Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$Find all possible values of $T = x + y + z$

Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.

Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.

There are $3$ clubs $A,B,C$ with non-empty members. For any triplet of members $(a, b, c)$ with $a \in A, b \in B, c \in C$, two of them are friend and two of them are not friend (here the friend relationship is bidirectional). Prove that one of these statements must be true 1. There exist one student from $A$ that knows all students from $B$ 2. There exist one student from $B$ that knows all students from $C$ 3. There exist one student from $C$ that knows all students from $A$

Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.

Let $ABC$ be a triangle inscribed in a circle ($\omega$) and $I$ is the incenter. Denote $D,E$ as the intersection of $AI,BI$ with ($\omega$). And $DE$ cuts $AC,BC$ at $F,G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of ($\omega$) at $A,B$ meet at $K$. Prove that three lines $AE,BD,KP$ are concurrent or parallel.

It is given a graph whose vertices are positive integers and an edge between numbers $a$ and $b$ exists if and only if $a + b + 1 | a^2 + b^2 + 1$. Is this graph connected?

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$is a perfect square.

In a school there are $40$ different clubs, each of them contains exactly $30$ children. For every $i$ from $1$ to $30$ define $n_i$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize $40$ new clubs with $30$ children in each of them such, that the analogical numbers $n_1, n_2,..., n_{30}$ will be the same for them.

Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

Let pairwise different positive integers $a,b, c$ with gcd$(a,b,c) = 1$ are such that $a | (b - c)^2, b | (c- a)^2, c | (a - b)^2$. Prove, that there is no non-degenerate triangle with side lengths $a, b$ and $c$.

Let be given a positive integer $n > 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that $$P(x)P(x^2) ... P(x^n) = P\left( x^{\frac{n(n+1)}{2}}\right)$$for all $x \in R$

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P(2x^2 - 4x)$ and $P(4x - 2x^2)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x),B(x)$ such that $A(x)B(x) = P(x)$ and $A(x) = B(x)$ has no root in $(-1.1)$

Let $ABC$ be a triangle, the circle having $BC$ as diameter cuts $AB,AC$ at $F,E$ respectively. Let $P$ a point on this circle. Let $C',B$' be the projections of $P$ upon the sides $AB,AC$ respectively. Let $H$ be the orthocenter of the triangle $AB'C'$. Show that $\angle EHF = 90^o$.

All of the numbers $1, 2,3,...,1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?

Some $n > 2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ... , lamp $k$ with $k+1$, ... , lamp $n$ with lamp $1$. At the beginning, all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice-versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$

A sequence $(a_1, a_2,...,a_k)$ consisting of pairwise different cells of an $n\times n$ board is called a cycle if $k \ge 4$ and cell ai shares a side with cell $a_{i+1}$ for every $i = 1,2,..., k$, where $a_{k+1} = a_1$. We will say that a subset $X$ of the set of cells of a board is malicious if every cycle on the board contains at least one cell belonging to $X$. Determine all real numbers $C$ with the following property: for every integer $n \ge 2$ on an $n\times n$ board there exists a malicious set containing at most $Cn^2$ cells.

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.

Find all triples of real numbers $(x, y,z)$ such that $$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$

Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.

Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

Find all functions $f : R^2 \to R$ that for all real numbers $x, y, z$ satisfies to the equation $f(f(x,z), f(z, y))= f(x, y) + z$

Let $ABC$ be a triangle, let $D$ be the touch point of the side $BC$ and the incircle of the triangle $ABC$, and let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the bisector of the angle $BAC$.

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$.

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.