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Subset G of the plane - IMO LongList 1992 TUR2

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September 2, 2010
geometrygeometric inequalitypoint setIMO ShortlistIMO Longlist

Problem Statement

Let {An∣n=1,2,⋯ }\{A_n | n = 1, 2, \cdots \} be a set of points in the plane such that for each nn, the disk with center AnA_n and radius 2n2^n contains no other point AjA_j . For any given positive real numbers a<ba < b and RR, show that there is a subset GG of the plane satisfying:
(i) the area of GG is greater than or equal to RR;
(ii) for each point PP in GG, a<∑n=1∞1∣AnP∣<b.a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.
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