Let Ω and O be the circumcircle and the circumcentre of an acute-angled triangle ABC with AB>BC. The angle bisector of ∠ABC intersects Ω at M=B. Let Γ be the circle with diameter BM. The angle bisectors of ∠AOB and ∠BOC intersect Γ at points P and Q, respectively. The point R is chosen on the line PQ so that BR=MR. Prove that BR∥AC.
(Here we always assume that an angle bisector is a ray.)Proposed by Sergey Berlov, Russia geometryangle bisectorIMO Shortlist