A 1957 Putnam exam problem

The video tutorial explores a geometric problem involving a finite set of points in a plane, where the maximum distance between any two points is 1. The goal is to prove that the number of pairs with this maximum distance is always less than or equal to the number of points. The tutorial walks through assumptions, counterexamples, and logical deductions to reach a contradiction, ultimately proving the statement. The problem is based on a Putnam exam question from 1957, and the tutorial provides a detailed analysis of the geometric properties involved.

Describe the significance of the angle being less than or equal to 60 degrees in the proof.

What conclusion can be drawn if the maximum distance pairs exceed the number of points?

Summarize the final conclusion of the proof regarding the number of maximum distance pairs.