RO-NMO 2014 Problem 111

Find all continuous functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ that satisfy: $\text{(i)}\text{id}+f$ is nondecreasing $\text{(ii)}$ There is a natural number $m$ such that $\text{id}+f+f^2\cdots +f^m$ is nonincreasing.

Here, $\text{id}$ represents the identity function, and ^ denotes functional power.