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

On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \mathrm{~cm}$. The actual distance between these two cities is $1050 \mathrm{~km}$. What is the scale of Tony's map?

Answer: $1:5 000 000$


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

What is the smallest positive integer that has exactly eight distinct positive divisors, where all divisors are powers of prime numbers?

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>.

For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$

Answer: $43$


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

How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct?

Answer: $5040$


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

A recipe calls for $5 \frac{3}{4}$ cups of flour and $2 \frac{1}{2}$ cups of sugar. If you make one-third of the recipe, how many cups of flour and sugar do you need? Express your answers as mixed numbers.

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


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>.

Let the common ratio of the geometric sequence $\{a_n\}$ be $q$, and the sum of the first $n$ terms be $S_n$. If $S_{n+1}$, $S_n$, and $S_{n+2}$ form an arithmetic sequence, find the value of $q$.

Answer: $-2$


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

Fix a sequence  $a_1,a_2,a_3\ldots$  of integers satisfying the following condition:for all prime numbers  $p$  and all positive integers  $k$ ,we have  $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of  $a_{2013}$ .

Answer: $13$


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

Find the remainder when $5^{2021}$ is divided by $17$.

Answer: $14$


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

Convert the binary number $110110100_2$ to base 4.

Answer: $31220_4$


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

In the Cartesian coordinate system $xOy$, let $D$ be the region represented by the inequality $|x| + |y| \leq 1$, and $E$ be the region consisting of points whose distance from the origin is no greater than 1. If a point is randomly thrown into $E$, the probability that this point falls into $D$ is.

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


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>.

Given a circle $C$: $(x+1)^{2}+y^{2}=r^{2}$ and a parabola $D$: $y^{2}=16x$ intersect at points $A$ and $B$, and $|AB|=8$, calculate the area of circle $C$.

Answer: $25\pi$


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

A rectangular sheet of paper is 20 cm long and 12 cm wide. It is folded along its diagonal. What is the perimeter of the shaded region?

Answer: $64$


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

Inside the circle \(\omega\), there is a circle \(\omega_{1}\) that touches it at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(L\), and \(M\) are collinear. Find the radius of circle \(\omega\) if the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively.

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>.

Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another.

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>.

Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive?

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>.

Compute \(\arccos(\cos 8.5)\). All functions are in radians.

Answer: $2.217$


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

Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 200$ such that $i^x+i^y$ is a real number.

Answer: $4950$


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

How many sequences of $0$s and $1$s of length $20$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no four consecutive $1$s?
A) 65
B) 75
C) 85
D) 86
E) 95

Answer: $86$


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

Around a circle, an isosceles trapezoid \(ABCD\) is described. The side \(AB\) touches the circle at point \(M\), and the base \(AD\) touches the circle at point \(N\). The segments \(MN\) and \(AC\) intersect at point \(P\), such that \(NP: PM=2\). Find the ratio \(AD: BC\).

Answer: $3:1$


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

Compute  $e^{\pi}+\pi^e$ . If your answer is  $A$  and the correct answer is  $C$ , then your score on this problem will be  $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$  (note that the closer you are to the right answer, the higher your score is).

*2017 CCA Math Bonanza Lightning Round #5.2*

Answer: $45.5999$


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

There are eight different symbols designed on  $n\geq 2$  different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any  $k$  symbols  $(1\leq k\leq 7)$  the number of shirts containing at least one of the  $k$  symbols is even. Determine the value of  $n$ .

Answer: $255$


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

Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2.

Answer: $1000111_2$


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

Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53.

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>.

The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\geq 2$. Determine the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer.

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>.

Given the function $f(x)=a\frac{x+1}{x}+\ln{x}$, the equation of the tangent line at the point (1, f(1)) is y=bx+5.

1. Find the real values of a and b.
2. Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$, where $e$ is the base of the natural logarithm.

Answer: $3+\ln{2}$


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

Given an arithmetic sequence $\{a_n\}$ with the first term $a$ ($a \in \mathbb{R}, a \neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\frac {a_{2n}}{a_{n}}= \frac {4n-1}{2n-1}$$.
(1) Find the general formula for $\{a_n\}$ and $S_n$;
(2) Does there exist positive integers $n$ and $k$ such that $S_n$, $S_{n+1}$, and $S_{n+k}$ form a geometric sequence? If so, find the values of $n$ and $k$; if not, explain why.

