ILL 1989 Problem 74

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