International Mathematics Competition for University Students
July 31 – August 6 2017, Blagoevgrad, Bulgaria
1. Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.
Proposed by: Alexandr Bolbot, Novosibirsk State University
Hint: Take square of $A^2=A^T$.