Question 54
We say that a polynomial $P$ is fantabulous if there exist real numbers $a_{0}, \ldots, a_{2022}$ such that
$$
P(X)=X^{2023}+a_{2022} X^{2022}+\cdots+a_{1} X+a_{0},
$$
if it has 2023 roots $r_{1}, \ldots, r_{2023}$ (not necessarily distinct) in $[0,1]$, and if $P(0)+P(1)=0$. Determine the maximum value that $r_{1} \cdot r_{2} \cdots r_{2023}$ can take for a fantabulous polynomial.
$$
P(X)=X^{2023}+a_{2022} X^{2022}+\cdots+a_{1} X+a_{0},
$$
if it has 2023 roots $r_{1}, \ldots, r_{2023}$ (not necessarily distinct) in $[0,1]$, and if $P(0)+P(1)=0$. Determine the maximum value that $r_{1} \cdot r_{2} \cdots r_{2023}$ can take for a fantabulous polynomial.