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N-gon must be a triangle if always exists a 60 degree angle

Source: Romanian TST 2000

February 14, 2011
geometrycombinatorics unsolvedcombinatorics

Problem Statement

Let P1P2…PnP_1P_2\ldots P_n be a convex polygon in the plane. We assume that for any arbitrary choice of vertices Pi,PjP_i,P_j there exists a vertex in the polygon PkP_k distinct from Pi,PjP_i,P_j such that ∠PiPkPj=60∘\angle P_iP_kP_j=60^{\circ}. Show that n=3n=3.
Radu Todor
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