Let $p, q$ be real numbers. Knowing that there are positive real numbers $a, b, c$, different two by two, such that $$p=\frac{a^2}{(b-c)^2}+\frac{b^2}{(a-c)^2}+\frac{c^2}{(a-b)^2},$$$$q=\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}+\frac{1}{(b-a)^2}$$calculate the value of $$\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}$$in terms of $p, q$.

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

Find all integers $ n \ge 2 $ for which it is possible to divide any triangle $ T $ in triangles $ T_1, T_2, \cdots, T_n $ and choose medians $ m_1, m_2, \cdots, m_n $, one in each of these triangles, so that these $ n $ medians have equal length.

Find all positive integers $a, b$, and $c$ such that the numbers $$\frac{a+1}{b}, \frac{b+1}{c} \quad \text{and} \quad \frac{c+1}{a}$$are positive integers.

Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with$$f(x+f(y))=f(x)+a\lfloor y \rfloor $$for all $x,y\in \mathbb{R}$

There is a finite set of points in the plane, where each point is painted in any of $ n $ different colors $ (n \ge 4) $. It is known that there is at least one point of each color and that the distance between any pair of different colored points is less than or equal a 1. Prove that it is possible to choose 3 colors so that, by removing all points of those colors, the remaining set of points can be covered with a radius circle $ \frac {1} {\sqrt {3}} $.

A new chess piece named $ mammoth $ moves like a bishop l (that is, in a diagonal), but only in 3 of the 4 possible directions. Different $ mammoths $ in the board may have different missing addresses. Find the maximum number of $ mammoths $ that can be placed on an $ 8 \times 8 $ chessboard so that no $ mammoth $ can be attacked by another.

Let $\Gamma$ be the circumcircle of a triangle $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.