Answer: $k=3$


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 sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$.

Answer: $902$


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

Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue.

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>.

So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother.

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 the polar coordinate system, the equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur.

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


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

Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas?

Answer: $450$


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

For  $a>0$ , let  $f(a)=\lim_{t\to\+0} \int_{t}^1 |ax+x\ln x|\ dx.$  Let  $a$  vary in the range  $0 <a< +\infty$ , find the minimum value of  $f(a)$ .

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


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>.

Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous Online tournament are also open **[here](https://www.stanfordmathtournament.com/competitions/smt-2024-online)**. Also if you’re coming in-person and want to get some pre-competition action, submit your solutions to PoTW [here](https://forms.gle/2FpDoHFeVHg2aqWS6), and the top 3 people with the most correct solutions will receive surprise prize at the tournament :)**Easy:** Consider the curves  $x^2 + y^2 = 1$  and  $2x^2 + 2xy + y^2 - 2x -2y = 0$ . These curves intersect at two points, one of which is  $(1, 0)$ . Find the other one. **Hard:** Tony the mouse starts in the top left corner of a  $3x3$  grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center.

Again feel free to reach out to us at [stanford.math.tournament@gmail.com](mailto:stanford.math.tournament@gmail.com) with any questions, or ask them in this thread!

Best,

[The SMT Team](https://www.stanfordmathtournament.com/our-team)

Answer: $\frac{1}{7}$


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 regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? [asy]
size(175);
defaultpen(linewidth(0.8));
path octagon;
string labels[]={"A","B","C","D","E","F","G","H","I"};
for(int i=0;i<=7;i=i+1)
{
pair vertex=dir(135-45/2-45*i);
octagon=octagon--vertex;
label(" $"+labels[i]+"$ ",vertex,dir(origin--vertex));
}
draw(octagon--cycle);
dot(origin);
label(" $J$ ",origin,dir(0));
[/asy]

Answer: $1152$


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

A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths?

Answer: $60$


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

\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \).

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>.

From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes.

Answer: $60$


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

Let  $ABC$  be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let  $N = 2017$ , and define the  $A$ -*ntipodes* to be the points  $A_1,\dots, A_N$  to be the points on segment  $BC$  such that  $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$ , and similarly define the  $B$ ,  $C$ -ntipodes. A line  $\ell_A$  through  $A$  is called a *qevian* if it passes through an  $A$ -ntipode, and similarly we define qevians through  $B$  and  $C$ . Compute the number of ordered triples  $(\ell_A, \ell_B, \ell_C)$  of concurrent qevians through  $A$ ,  $B$ ,  $C$ , respectively.

*Proposed by Brandon Wang*

Answer: $2017^3 - 2$


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>.

Find the greatest root of the polynomial $f(x) = 16x^4 - 8x^3 + 9x^2 - 3x + 1$.

Answer: $0.5$


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

Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill.

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>.

Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls.

Answer: $\frac{35}{128}$


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

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. Determine the largest possible area of pentagon $ABCDE$.

Answer: $9 + 3\sqrt{2}$


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

Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \( 20 \text{ cm} \), what is the outside perimeter of the mosaic?

Answer: $240$


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

Given a square ABCD with a side length of 2, points M and N are the midpoints of sides BC and CD, respectively. If vector $\overrightarrow {MN}$ = x $\overrightarrow {AB}$ + y $\overrightarrow {AD}$, find the values of xy and $\overrightarrow {AM}$ • $\overrightarrow {MN}$.

Answer: $-1$


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

Given the function \( f(x) = x^3 + ax^2 + bx + c \) where \( a, b, \) and \( c \) are nonzero integers, if \( f(a) = a^3 \) and \( f(b) = b^3 \), what is the value of \( c \)?

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>.

Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number.

Answer: $38$


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

If $|x| + x + y = 14$ and $x + |y| - y = 16,$ find $x + y.$

Answer: $-2$


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

Given \( 0 \leq m-n \leq 1 \) and \( 2 \leq m+n \leq 4 \), when \( m - 2n \) reaches its maximum value, what is the value of \( 2019m + 2020n \)?

