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Polynomials having a large number of roots!

Source: 2024 Vietnam National Olympiad - Problem 5

January 6, 2024
algebrapolynomial

Problem Statement

For each polynomial P(x)P(x), define P1(x)=P(x),∀x∈R,P_1(x)=P(x), \forall x \in \mathbb{R}, P2(x)=P(P1(x)),∀x∈R,P_2(x)=P(P_1(x)), \forall x \in \mathbb{R}, ...... P2024(x)=P(P2023(x)),∀x∈R.P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}. Let a>2a>2 be a real number. Is there a polynomial PP with real coefficients such that for all t∈(−a,a)t \in (-a, a), the equation P2024(x)=tP_{2024}(x)=t has 220242^{2024} distinct real roots?
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