Find all polynomials $P(x)$ with integer coefficients such that $a^2+b^2-c^2$ divides $P(a)+P(b)-P(c)$, for all integers $a,b,c$.

Is there exist a sequence $a_0,a_1,a_2,\cdots $ consisting of non-zero integers that satisfies the following condition? [b]Condition[/b]: For all integers $n$ ($\ge 2020$), equation $$a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0$$ has a real root with its absolute value larger than $2.001$.

Find all $f:\mathbb{Z}\to\mathbb{Z}$ such that \[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\] holds for all $x,y\in\mathbb{Z}$. [i]Proposed by usjl[/i]

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

[b]p1.[/b] $p, q, r$ are prime numbers such that $p^q + 1 = r$. Find $p + q + r$. [b]p2.[/b] $2014$ apples are distributed among a number of children such that each child gets a different number of apples. Every child gets at least one apple. What is the maximum possible number of children who receive apples? [b]p3.[/b] Cathy has a jar containing jelly beans. At the beginning of each minute he takes jelly beans out of the jar. At the $n$-th minute, if $n$ is odd, he takes out $5$ jellies. If n is even he takes out $n$ jellies. After the $46$th minute there are only $4$ jellies in the jar. How many jellies were in the jar in the beginning? [b]p4.[/b] David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work? [b]p5.[/b] Let $a, b, c, d$ be positive real numbers such that $$a^2 + b^2 = 1$$ $$c^2 + d^2 = 1;$$ $$ad - bc =\frac17$$ Find $ac + bd$. [b]p6.[/b] Three circles $C_A,C_B,C_C$ of radius $1$ are centered at points $A,B,C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap. [b]p7.[/b] Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$? [b]p8.[/b] Let $S$ be the set of integers $1 \le N \le 2^{20}$ such that $N = 2^i + 2^j$ where $i, j$ are distinct integers. What is the probability that a randomly chosen element of $S$ will be divisible by $9$? [b]p9.[/b] Given a two-pan balance, what is the minimum number of weights you must have to weigh any object that weighs an integer number of kilograms not exceeding $100$ kilograms? [b]p10.[/b] Alex, Michael and Will write $2$-digit perfect squares $A,M,W$ on the board. They notice that the $6$-digit number $10000A + 100M +W$ is also a perfect square. Given that $A < W$, find the square root of the $6$-digit number. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$. Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$. a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$. b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.

Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$. Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is: ${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $

Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$

Show that for any real $x\in[0,1]$ the inequality \[\frac{(1-x)x^2}{(1+x)^3}<\frac{1}{25}\] holds.

Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.

Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ . (In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)

Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$

Let $\mathbb{R}$ denote the set of all real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, \[f(xy+xf(x))=f(x)\left(f(x)+f(y)\right).\]

The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$? $ \textbf{(A)}\ 78 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 98 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 118$

Find all pairs of functions $ f : \mathbb R \to \mathbb R$, $g : \mathbb R \to \mathbb R$ such that \[f \left( x + g(y) \right) = xf(y) - y f(x) + g(x) \quad\text{for all } x, y\in\mathbb{R}.\]

Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

Let $f(n) = n + \lfloor \sqrt{n} \rfloor$. Prove that for every positive integer $m$, the integer sequence $m, f(m), f(f(m)), \dots$ contains at least one square of an integer.

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:\\ $\circ$ The first member $s_1$ is 2.\\ $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.\\ The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$. What are the 4 numbers that immediately follow $s_k = 2016$ in this sequence?

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that [b](i)[/b] $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and [b](ii)[/b] $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$