flanked by gold and silver watches.
- The three watches in a row must have different bracelets, gemstones, and clasps.

How many ways are there to make Anton happy?

Answer: $72$


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

If the positive numbers $a$ and $b$ satisfy the equation $2+\log_{2}a=3+\log_{3}b=\log_{6}(a+b)$, find the value of $\frac{1}{a}+\frac{1}{b}$.

Answer: $108$


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

The sequence is formed by taking all positive multiples of 4 that contain at least one digit that is either a 2 or a 3. What is the $30^\text{th}$ term of this sequence?

Answer: $132$


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

Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $ABC$, and $\frac{a-c}{b-\sqrt{2}c}=\frac{sin(A+C)}{sinA+sinC}$.
$(Ⅰ)$ Find the measure of angle $A$;
$(Ⅱ)$ If $a=\sqrt{2}$, $O$ is the circumcenter of triangle $ABC$, find the minimum value of $|3\overrightarrow{OA}+2\overrightarrow{OB}+\overrightarrow{OC}|$;
$(Ⅲ)$ Under the condition of $(Ⅱ)$, $P$ is a moving point on the circumcircle of triangle $ABC$, find the maximum value of $\overrightarrow{PB}•\overrightarrow{PC}$.

Answer: $\sqrt{2} + 1$


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

Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $c\cos A=5$, $a\sin C=4$.

(1) Find the length of side $c$;
(2) If the area of $\triangle ABC$, $S=16$, find the perimeter of $\triangle ABC$.

Answer: $13+ \sqrt {41}$


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

Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line.

Answer: $-10$


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

Given complex numbers $z_1 = \cos\theta - i$ and $z_2 = \sin\theta + i$, the maximum value of the real part of $z_1 \cdot z_2$ is \_\_\_\_\_\_, and the maximum value of the imaginary part is \_\_\_\_\_\_.

Answer: $\sqrt{2}$


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

There are  $10$  girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

Answer: $512$


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>.

Given that $\overrightarrow{OA}=(1,0)$, $\overrightarrow{OB}=(1,1)$, and $(x,y)=λ \overrightarrow{OA}+μ \overrightarrow{OB}$, if $0\leqslant λ\leqslant 1\leqslant μ\leqslant 2$, then the maximum value of $z= \frac {x}{m}+ \frac{y}{n}(m > 0,n > 0)$ is $2$. Find the minimum value of $m+n$.

Answer: $\frac{5}{2}+ \sqrt{6}$


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

Given an isosceles right triangle ∆PQR with hypotenuse PQ = 4√2, let S be the midpoint of PR. On segment QR, point T divides it so that QT:TR = 2:1. Calculate the area of ∆PST.

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


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

Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq.

Answer: $\frac{3}{28}\left(10-\sqrt{2}\right)$


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

Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints.

Answer: $9\pi$


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

Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .

*2021 CCA Math Bonanza Lightning Round #3.4*

Answer: $164$


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

Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x-1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?

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>.

Begin by adding 78.652 to 24.3981. After adding, subtract 0.025 from the result. Finally, round the answer to the nearest hundredth.

Answer: $103.03$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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, with $C= \dfrac {\pi}{3}$, $b=8$. The area of $\triangle ABC$ is $10 \sqrt {3}$.
(I) Find the value of $c$;
(II) Find the value of $\cos (B-C)$.

Answer: $\dfrac {13}{14}$


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>.

If \(\sin A = \frac{p}{5}\) and \(\frac{\cos A}{\tan A} = \frac{q}{15}\), find \(q\).

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>.

 $\textbf{Problem 5.}$  Miguel has two clocks, one clock advances  $1$  minute per day and the other one goes  $15/10$  minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously?

Answer: $1440$


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

You have $7$ red shirts, $8$ green shirts, $10$ pairs of pants, $10$ blue hats, $10$ red hats, and $5$ scarves (each distinct). How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one scarf, without having the same color of shirts and hats?

