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

(I think this is easy for a IMO 3)

## IMO 1989/3

### IMO 1989/3

"Questions we can't answer are far better than answers we can't question"