Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$ $$x^{2p} - 2px^m - p^2x^n - 1$$
Source: 2025 Korea Winter Program Practice Test P8
Tags: algebra, polynomial
Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$ $$x^{2p} - 2px^m - p^2x^n - 1$$