2016 IMO Shortlist A4

Denote by $\mathbb{R}^{+}$the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that $$x f\left(x^{2}\right) f(f(y))+f(y f(x))=f(x y)\left(f\left(f\left(x^{2}\right)\right)+f\left(f\left(y^{2}\right)\right)\right)$$ for all positive real numbers $x$ and $y$.

用 $\mathbb{R}^{+}$ 表示所有正实数的集合。查找所有函数 $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$使得 $$x f\left(x^{2}\right) f(f(y))+f(y f(x))=f(x y)\left(f\left(f\left(x^{2}\right)\right)+f\left(f\left(y^{2}\right)\right)\right)$$ 对于所有正实数 $x$ 和 $y$。