Question 16
Let $ ABC $ be an acute-angled triangle with $ AB \ne AC $. The circle with diameter $ BC $ intersects the sides $ AB $ and $ AC $ at $ M $ and $ N $, respectively. Denote by $ O $ the midpoint of $ BC $. The bisectors of the angles $ BAC $ and $ MON $ intersect at $ R $. Prove that the circumcircles of the triangles $ BMR $ and $ CNR $ have a common point lying on the line segment $ BC $.