2010 IMO Shortlist A5

Denote by $\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$: $$f\left(f(x)^{2} y\right)=x^{3} f(x y)$$ (Switzerland) Answer. The only such function is $f(x)=\frac{1}{x}$.

用 $\mathbb{Q}^{+}$ 表示所有正有理数的集合。确定所有函数 $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ 对于所有 $x, y \in \mathbb{Q}^{+}$ 满足以下方程： $$f\left(f(x)^{2} y\right)=x^{3} f(x y)$$ (瑞士) 答案。唯一这样的函数是 $f(x)=\frac{1}{x}$。