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Five points with distnaces exceeding 2

Source: 44th International Tournament of Towns, Senior O-Level P5, Spring 2023

January 9, 2024
geometrydistances

Problem Statement

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space?
Alexey Tolpygo