RO-TST 2015 Problem 1

Let $ABC$ be a triangle, let $O$ be its circumcenter, let $A'$ be the orthogonal projection of $A$ on the line $BC$ , and let $X$ be a point on the open ray $AA'$ emanating from $A$ . The internal bisectrix of the angle $BAC$ meets the circumcircle of $ABC$ again at $D$ . Let $M$ be the midpoint of the segment $DX$ . The line through $O$ and parallel to the line $AD$ meets the line $DX$ at $N$ . Prove that the angles $BAM$ and $CAN$ are equal.