Question 45
Find all functions $f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $0 \lt x, y, z$ the following numbers are the side lengths of a triangle:
$$
x + f(y),\ f(f(y)) + z,\ f(f(z)) + f(x)
$$
and for every positive number $a$ there exists $0 \lt b$ such that $f(b) \lt a$.
$$
x + f(y),\ f(f(y)) + z,\ f(f(z)) + f(x)
$$
and for every positive number $a$ there exists $0 \lt b$ such that $f(b) \lt a$.