Problem

Source: Kazakhstan National Olympiad 11, Problem 3.

Tags: functional equation, algebra



Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)