(Fixed points and limits) Find all functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ such that for all $x, y>0, f(x f(y))=y f(x)$ and $\lim _{x \rightarrow+\infty} f(x)=0$.

(IMO 1983)
With $x=y$, we see that for all $x>0, x f(x)$ is a fixed point of f. If $z$ is a fixed point of $\mathrm{f}$, with $x=1$ and $y=z$, we get $z=z f(1) \neq 0$ so 1 is also a fixed point.

Moreover, a quick induction shows that for all $n \in \mathbb{N}, a^{n}$ is also a fixed point of $f$. The substitution $y=a=\frac{1}{x}$ gives $a f\left(\frac{1}{a}\right)=f\left(\frac{1}{a} f(a)\right)=f(1)=1$, so $a^{-1}$ is also a fixed point of $f$. We deduce that, for all $\mathfrak{m} \in \mathbb{Z}, f\left(a^{\mathfrak{m}}\right)=a^{m}$. The condition on the limit at $+\infty$ imposes $a=1$. Therefore, for all positive real numbers $x, f(x)=\frac{1}{x}$, which is indeed a solution to the problem.