Question 5
For a polynomial $P(x)$, define $P_1(x)=P(x)$ and $P_{j+1}(x)=P(P_j(x))$ for $1\le j\le2023$, for all real $x$. Let $a\gt 2$ be real. Does there exist a real-coefficient polynomial $P(x)$ such that for every $t\in(-a,a)$, the equation $P_{2024}(x)=t$ has exactly $2^{2024}$ distinct real roots?