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Regular 7-gon IMO LongList 1992 PRK1

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September 2, 2010
geometrypolygoncircumcirclegeometric inequalityIMO ShortlistIMO Longlist

Problem Statement

Let a regular 77-gon A0A1A2A3A4A5A6A_0A_1A_2A_3A_4A_5A_6 be inscribed in a circle. Prove that for any two points P,QP, Q on the arc A0A6A_0A_6 the following equality holds: ∑i=06(−1)iPAi=∑i=06(−1)iQAi.\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .
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