IMO 1969 LL YUG70

Origin: YUG

Problem

A park has the shape of a convex pentagon of area 5 \sqrt 3 ha (= 50000 \sqrt 3 m2). A man standing at an interior point O of the park notices that he stands at a distance of at most 200 m from each vertex of the pentagon. Prove that he stands at a distance of at least 100 m from each side of the pentagon.