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Configurations for which this maximum is attained

Source: IMO Longlist 1989, Problem 74

September 18, 2008
geometry3D geometryspherecombinatoricsmaximizationgeometric inequalityIMO Shortlist

Problem Statement

For points A1,…,A5 A_1, \ldots ,A_5 on the sphere of radius 1, what is the maximum value that min1≤i,j≤5AiAj min_{1 \leq i,j \leq 5} A_iA_j can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set {A1,…,A5} \{A_1, \ldots ,A_5\} for which this maximum is attained.)
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