OME 2019

An integer set T is orensan if there exist integersa<b<c, where aand c are part of T, but b is not part of T. Count the number of subsets T of {1,2,...,2019} which are orensan.

Determine if there exists a finite set $S$ formed by positive prime numbers so that for each integer $n\geq2$ , the number $2^2 + 3^2 +...+ n^2$ is a multiple of some element of $S$ .

The real numbers $a$ , $b$ and $c$ verify that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct real roots; these roots are equal to $\tan y$ , $\tan 2y$ and $\tan 3y$ , for some real number $y$ . Find all possible values of $y$ , $0\leq y < \pi$ .

Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$

We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$ .Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$ Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$ .

In the scalene triangle $ABC$ , the bisector of angle A cuts side $BC$ at point $D$ . The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$ . Prove that angles $EDG$ and $ADF$ are equal.