Problem 6

In the plane, let a set of

n

points,

n\\ge3

, be given, and join every pair by a segment. Let

d

be the length of the longest segment. A diameter is any joining segment of length

d

. Prove that the number of diameters is at most

n

Step 4 of 4: Finish by induction

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E-1\\le n-1\\Longrightarrow E\\le n.

Detailed analysis

Remove $C$ and its sole incident diameter $AC$ . The remaining $n-1$ points still have all their diameters among the remaining segments, so induction gives at most $n-1$ of them. Restoring $AC$ gives $E\le n$ , contradicting $E>n$ and proving the claim.