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Functional equation

Source: Turkish TST 2011 Problem 1

July 23, 2011
functionalgorithminductionnumber theoryEuclidean algorithmalgebrafunctional equation

Problem Statement

Let Q+\mathbb{Q^+} denote the set of positive rational numbers. Determine all functions f:Q+→Q+f: \mathbb{Q^+} \to \mathbb{Q^+} that satisfy the conditions
f(xx+1)=f(x)x+1andf(1x)=f(x)x3 f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}
for all x∈Q+.x \in \mathbb{Q^+}.
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