IMO 1984 LL AUS2

Given a regular convex 2m-sided polygon P, show that there is

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IMO 1984 LL AUS2

Origin: AUS

Problem

Given a regular convex 2m-sided polygon P, show that there is a 2m-sided polygon \pi with the same vertices as P (but in different order) such that \pi has exactly one pair of parallel sides.