BALKAN 2006 Problem 3

Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$ , $n+\frac 1{pm}$ , $p+\frac 1{mn}$ are integers.

Valentin Vornicu, Romania

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.