PL-OM 1994 Problem 1

Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$ , $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

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.