In a plane a set of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
points (
![n \geq 3](/media/m/5/4/8/54807b3bf99aa939833fe57bf8d891d3.png)
) is give. Each pair of points is connected by a segment. Let
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
. Prove that the number of diameters of the given set is at most
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
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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 $n$.