Find the continuous functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ such that for all $x, y$

$$
f(x+y)(f(x)+f(y))=f(x) f(y)
$$
The functional equation can be rewritten as

$$
\frac{1}{f(x)}+\frac{1}{f(y)}=\frac{1}{f(x+y)}
$$

Thus, by setting \( g(x) = \frac{1}{f(x)} \), we see that \( g \) satisfies the Cauchy equation \( g(x+y) = g(x) + g(y) \). Since we also know that \( g \) is continuous, the only solutions are \( g(x) = \alpha x \), and thus \( f(x) = \frac{1}{\alpha x} \). To ensure the function is positive, it suffices to take \( \alpha > 0 \).