Answer: $7500$


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

Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible values of \(a\), given that \(a + b = 114.5\).

Answer: $\frac{190}{3}$


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 four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g.  $3311$  and  $2222$  are stutterer numbers. Find all stutterer numbers that are square numbers.

Answer: $7744$


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

Let the domain of the function $f(x)$ be $R$. $f(x+1)$ is an odd function, and $f(x+2)$ is an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, calculate $f(\frac{9}{2})$.

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


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

We call a four-digit number with the following property a "centered four-digit number": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit number is exactly in the middle position. For example, 2021 is a "centered four-digit number". How many "centered four-digit numbers" are there, including 2021?

Answer: $90$


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

In the Cartesian coordinate system, the coordinates of point P(a,b) satisfy $a \neq b$, and both $a$ and $b$ are elements of the set $\{1,2,3,4,5,6\}$. Additionally, the distance from point P to the origin, $|OP|$, is greater than or equal to 5. The number of such points P is ______.

Answer: $20$


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

In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\overrightarrow{m}\left(a, \sqrt{3b}\right)$ is parallel to $\overrightarrow{n}=\left(\cos A,\sin B\right)$.

(1) Find $A$;

(2) If $a= \sqrt{7},b=2$, find the area of $∆ABC$.

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


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

Given the sequence $103, 1003, 10003, 100003, \dots$, find how many of the first 2048 numbers in this sequence are divisible by 101.

Answer: $20$


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

Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: $0.293$


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

In triangle $ABC$, $AB = 5$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$.

Answer: $\sqrt{0.45}$


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

What is the smallest positive value of $m$ so that the equation $15x^2 - mx + 315 = 0$ has integral solutions?

Answer: $150$


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

Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have.

Answer: $10$


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

Given that the area of obtuse triangle $\triangle ABC$ is $2 \sqrt {3}$, $AB=2$, $BC=4$, find the radius of the circumcircle.

Answer: $\dfrac {2 \sqrt {21}}{3}$


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 fly trapped inside a rectangular prism with dimensions $1$ meter, $2$ meters, and $3$ meters decides to tour the corners of the box. It starts from the corner $(0,0,0)$ and ends at the corner $(0,0,3)$, visiting each of the other corners exactly once. Determine the maximum possible length, in meters, of its path assuming that it moves in straight lines.
A) $\sqrt{14} + 4 + \sqrt{13} + \sqrt{5}$
B) $\sqrt{14} + 4 + \sqrt{5} + \sqrt{10}$
C) $2\sqrt{14} + 4 + 2$
D) $\sqrt{14} + 6 + \sqrt{13} + 1$
E) $\sqrt{14} + 6 + \sqrt{13} + \sqrt{5}$

Answer: $\sqrt{14} + 6 + \sqrt{13} + \sqrt{5}$


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>.

Xiaoming takes 100 RMB to the store to buy stationery. After returning, he counts the money he received in change and finds he has 4 banknotes of different denominations and 4 coins of different denominations. The banknotes have denominations greater than 1 yuan, and the coins have denominations less than 1 yuan. Furthermore, the total value of the banknotes in units of "yuan" must be divisible by 3, and the total value of the coins in units of "fen" must be divisible by 7. What is the maximum amount of money Xiaoming could have spent?

(Note: The store gives change in denominations of 100 yuan, 50 yuan, 20 yuan, 10 yuan, 5 yuan, and 1 yuan banknotes, and coins with values of 5 jiao, 1 jiao, 5 fen, 2 fen, and 1 fen.)

Answer: $63.37$


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>.

Let $a,$ $b,$ $c,$ $d$ be distinct integers, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$  Find the smallest possible value of
\[|a + b \omega + c \omega^2 + d \omega^3|.\]

Answer: $\sqrt{4.5}$


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

As shown in the diagram, points \(E\), \(F\), \(G\), and \(H\) are located on the four sides of the square \(ABCD\) as the trisection points closer to \(B\), \(C\), \(D\), and \(A\) respectively. If the side length of the square is 6, what is the area of the shaded region?

