CZ-SK-MO 2001 Problem 1

Determine all polynomials $P$ such that for every real number $x$ , $P(x)^2 +P(-x) = P(x^2)+P(x)$

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.