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IMO LongList 1987 - Prove that there exist x, y

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September 6, 2010
functioncalculusderivativealgebra proposedalgebra

Problem Statement

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