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f(x) (f (\sqrt[3]((2+x) /(2-x) ) ^2= x^3/ 4

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2013 Shortlist F2 day1 (F = Functions)

September 26, 2021
functional equationfunctionalalgebra

Problem Statement

Find all functions f:R−{0,2}→Rf:R -\{0,2\} \to R that satisfy for all x≠0,2x \ne 0,2 f(x)⋅(f(2+x2−x3))2=x34f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}
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