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min of sum of square of distances of 12 points with max distance 1

Source: Ukraine TST 2018 p5

April 29, 2020
combinatorial geometrygeometryinequalitiesgeometric inequality

Problem Statement

Find the smallest positive number λ\lambda such that for an arbitrary 1212 points on the plane P1,P2,...P12P_1,P_2,...P_{12} (points may coincide), with distance between arbitrary two of them does not exceeds 11, holds the inequality ∑1≤i≤j≤12PiPj2≤λ\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda
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