Problem

Source: 2024 Vietnam National Olympiad - Problem 5

Tags: algebra, polynomial



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