IMO 1987 LL VIE73

Let f(x) be a periodic function of period T > 0 defined over R.

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IMO 1987 LL VIE73

Origin: VIE

Problem

Let f(x) be a periodic function of period T > 0 defined over R. Its first derivative is continuous on R. Prove that there exist x, y \in[0, T ) such that x ̸= y and f(x)f ′(y) = f(y)f ′(x).