IR-TST 2023 Problem 9

Find all function $f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$x+f(y) , f(f(y)) + z , f(f(z))+f(x)$$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$ .

Proposed by Amirhossein Zolfaghari