IMO 2004 Problem 1
IMO 2004 Problem 1 is a plane geometry problem from the International Mathematical Olympiad 2004, Problem 1: two circumcircles meeting on BC.
Problem
Let $ABC$ be an acute-angled triangle with $AB \neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
Source International Mathematical Olympiad 2004, Problem 1
The diagram
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