IR-TST 2010 Problem 3

Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$ : $p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0$

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.