VMO 2017 Problem 7

Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$ . The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ( $E\ne A, F\ne A$ ).

a) Let $K$ be the intersection of $BE$ and $CF$ . Prove that $AK,BC,OG$ are concurrent.

b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$ ) of $(I)$ . $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$ . Assume that $MN$ intersects $(O)$ at $P$ nad $Q$ . Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$ , the circumcircle $(GPQ)$ always passes through two fixed points.