(OIM 1983)

Find all functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ such that, for all $x, y>0, f(y f(x))=x f(y)$ and $\lim _{x \rightarrow+\infty} f(x)=0$.
Let's focus on the fixed points of $f$: for all $x$, $x f(x)$ is a fixed point of $f$. If 1 is the only fixed point, then $f(x) = 1 / x$ for all $x$.

Otherwise, let $z$ be a fixed point other than 1. If $z'$ is a fixed point, then $1 / z'$ is also a fixed point (set $x = z'$ and $y = 1 / z'$ in the initial equation). Therefore, we can assume that $z > 1$. We prove by induction on $n$ that if $z$ is a fixed point, then $z^{2^n}$ is also a fixed point: if $z^{2^n}$ is a fixed point, it suffices to choose $x = y = z^{2^n}$ to see that $z^{2^{n+1}}$ is also a fixed point. Thus, we have a sequence of arbitrarily large fixed points for large $n$.