# A certain type of points in the plane

In the plane a point $O$ and a sequence of points $P_1 , P_2 , P_3 , ...$ are given. The distances $OP_1 , OP_2 , OP_3 , ...$ are $r_1 , r_2 , r_3 , ....$ , where $r_1 ≤ r_2 ≤ r_3 ≤$ ... . Let $α$ satisfy $0 < α < 1$ . Suppose that for every $n$ the distance from the point $P_n$ to another point of the sequence is greater than or equal to $r_n ^α$ . Then does there exist an exponent $β$ , such that for some $C$ independent of $n$ ,

$r_n$ ≥ $C$ $n^β$ , for all $n$ = $1 , 2 ,$ ...

?

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I don't know. Where does this problem come from? –  Gerry Myerson Nov 8 '12 at 11:54
@ Gerry: This was one of the longlisted problems of IMO 1967 , I found it in a problem book , where it is said to be non-elementary and not even any hint to the solution is given . –  Souvik Dey Nov 9 '12 at 5:01