MEMO 2021

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$ . The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$ , respectively. Lines $EF$ and $BC$ meet at $G$ . Prove that $AD$ and $DG$ are perpendicular.

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$ . Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$ . Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$ . The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$ , such that the points $P, B, C$ , and $Q$ lie in this order on the line $BC$ . Prove that $PB=QC$ .

Proposed by Dominik Burek, Poland

Find all pairs $(n, p)$ of positive integers such that $p$ is prime and $$1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2).$$

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$ .