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>.

In triangle $PQR,$ $PQ = 24,$ $QR = 25,$ $PR = 7,$ and point $H$ is the intersection of the altitudes (the orthocenter). Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $PQR$ and $P'Q'R'$?

Answer: $84$


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

Let  $A$  be a subset of  $\{1,2,\ldots,2020\}$  such that the difference of any two distinct elements in  $A$  is not prime. Determine the maximum number of elements in set  $A$ .

Answer: $505$


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

Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel.

Answer: $87880$


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

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

Answer: $291$


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

Evaluate the expression: \[ M = \frac{\sqrt{\sqrt{7} + 3} + \sqrt{\sqrt{7} - 3}}{\sqrt{\sqrt{7} + 2}} + \sqrt{5 - 2\sqrt{6}}, \]
and determine the value of $M$.
A) $\sqrt{9}$
B) $\sqrt{2} - 1$
C) $2\sqrt{2} - 1$
D) $\sqrt{\frac{7}{2}}$
E) $1$

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


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

Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win of 30 rubles, and there are no other prizes. The Scientist decided to take a risk since he has nothing to lose. Find the probability that the Absent-Minded Scientist will be able to win enough money to buy a bus ticket.

Answer: $0.19$


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

Let sets $X$ and $Y$ have $30$ and $25$ elements, respectively, and there are at least $10$ elements in both sets. Find the smallest possible number of elements in $X \cup Y$.

Answer: $45$


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

Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable.

Answer: $144$


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

In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$$, (where $θ$ is the parameter).
(Ⅰ) Find the Cartesian equation of line $l$ and the general equation of curve $C$;
(Ⅱ) Curve $C$ intersects the x-axis at points $A$ and $B$, with $x_A < x_B$, $P$ is a moving point on line $l$, find the minimum perimeter of $\triangle PAB$.

Answer: $2+ \sqrt {34}$


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>.

Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).

Answer: $1013$


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

In the triangular pyramid $A-BCD$, where $AB=AC=BD=CD=BC=4$, the plane $\alpha$ passes through the midpoint $E$ of $AC$ and is perpendicular to $BC$, calculate the maximum value of the area of the section cut by plane $\alpha$.

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


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

What is the sum of every third odd number between $100$ and $300$?

Answer: $6800$


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

$14 N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $N$.

Answer: $35964$


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

In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many scenarios are there where both A and B grab a red envelope? (Answer with a number)

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>.

Compute the following expression:
\[
\frac{(1 + 15) \left( 1 + \dfrac{15}{2} \right) \left( 1 + \dfrac{15}{3} \right) \dotsm \left( 1 + \dfrac{15}{17} \right)}{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{15} \right)}.
\]

Answer: $496$


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

If $M = 1! \times 2! \times 3! \times 4! \times 5! \times 6! \times 7! \times 8! \times 9!$, calculate the number of divisors of $M$ that are perfect squares.

Answer: $672$


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

Find the limit, when  $n$  tends to the infinity, of   $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$

Answer: $\sqrt{3}$


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

Given the function $f(x) = x^2 - 3x - 1$, find the derivative of $f(2)$ and $f'(1)$.

Answer: $-1$


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

Given $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, find the value of $g(-1)$, where $g(x) = f(x) + 2$.

Answer: $-1$


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

Let  $a$  and  $b$  be nonzero real numbers such that  $\tfrac{1}{3a}+\tfrac{1}{b}=2011$  and  $\tfrac{1}{a}+\tfrac{1}{3b}=1$ .  What is the quotient when  $a+b$  is divided by  $ab$ ?

Answer: $1509$


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

Given $f(x)= \begin{cases} 2a-(x+ \frac {4}{x}),x < a\\x- \frac {4}{x},x\geqslant a\\end{cases}$.
(1) When $a=1$, if $f(x)=3$, then $x=$ \_\_\_\_\_\_;
(2) When $a\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=$ \_\_\_\_\_\_.

