ILL 1987 Problem 73

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$ . Its first derivative is continuous on $\mathbb R$ . Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and $$f(x)f'(y)=f'(x)f(y).$$

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