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f(xf(y)) f(y) = f (xy/(x + y) )

Source: Switzerland - 2007 Swiss MO Final Round p5

December 26, 2022
functionalfunctional equationalgebra

Problem Statement

Determine all functions f:R≥0→R≥0f : R_{\ge 0} \to R_{\ge 0} with the following properties: (a) f(1)=0f(1) = 0, (b) f(x)>0f(x) > 0 for all x>1x > 1, (c) For all x,y≥0x, y\ge 0 with x+y>0x + y > 0 holds f(xf(y))f(y)=f(xyx+y)f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)
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