Answer: $- \frac {11}{6}$


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 the ratio of the legs of a right triangle is $1:3$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $1:3$
B) $1:9$
C) $3:1$
D) $9:1$

Answer: $9:1$


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>.

A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, forming a tray with taped edges. Determine the height of this tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges.

Answer: $5\sqrt{2}$


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

A cube \(ABCDA_1B_1C_1D_1\) has edge length 1. Point \(M\) is taken on the side diagonal \(A_1D\), and point \(N\) is taken on \(CD_1\), such that the line segment \(MN\) is parallel to the diagonal plane \(A_1ACC_1\). Find the minimum length of \(MN\).

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


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

Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, if $g(x) = f(x) + 2$, then $g(-1) = \_\_\_\_\_\_\_$.

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>.

Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers.

Answer: $20$


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

In right triangle \( \triangle ABC \) where \(\angle ACB = 90^\circ\), \(CA = 3\), and \(CB = 4\), there is a point \(P\) inside \(\triangle ABC\) such that the sum of the distances from \(P\) to the three sides is \(\frac{13}{5}\). Find the length of the locus of point \(P\).

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


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>.

What is the value of $\frac{(2200 - 2096)^2}{121}$?

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>.

In an acute-angled triangle  $ABC$ , the point  $O$  is the center of the circumcircle, and the point  $H$  is the orthocenter. It is known that the lines  $OH$  and  $BC$  are parallel, and  $BC = 4OH $ . Find the value of the smallest angle of triangle  $ ABC $ .

(Black Maxim)

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>.

A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game.

Answer: $21$


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

A triangle has altitudes of lengths 15, 21, and 35. Find its area.

Answer: $210$


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

The cells of a $20 \times 20$ table are colored in $n$ colors such that for any cell, in the union of its row and column, cells of all $n$ colors are present. Find the greatest possible number of blue cells if:

(a) $n=2$;

(b) $n=10$.

Answer: $220$


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

Suppose that  $S$  is a series of real numbers between  $2$  and  $8$  inclusive, and that for any two elements  $y > x$  in  $S,$    $$ 98y - 102x - xy \ge 4. $$  What is the maximum possible size for the set  $S?$   $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

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>.

In a WeChat group, five people, namely A, B, C, D, and E, are playing a game of grabbing red packets. There are $4$ red packets in total, each person can grab at most one, and all red packets will be grabbed. Among the $4$ red packets, there are two $2$ yuan packets, one $3$ yuan packet, and one $4$ yuan packet. (Assume that red packets with the same amount are considered the same.) The number of ways that both A and B can grab a red packet is _______ (answer with a number).

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>.

Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares.

![Image](https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png)

What is the largest number of such knights that can be placed on the  $8\times 8$  chessboard without them attacking each other?

*Proposed by Arsenii Nikolaiev*

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 there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.

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>.

In the pattern of numbers shown, every row begins with a 1 and ends with a 2. Each of the numbers, not on the end of a row, is the sum of the two numbers located immediately above and to the right, and immediately above and to the left. For example, in the fourth row the 9 is the sum of the 4 and the 5 in the third row. If this pattern continues, the sum of all of the numbers in the thirteenth row is:

Answer: $12288$


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

In triangle $ABC$, where the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, it is given that $2 \sqrt {3}ac\sin B = a^{2} + b^{2} - c^{2}$.
$(1)$ Determine the size of angle $C$;
$(2)$ If $b\sin (\pi - A) = a\cos B$ and $b= \sqrt {2}$, find the area of $\triangle ABC$.

Answer: $\frac{\sqrt {3} + 1}{4}$


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>.

In a plane Cartesian coordinate system, points where both the vertical and horizontal coordinates are integers are called lattice points. The number of lattice points $(x, y)$ satisfying the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is:

Answer: $16$