Answer: $2019$


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

In a modified game similar to Deal or No Deal, participants choose a box at random from a set of 30 boxes, each containing one of the following values:
\begin{tabular}{|c|c|}
\hline
\$0.50 & \$2,000 \\
\hline
\$2 & \$10,000 \\
\hline
\$10 & \$20,000 \\
\hline
\$20 & \$40,000 \\
\hline
\$50 & \$100,000 \\
\hline
\$100 & \$200,000 \\
\hline
\$500 & \$400,000 \\
\hline
\$1,000 & \$800,000 \\
\hline
\$1,500 & \$1,000,000 \\
\hline
\end{tabular}
After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate to have at least a 50% chance of holding a box containing at least \$200,000?

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 the ancient Chinese mathematical text "The Mathematical Classic of Sunzi", there is a problem stated as follows: "Today, a hundred deer enter the city. Each family takes one deer, but not all are taken. Then, three families together take one deer, and all deer are taken. The question is: how many families are there in the city?" In this problem, the number of families in the city is ______.

Answer: $75$


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

Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown. The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it. What is the greatest possible integer that she can write on the top cube?

Answer: $118$


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

Given the equations:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 3,
\]
find the value of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\).

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>.

Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\]

Answer: $4044$


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

A bridge needs to be constructed over a river with a required elevation of 800 feet from one side to the other. Determine the additional bridge length required if the gradient is reduced from 2% to 1.5%.

Answer: $13333.33$


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

It is known that the vertex of angle $\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \frac {1}{2}x$ ($x\leq0$).
(Ⅰ) Find the value of $\cos\left( \frac {\pi}{2}+\theta \right)$;
(Ⅱ) If $\cos\left( \alpha+ \frac {\pi}{4} \right)=\sin\theta$, find the value of $\sin\left(2\alpha+ \frac {\pi}{4}\right)$.

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


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

A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5?

Answer: $\frac{1}{15}$


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

Medians $\overline{DP}$ and $\overline{EQ}$ of isosceles $\triangle DEF$, where $DE=EF$, are perpendicular. If $DP= 21$ and $EQ = 28$, then what is ${DE}$?

Answer: $\frac{70}{3}$


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>.

There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there?

Answer: $36$


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

In the center of a square, there is a police officer, and in one of the vertices, there is a gangster. The police officer can run throughout the whole square, while the gangster can only run along its sides. It is known that the ratio of the maximum speed of the police officer to the maximum speed of the gangster is: 0.5; 0.49; 0.34; 1/3. Can the police officer run in such a way that at some point he will be on the same side of the square as the gangster?

Answer: $1/3$


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

A phone number \( d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7} \) is called "legal" if the number \( d_{1} d_{2} d_{3} \) is equal to \( d_{4} d_{5} d_{6} \) or to \( d_{5} d_{6} d_{7} \). For example, \( 234-2347 \) is a legal phone number. Assume each \( d_{i} \) can be any digit from 0 to 9. How many legal phone numbers exist?

Answer: $19990$


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

In the Cartesian coordinate system $(xOy)$, let the line $l: \begin{cases} x=2-t \\ y=2t \end{cases} (t \text{ is a parameter})$, and the curve $C_{1}: \begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is a parameter})$. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis:

(1) Find the polar equations of $C_{1}$ and $l$:

(2) Let curve $C_{2}: \rho=4\sin\theta$. The curve $\theta=\alpha(\rho > 0, \frac{\pi}{4} < \alpha < \frac{\pi}{2})$ intersects with $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. If the midpoint of segment $AB$ lies on line $l$, find $|AB|$.

Answer: $\frac{4\sqrt{10}}{5}$


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

On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, when A and C meet for the first time, B is also exactly halfway between them. If B's speed is four times that of A's, how many meters has A run when B and C started?

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>.

Given the parametric equation of line $l$ is $\begin{cases} & x=1+3t \\ & y=2-4t \end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l$.

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


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

Given the letters a, b, c, d, e arranged in a row, find the number of arrangements where both a and b are not adjacent to c.

