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IMO LongList 1967, Sweden 5

Source: IMO LongList 1967, Sweden 5

December 16, 2004
geometrypoint seteuclidean distanceIMO ShortlistIMO Longlist

Problem Statement

In the plane a point OO is and a sequence of points P1,P2,P3,…P_1, P_2, P_3, \ldots are given. The distances OP1,OP2,OP3,…OP_1, OP_2, OP_3, \ldots are r1,r2,r3,…r_1, r_2, r_3, \ldots Let α\alpha satisfies 0<α<1.0 < \alpha < 1. Suppose that for every nn the distance from the point PnP_n to any other point of the sequence is ≥rnα.\geq r^{\alpha}_n. Determine the exponent β\beta, as large as possible such that for some CC independent of nn rn≥Cnβ,n=1,2,…r_n \geq Cn^{\beta}, n = 1,2, \ldots
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