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p(x^2) + q(x^2) = p(x)q(x) for integer polynomials

Source: 2021 Dürer Math Competition Regional E+4 https://artofproblemsolving.com/community/c2773609_2021_

January 7, 2022
algebrapolynomial

Problem Statement

Find all pairs of polynomials (p,q)(p, q) with integer coefficients that satisfy the equation p(x2)+q(x2)=p(x)q(x)p(x^2) + q(x^2) = p(x)q(x) such that pp is of degree nn and has nn nonnegative real roots (with multiplicity).
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