IZhO 2022 Problem 1

Non-zero polynomials $P(x)$ , $Q(x)$ , and $R(x)$ with real coefficients satisfy the identities $$P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0.$$ Prove that the degrees of the three polynomials are all even.