IR-TST 2015 Problem 1

Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $P(x)^3+Q(x)^3=x^{12}+1.$

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.