2017 IMO Shortlist A8

Assume that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the following condition: For every $x, y \in \mathbb{R}$ such that $(f(x)+y)(f(y)+x)>0$, we have $f(x)+y=f(y)+x$. Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$. (Netherlands)

假设函数 $f: \mathbb{R} \rightarrow \mathbb{R}$ 满足以下条件：对于 \mathbb{R}$中的每个$x, y \in \mathbb{R}$使得$(f(x)+y)(f(y)+x)>0$，我们有$f(x)+y=f(y)+x$。证明只要$x>y$，$f(x)+y \leq f(y)+x$。 (荷兰)