Question 135
Find all $a \in \mathbb{R}_{\ge 0}$ such that $f(a) = 0$ for any function $f: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ satisfying the equality
$$
f(f(x) + f(y)) = y f(1 + y f(x))
$$
for all nonnegative real numbers $x$ and $y$.
$$
f(f(x) + f(y)) = y f(1 + y f(x))
$$
for all nonnegative real numbers $x$ and $y$.