Question 163
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 $a \gt 2$ 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?
$$
P_1(x) = P(x), \forall x \in \mathbb{R},
$$
$$
P_2(x) = P(P_1(x)), \forall x \in \mathbb{R},
$$
$$
P_{2024}(x) = P(P_{2023}(x)), \forall x \in \mathbb{R}.
$$
Let $a \gt 2$ 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?