RO-NMO 2008 Problem 114

Let $A=(a_{ij})_{1\leq i,j\leq n}$ be a real $n\times n$ matrix, such that $a_{ij} + a_{ji} = 0$ , for all $i,j$ . Prove that for all non-negative real numbers $x,y$ we have $$\det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.$$