IMO 1967 LL SWE52

Origin: SWE

Problem

In the plane a point O and a sequence of points P1, P2, P3, . . . are given. The distances OP1, OP2, OP3, . . . are r1, r2, r3, . . . , where r1 \leq r2 \leqr3 \leq\cdot \cdot \cdot . Let \alpha satisfy 0 < \alpha < 1. Suppose that for every n the distance from the point Pn to any other point of the sequence is greater than or equal to r\alpha n. Determine the exponent \beta, as large as possible, such that for some C independent of n,2 rn \geqCn\beta, n = 1, 2, . . . .

Solution

This problem is not elementary. The solution offered by the proposer was not quite clear and complete (the existence was not proved).