KöMaL 2023 A867

Let $p(x)$ be a monic integer polynomial (polynomial with integer coefficients and leading coefficient 1) of degree $n$ that has $n$ real roots, $\alpha_1, \alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$ (i.e. it's not possible to find an integer polynomial different from constant 1 and $-1$ that divides both $p(x)$ and $q(x)$). Prove that $\sum\limits_{i=1}^{n} \big|q(\alpha_i)\big|\ge n$.