IMO 1978 LL GBR20

Origin: GBR

Problem

Let O be the center of a circle. Let OU, OV be perpendicular radii of the circle. The chord PQ passes through the midpoint M of UV . Let W be a point such that PM = PW, where U, V, M, W are collinear. Let R be a point such that PR = MQ, where R lies on the line PW. Prove that MR = UV . Alternative version: A circle S is given with center O and radius r. Let M be a point whose distance from O is r \sqrt