IMO 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 every point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P.$

Prove that: $$k < \frac {1}{2} + \sqrt {2 \cdot n}$$