ILL 1992 Problem 73

Let $\{A_n | n = 1, 2, \cdots \}$ be a set of points in the plane such that for each $n$ , the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . 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 $G$ , $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$