Answer: $36$


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

How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)?

Answer: $36$


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

Find the number of pairs of natural numbers \((x, y)\) such that \(1 \leq x, y \leq 1000\) and \(x^2 + y^2\) is divisible by 5.

Answer: $200000$


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

Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \dfrac{2\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord of length $2$.

(Ⅰ) Find the standard equation of the ellipse $C$;

(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, $O$ be the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Investigate whether the value of $\dfrac{|MN|}{|OQ|^2}$ is a constant. If it is, find this constant; if not, please explain why.

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


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

Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,

$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}$

Rearrange the numbers in the sequence $\{a_k\}_{k=1}^{11}$ in decreasing order to produce a new sequence $\{b_k\}_{k=1}^{11}$. Find the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k$.

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>.

Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2c+b-2a\cos B=0$.
$(1)$ Find angle $A$;
$(2)$ If $a=2\sqrt{3}$, $\overrightarrow{BA}\cdot \overrightarrow{AC}=\frac{3}{2}$, and $AD$ is the median of $\triangle ABC$, find the length of $AD$.

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


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

Find the number of different possible rational roots of the polynomial:
\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\]

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>.

The number of minutes in a week is closest to:

Answer: $10000$


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

We define the polynomial   $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$   Find the largest prime divisor of  $P (2)$ .

Answer: $61$


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

What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43?

Answer: $19$


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

Given:
$$ \frac{ \left( \frac{1}{3} \right)^2 + \left( \frac{1}{4} \right)^2 }{ \left( \frac{1}{5} \right)^2 + \left( \frac{1}{6} \right)^2} = \frac{37x}{73y} $$
Express $\sqrt{x} \div \sqrt{y}$ as a common fraction.

Answer: $\frac{75 \sqrt{73}}{6 \sqrt{61} \sqrt{37}}$


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>.

Consider real numbers  $A$ ,  $B$ , \dots,  $Z$  such that \[
  EVIL = \frac{5}{31}, \;
  LOVE = \frac{6}{29}, \text{ and }
  IMO  = \frac{7}{3}.
\] If  $OMO = \tfrac mn$  for relatively prime positive integers  $m$  and  $n$ , find the value of  $m+n$ .

*Proposed by Evan Chen*

Answer: $579$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative.

Answer: $46$


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

Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the perpendiculars $C$ and $E$ on $r$ from points $J$ and $B$ is $180 m$. What is the shortest distance Juca can travel to return to his tent, passing through the river?

Answer: $180\sqrt{2}$


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>.

What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$?

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


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

Given the ellipse $C\_1: \frac{x^2}{8} + \frac{y^2}{4} = 1$ with left and right foci $F\_1$ and $F\_2$, a line $l\_1$ is drawn through point $F\_1$ perpendicular to the $x$-axis. Line $l\_2$ is perpendicular to $l\_1$ at point $P$, and the perpendicular bisector of segment $PF\_2$ intersects $l\_2$ at point $M$.

(I) Find the equation of the trajectory $C\_2$ of point $M$;
(II) Two perpendicular lines $AC$ and $BD$ are drawn through point $F\_2$, intersecting ellipse $C\_1$ at points $A$, $C$ and $B$, $D$ respectively. Find the minimum value of the area of quadrilateral $ABCD$.

Answer: $\frac{64}{9}$


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

A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How many times did the clock chime "cuckoo" during this period?

Answer: $43$


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

Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\overrightarrow{a}+y\overrightarrow{b}|=1$ and $xy\geqslant 0$, the inequality $|x+2y|\leqslant \frac{8}{\sqrt{15}}$ holds. Find the minimum value of $\overrightarrow{a}\cdot\overrightarrow{b}$.

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>.

We have  $n$  positive integers greater than  $1$  and less than  $10000$  such that neither of them is prime but any two of them are relative prime. Find the maximum value of  $n $ .

Answer: $25$


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

Given that the terminal side of angle $\alpha$ ($0 < \alpha < \frac{\pi}{2}$) passes through the point $(\cos 2\beta, 1+\sin 3\beta \cos \beta - \cos 3\beta \sin \beta)$, where $\frac{\pi}{2} < \beta < \pi$ and $\beta \neq \frac{3\pi}{4}$, calculate $\alpha - \beta$.

