Denote by
![\mathbb{Q}^+](/media/m/f/3/b/f3b4bcdab566d6231aff93d231bd91c9.png)
the set of all positive rational numbers. Determine all functions
![f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+](/media/m/e/1/c/e1c944589714193f84d36588db0e51b0.png)
which satisfy the following equation for all
![f\left( f(x)^2y \right) = x^3 f(xy).](/media/m/3/b/5/3b5c7ba8b5b62331bfc8f44068fd9038.png)
Proposed by Thomas Huber, Switzerland
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Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ $$f\left( f(x)^2y \right) = x^3 f(xy).$$
Proposed by Thomas Huber, Switzerland