ILL 1977 Problem 49

Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are 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.