### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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### Problem 9

Problem 9. Determine all pairs $\displaystyle P(x)$, $\displaystyle Q(x)$ of complex polynomials with leading coefficient $\displaystyle 1$ such that $\displaystyle P(x)$ divides $\displaystyle Q(x)^2+1$ and $\displaystyle Q(x)$ divides $\displaystyle P(x)^2+1$.

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