Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. Brazitikos Silouanos, Greece

Prove that if $p$ and $q$ are two prime numbers, such that $$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$then $p=q$.

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.

Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.

Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Show that $$\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}.$$When does the equality take place?

Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$. $Turkey$

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of $$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$

On a board there are written the integers from $1$ to $119$. Two players, $A$ and $B$, make a move by turn. A $move$ consists in erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player $A$ makes the first move. Find the highest integer $k$, such that the player $A$ can be sure that his score is not smaller than $k$.

In a convex quadrilateral $ABCD$ the angles $BAD$ and $BCD$ are equal. Points $M$ and $N$ lie on the sides $(AB)$ and $(BC)$ such that the lines $MN$ and $AD$ are parallel and $MN=2AD$. The point $H$ is the orthocenter of the triangle $ABC$ and the point $K$ is the midpoint of $MN$. Prove that the lines $KH$ and $CD$ are perpendicular.

Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.