International Mathematical Olympiad 1989 Problem 3

Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in the plane such that

(i) No three points of $S$ are collinear, and

(ii) For any point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P$.

Prove that:

$$k < \frac{1}{2} + \sqrt{2n}.$$