Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.

Find all $ f: \mathbb{N}\to\mathbb{N}$ so that : $ f(n) \plus{} f(n \plus{} 1) \equal{} f(n \plus{} 2)f(n \plus{} 3) \minus{} 1996$

Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$. Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and \[a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}\] for all $n\geq 2$. Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.

On the interval $[0,1]$ are given functions $S(x) = 1 - x$ and $T(x) = x/2$. Does there exist a function of the form $f = g_1\circ g_2\circ ... \circ g_n$, where $n \in N$ and each $g_k$ is either $S(x)$ or $T(x)$, such that $$f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?$$

For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$. Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation \[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]

A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$ and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.

[u]Round 1[/u] [b]p1.[/b] What is the remainder when $2021$ is divided by $102$? [b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p3.[/b] In how many ways can $59$ be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.) [u]Round 2 [/u] [b]p4.[/b] Two positive integers have a sum of $60$. Their least common multiple is $273$. What is the positive diffeerence between the two numbers? [b]p5.[/b] How many ways are there to distribute $13$ identical apples among $4$ identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples. [b]p6.[/b] In square $ABCD$ with side length $5$, $P$ lies on segment $AB$ so that $AP = 3$ and $Q$ lies on segment $AD$ so that $AQ = 4$. Given that the area of triangle $CPQ$ is $x$, compute $2x$. [u]Round 3 [/u] [b]p7.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers such that $2a+3b+5c = 15$. [b]p8.[/b] What is the greatest integer $n \le 15$ such that $n + 1$ and $n^2 + 3$ are both prime? [b]p9.[/b] For positive integers $a, b$, and $c$, suppose that $gcd \,\,(a, b) = 21$, $gcd \,\,(a, c) = 10$, and $gcd \,\,(b,c) = 11$. Find $\frac{abc}{lcm \,\,(a,b,c)}$ . (Note: $gcd$ is the greatest common divisor function and $lcm$ is the least common multiple function.) [u]Round 4[/u] [b]p10.[/b] The vertices of a square in the coordinate plane are at $(0, 0)$, $(0, 6)$, $(6, 0)$, and $(6, 6)$. Line $\ell$ intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines $\ell$ are there that divide the square into two regions, one of them having an area of $12$? [b]p11.[/b] Let $f(n)$ be defined as follows for positive integers $n$: $f(1) = 0$, $f(n) = 1$ if $n$ is prime, and $f(n) = f(n - 1) + 1$ otherwise. What is the maximum value of $f(n)$ for $n \le 120$? [b]p12.[/b] The graph of the equation $y = x^3 + ax^2 + bx + c$ passes through the points $(2,4)$, $(3, 9)$, and $(4, 16)$. What is $b$? PS. You should use hide for answers. Rounds 5- 8 have been posted [url=https://artofproblemsolving.com/community/c3h2949415p26408227]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any real numbers $ a,b, $ there exists $ c\in [-2,1] $ such that $ \big| c^3+ac+b\big| \ge 1. $

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

Does there exist an irreducible two variable polynomial $f(x,y)\in \mathbb{Q}[x,y]$ such that it has only four roots $(0,1),(1,0),(0,-1),(-1,0)$ on the unit circle.

Estimate the value of $e^f$ , where $f = e^e$ .

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?

Consider the complex numbers $x,y,z$ such that $|x|=|y|=|z|=1$. Define the number $$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$ $\textbf{(a)}$ Prove that $a$ is a real number. $\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$. [i] (Stefan Bălăucă & Vlad Robu)[/i]

Let $a,b,c$ be positive real numbers that satisfy the condition $a + b + c = 1$. Prove the inequality $$\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123$$