Let
and
be positive integers and let
be a set of
points in the plane such that
i.) no three points of
are collinear, and
ii.) for every point
of
there are at least
points of
equidistant from
Prove that:
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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:
$$k < \frac {1}{2} + \sqrt {2 \cdot n}$$
Školjka