SMO 2009 Problem 5

Find all integers x,y,z with $2\leq x\leq y\leq z$ st $xy \equiv 1$ (mod z) $xz\equiv 1$ (mod y) $yz \equiv 1$ (mod x)

Dataset text AI-MO AoPS. 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.