VMO 2017 Problem 3

Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$ , respectively. $AH$ intersects $(O)$ at $D$ ( $D\ne A$ ).

a) Let $I$ be the midpoint of $AH$ , $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$ . Prove that $MN\perp OH$ .

b) The lines $DE$ , $DF$ intersect $(O)$ at $P,Q$ respectively ( $P\ne D,Q\ne D$ ). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ( $R\ne A, S\ne A$ ). Prove that $BP,CQ,RS$ are concurrent.