Question 186
Let $\mathbb{R}_{\gt 0}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}_{\gt 0}$ such that, for every $x \in \mathbb{R}_{\gt 0}$, there exists a unique $y \in \mathbb{R}_{\gt 0}$ satisfying
$$
x f(y)+y f(x) \leqslant 2 .
$$
$$
x f(y)+y f(x) \leqslant 2 .
$$