In a plane a set of points () is give. Each pair of points is connected by a segment. Let be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length . Prove that the number of diameters of the given set is at most .
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In a plane a set of $n$ points ($n \geq 3$) is give. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.
Source: Međunarodna matematička olimpijada, shortlist 1965
Školjka 
points (
) is give. Each pair of points is connected by a segment. Let 
be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length