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Geometric inequality with 12 points

Source: China Team Selection Test 2016 Test 2 Day 1 Q2

March 20, 2016
geometryinequalitiesgeometric inequality

Problem Statement

Find the smallest positive number λ\lambda, such that for any 1212 points on the plane P1,P2,…,P12P_1,P_2,\ldots,P_{12}(can overlap), if the distance between any two of them does not exceed 11, then ∑1≤i<j≤12∣PiPj∣2≤λ\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda.
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