IMC 2018 Problem 9

Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$ .

Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro