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IMC 2013 Problem 2

Problem 2. Let ${f:\Bbb{R} \rightarrow \Bbb{R}}$ be a twice differentiable function. Suppose ${f(0)=0}$. Prove that there exists ${\xi \in (-\pi/2,\pi/2)}$ such that

$\displaystyle f''(\xi)=f(\xi)(1+2\tan^2 \xi).$