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Unit circle and n points such that sum of vectors is zero

Source: IMO LongList 1982 - P39

April 20, 2011
vectorinequalitiesanalytic geometrygeometry unsolvedgeometry

Problem Statement

Let SS be the unit circle with center OO and let P1,P2,…,PnP_1, P_2,\ldots, P_n be points of SS such that the sum of vectors vi=OPi⟶v_i=\stackrel{\longrightarrow}{OP_i} is the zero vector. Prove that the inequality ∑i=1nXPi≥n\sum_{i=1}^n XP_i \geq n holds for every point XX.
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