IMO 1992 LL TUR73

Origin: TUR

Problem

Let {An | n = 1, 2, . . .} be a set of points in the plane such that for each n, the disk with center An and radius 2n contains no other point Aj. For any given positive real numbers a < b and R, show that there is a subset G of the plane satisfying: (i) the area of G is greater than or equal to R; (ii) for each point P in G, a < \infty n=1 |AnP | < b.