BY-MO 2001 Problem 9

The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$ . Let $O$ be the intersection of $AC$ and $BD$ . Circle $S_2$ passes through $D$ and $O$ , intersecting $AD$ and $CD$ at $M$ and $N$ , respectively. Lines $OM$ and $AB$ intersect at $R$ , lines $ON$ and $BC$ intersect at $T$ , and $R$ and $T$ lie on the same side of line $BD$ as $A$ . Prove that $O$ , $R$ , $T$ , and $B$ are concyclic.