Question 8
Find all positive integer $n \gt 1$ such that: there exist some real coefficient polynomial $P(x)$ of degree $n$, having the leading coefficient as $1$ and there exist distinct real numbers $r, s, t$ with the sum is $-2023$ and $P(k) \in \{r, s, t\}$ for all $k = 1, 2, 3, \dots, 3n - 1, 3n$.