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Two polynomials with specific number of roots

Source: BMO SL 2023 A5

May 3, 2024
algebra

Problem Statement

Are there polynomials P,QP, Q with real coefficients, such that P(P(x))⋅Q(Q(x))P(P(x))\cdot Q(Q(x)) has exactly 20232023 distinct real roots and P(Q(x))⋅Q(P(x))P(Q(x)) \cdot Q(P(x)) has exactly 20242024 distinct real roots?
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