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1992-gon with side lengths 1, 2, 3, ..., 1992 circumscribed

Source: IMO Shortlist 1992, Problem 8

August 13, 2008
algebraconvex polygonincirclepolygonAdditive combinatoricsIMO Shortlist

Problem Statement

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: (i) its side lengths are 1,2,3,…,1992 1, 2, 3, \ldots, 1992 in some order; (ii) the polygon is circumscribable about a circle. Alternative formulation: Does there exist a 1992-gon with side lengths 1,2,3,…,1992 1, 2, 3, \ldots, 1992 circumscribed about a circle? Answer the same question for a 1990-gon.
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