MEMO 2019

Let $ABC$ be an acute-angled triangle such that $AB<AC$ . Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$ . Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$ . Finally, let $M$ be the midpoint of the side $AB$ . Prove that $\angle APD=\angle MPB$ .

Proposed by Dominik Burek, Poland

Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$ . Denote by $D$ the midpoint of the shorter arc $AB$ of $c$ . Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$ . Prove that $X, Y$ and $P$ are collinear.

Proposed by Dominik Burek, Poland

Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$ . Prove that $c(a-1)+b$ does not divide $c(b-1)+a$ .

Proposed by Dominik Burek, Poland

Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$ . Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$ .

Proposed by Alain Rossier, Switzerland