RO-NMO 2005 Problem 83

Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$ , respectively $BC$ , and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$ . Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$ , with respectively $B'C$ and $C'N$ .

a) Prove that $AC' \perp (OPM)$ ; b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$ .

Mircea Fianu