OME 2016

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$ , $a_2=g_2$ and $a_{10}=g_3$ . Find with proof that, for every positive integer $p$ , there is a positive integer $m$ , such that $g_p=a_m$ .

Given a positive prime number $p$ . Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$ , if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$ .

In the circumscircle of a triangle $ABC$ , let $A_1$ be the point diametrically opposed to the vertex $A$ . Let $A'$ the intersection point of $AA'$ and $BC$ . The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$ , respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$ .

Let $m$ be a positive integer and $a$ and $b$ be distinct positive integers strictly greater than $m^2$ and strictly less than $m^2+m$ . Find all integers $d$ such that $m^2 < d < m^2+m$ and $d$ divides $ab$ .