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


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>.

Vasya and Petya are participating in a school sports-entertainment game. They only have one pair of roller skates between the two of them, and need to traverse a distance of 3 km as quickly as possible. They start simultaneously, with one running and the other roller-skating. At any moment, the one on roller skates can leave them for the other and continue running without them. This exchange can occur as many times as desired. Find the minimum time for both friends to complete the distance (which is determined by who finishes last), given that Vasya's running and roller skating speeds are 4 km/h and 8 km/h respectively, while Petya's speeds are 5 km/h and 10 km/h. Assume that no time is lost during the exchange of roller skates.

Answer: $0.5$


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

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

Answer: $20\sqrt{5}$


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

The maximum and minimum values of the function y=2x^3-3x^2-12x+5 on the interval [0,3] need to be determined.

Answer: $-15$


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

Given the sequence 2, $\frac{5}{3}$, $\frac{3}{2}$, $\frac{7}{5}$, $\frac{4}{3}$, ..., then $\frac{17}{15}$ is the \_\_\_\_\_ term in this sequence.

Answer: $14$


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

A six-digit palindrome is a positive integer with respective digits $abcdcba$, where $a$ is non-zero. Let $T$ be the sum of all six-digit palindromes. Calculate the sum of the digits of $T$.

Answer: $20$


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

Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\).

Answer: $15$


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

If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 30$ and \[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{240}{pqr} = 1,\] compute $pqr$.

Answer: $1080$


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

In a certain sequence, the first term is $a_1 = 101$ and the second term is $a_2 = 102$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 2$ for all $n \geq 1$. Determine $a_{50}$.

Answer: $117$


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

Given a polynomial with integer coefficients,
\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\]
find the number of different possible rational roots of this polynomial.

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>.

Find the values of the real number \( a \) such that all the roots of the polynomial in the variable \( x \),

\[ x^{3}-2x^{2}-25x+a \]

are integers.

Answer: $-50$


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

Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\omega$ with radius $\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M$, where $A$ is at the top left corner of the square, find the length of $AP$.

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


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

In triangle \(ABC\), \(BK\) is the median, \(BE\) is the angle bisector, and \(AD\) is the altitude. Find the length of side \(AC\) if it is known that lines \(BK\) and \(BE\) divide segment \(AD\) into three equal parts and the length of \(AB\) is 4.

Answer: $2\sqrt{3}$


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

A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the locomotive is $100$ km/h. At what speed should the locomotive travel to minimize the total cost of traveling from city A to city B?

Answer: $20 \sqrt[3]{5}$


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

In the three-dimensional Cartesian coordinate system, the equation of the plane passing through $P(x_{0}, y_{0}, z_{0})$ with normal vector $\overrightarrow{m}=(a, b, c)$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$, and the equation of the line passing through $P(x_{0}, y_{0}, z_{0})$ with direction vector $\overrightarrow{n}=(A, B, C)$ is $\frac{x-x_{0}}{A}=\frac{y-y_{0}}{B}=\frac{z-z_{0}}{C}$. Read the above information and solve the following problem: Given the equation of the plane $\alpha$ as $3x+y-z-5=0$, and the equation of the line $l$ passing through the point $P(0,0,0)$ as $x=\frac{y}{2}=-z$, then one direction vector of the line $l$ is ______, and the cosine value of the angle between the line $l$ and the plane $\alpha$ is ______.

Answer: $\frac{\sqrt{55}}{11}$


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>.

Find maximum value of number  $a$  such that for any arrangement of numbers  $1,2,\ldots ,10$  on a circle, we can find three consecutive numbers such their sum bigger or equal than  $a$ .

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>.

Given that the radius of a sphere is 24cm, and the height of a cone is equal to the diameter of this sphere, and the surface area of the sphere is equal to the surface area of the cone, then the volume of this cone is \_\_\_\_\_\_ cm3.

Answer: $12288\pi$


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>.

How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?

Answer: $1535$


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

Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction.

Answer: $\frac{45}{8}$


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

Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits.

Answer: $972538$


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

There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position?

