US-TST 2016 Problem 3

Let $p$ be a prime number. Let $\mathbb{F}_{p}$ denote the integers modulo $p$, and let $\mathbb{F}_{p}[x]$ be the set of polynomials with coefficients in $\mathbb{F}_{p}$. Define $\Psi: \mathbb{F}_{p}[x] \rightarrow \mathbb{F}_{p}[x]$ by $$\Psi\left(\sum_{i=0}^{n} a_{i} x^{i}\right)=\sum_{i=0}^{n} a_{i} x^{x^{i}} .$$ Prove that for nonzero polynomials $F, G \in \mathbb{F}_{p}[x]$, $$\Psi(\operatorname{gcd}(F, G))=\operatorname{gcd}(\Psi(F), \Psi(G))$$