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Let n and k be two natural numbers and let S be a set of n points such that
(a) no three points of S are collinear.
(b) for any point P of S there are at least k points of S which are equidistant from P.
Prove that k<1/2+(2n)^(1/2)

Question LaTeXed:
Let $n$ and $k$ be two natural numbers and let $S$ be a set of n points such that
(a) no three points of $S$ are collinear.
(b) for any point $P$ of $S$ there are at least $k$ points of $S$ which are equidistant from $P$.
Prove that $k< \frac{1}{2}+(2n)^{\frac{1}{2}}$
Better:Prove that $k< \frac{1}{2}+\sqrt{2n}$

Already posted: viewtopic.php?f=17&t=3262&p=15746#p15746