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Distance of points

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September 9, 2010
geometrygeometric inequalityoptimizationminimizationmaximizationIMO Shortlistpoint set

Problem Statement

Let P1,P2,…,PnP_1, P_2, \dots , P_n be distinct points of the plane, n≥2n \geq 2. Prove that max⁡1≤i<j≤nPiPj>32(n−1)min⁡1≤i<j≤nPiPj \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j
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