ILL 1969 Problem 39

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

Dataset text AoPS wiki problems. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.