Let $a, b, c$ be distinct real numbers and $x$ be a real number. Given that three numbers among $ax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a$ coincide, prove that $x=1$.

Two distinct positive integers are called "relatively consistent" if the larger one can be written as a sum of some distinct positive divisors of the other one. Show that there exist 2018 positive integers such that any two of them are "relatively consistent"

Let $H$ be the orthocenter of an acute angled triangle $ABC$. Circumcircle of the triangle $ABC$ and the circle of diameter $[AH]$ intersect at point $E$, different from $A$. Let $M$ be the midpoint of the small arc $BC$ of the circumcircle of the triangle $ABC$ and let $N$ the midpoint of the large arc $BC$ of the circumcircle of the triangle $BHC$ Prove that points $E, H, M, N$ are concyclic.

$n\geq3$ boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)

Let $a_1, a_2, ... , a_{1000}$ be a sequence of integers such that $a_1=3, a_2=7$ and for all $n=2, 3, ... , 999$ $a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)$. Find the number of indices $1\leq n\leq 1000$ for which $a_n+2018$ is a perfect square.

A point $E$ is located inside a parallelogram $ABCD$ such that $\angle BAE = \angle BCE$. The centers of the circumcircles of the triangles $ABE,ECB, CDE$ and $DAE$ are concyclic.

In the round robin chess tournament organized in a school every two students played one match among themselves. Find the minimal possible number of students in the school if each girl student has at least 21 wins in matches against boy students and each boy student has at least 12 wins in matches against girl students.

Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$