Question 3
Given a set $S$ in the plane containing $n$ points and satisfying the conditions
(i) no three points of $S$ are collinear,
(ii) for every point $P$ of $S$ there exist at least $k$ points in $S$ that have the same distance to $P$,
prove that the following inequality holds:
$$k\lt \frac12+\sqrt{2n}.$$
(i) no three points of $S$ are collinear,
(ii) for every point $P$ of $S$ there exist at least $k$ points in $S$ that have the same distance to $P$,
prove that the following inequality holds:
$$k\lt \frac12+\sqrt{2n}.$$