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x_1^2 +x_2^2 +...+x_{2000}^2<9, points interior in circle of radius 1

Source: Switzerland - Swiss TST 2000 p15

February 19, 2020
geometrypointscombinatorial geometryGeometric Inequalitiescombinatorics

Problem Statement

Let S={P1,P2,...,P2000}S = \{P_1,P_2,...,P_{2000}\} be a set of 20002000 points in the interior of a circle of radius 11, one of which at its center. For i=1,2,...,2000i = 1,2,...,2000 denote by xix_i the distance from PiP_i to the closest point Pj≠PiP_j \ne P_i. Prove that x12+x22+...+x20002<9x_1^2 +x_2^2 +...+x_{2000}^2<9 .
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