IMC 2013 Problem 2

Let ${f:{\cal R} \to {\cal R}}$ be a twice differentiable function. Suppose ${f\left( 0 \right) = 0}$ . Prove there exists ${\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}$ such that $${f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.$$

Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.