Answer: $50$


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

Define a sequence by \( a_0 = \frac{1}{3} \) and \( a_n = 1 + (a_{n-1} - 1)^3 \). Compute the infinite product \( a_0 a_1 a_2 \dotsm \).

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


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

A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar?

Answer: $16$


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

Let $\\((2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5\\)$. Evaluate the value of $\dfrac{a_0 + a_2 + a_4}{a_1 + a_3}$.

Answer: $-\dfrac{122}{121}$


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

Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x$,
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) When $x\in[0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$.

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>.

Find the least positive integer  $n$  such that the prime factorizations of  $n$ ,  $n + 1$ , and  $n + 2$  each have exactly two factors (as  $4$  and  $6$  do, but  $12$  does not).

Answer: $33$


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

Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of wheat has risen to 1.6 yuan per kilogram, so what he pays is equivalent to 80 yuan per mu of land. Through Li Qiang's calculations, you can find out how many mu of land this 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>.

Let  $n$  and  $k$  be integers satisfying  $\binom{2k}{2} + n = 60$ .  It is known that  $n$  days before Evan's 16th birthday, something happened.  Compute  $60-n$ .

*Proposed by Evan Chen*

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>.

Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF.

Answer: $2\sqrt{65}$


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

An acute isosceles triangle, \( ABC \), is inscribed in a circle. Through \( B \) and \( C \), tangents to the circle are drawn, meeting at point \( D \). If \( \angle ABC = \angle ACB = 3 \angle D \) and \( \angle BAC = k \pi \) in radians, then find \( k \).

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


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

Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number)

Answer: $420$


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

Given $f(x)=x^{2}-ax$, $g(x)=\ln x$, $h(x)=f(x)+g(x)$,
(1) Find the range of values for the real number $a$ such that $f(x) \geq g(x)$ holds true for any $x$ within their common domain;
(2) Suppose $h(x)$ has two critical points $x_{1}$, $x_{2}$, with $x_{1} \in (0, \frac{1}{2})$, and if $h(x_{1}) - h(x_{2}) > m$ holds true, find the maximum value of the real number $m$.

Answer: $\frac{3}{4} - \ln 2$


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

Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), calculate the value of \(\overrightarrow{CA} \cdot \overrightarrow{BE}\).

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>.

Compute the value of the following expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \]

Answer: $2046$


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

Points \(A\) and \(B\) are connected by two arcs of circles, convex in opposite directions: \(\cup A C B = 117^\circ 23'\) and \(\cup A D B = 42^\circ 37'\). The midpoints \(C\) and \(D\) of these arcs are connected to point \(A\). Find the angle \(C A D\).

Answer: $80$


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

Let \(a,\) \(b,\) \(c,\) \(d,\) \(e,\) \(f,\) \(g,\) and \(h\) be real numbers such that \(abcd = 8\) and \(efgh = 16.\) Find the minimum value of
\[
(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2.
\]

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>.

Determine the value of $l$ for which
\[\frac{9}{x + y + 1} = \frac{l}{x + z - 1} = \frac{13}{z - y + 2}.\]

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>.

Let $x_{0}$ be a zero of the function $f(x)=\sin \pi x$, and suppose it satisfies $|x_{0}| + f\left(x_{0}+ \frac{1}{2}\right) < 11$. Calculate the number of such zeros.

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 regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \( P \) within the circumscribed sphere, the probability that \( P \) lies within one of the five smaller spheres is closest to:

Answer: $0.2$


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>.

An influenza outbreak occurred in regions $A$, $B$, and $C$. The percentages of people with influenza in these three regions are $6\%$, $5\%$, and $4\%$ respectively. It is assumed that the populations of these three regions are in the ratio of $3:5:2$. Now, a person is randomly selected from these three regions.
$(1)$ Find the probability that this person has influenza.
$(2)$ If this person has influenza, find the probability that this person is from region $A$.

Answer: $\frac{18}{51}$


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

Teacher Tan awarded a stack of exercise books to the students who were named "Outstanding Students" in the math Olympiad class. If each student is awarded 3 books, there are 7 books left over; if each student is awarded 5 books, there are 9 books short. How many students received the award? How many exercise books are there in total?

Answer: $31$