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At least sqrt(n) distinct values - ILL 1990 CUB1

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September 18, 2010
geometrypoint seteuclidean distancecombinatorial geometryIMO ShortlistIMO Longlist

Problem Statement

AA and BB are two points in the plane α\alpha, and line rr passes through points A,BA, B. There are nn distinct points P1,P2,…,PnP_1, P_2, \ldots, P_n in one of the half-plane divided by line rr. Prove that there are at least n\sqrt n distinct values among the distances AP1,AP2,…,APn,BP1,BP2,…,BPn.AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.
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