CZ-SK-MO 2006 Problem 5

Find all triples $(p,q,r)$ of pairwise distinct primes such that $p\mid q+r, q\mid r+2p, r\mid p+3q.$

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.