Question 118
Find all increasing functions $f: \mathbb{R}^+ \cup \{0\} \to \mathbb{R}^+ \cup \{0\}$, such that for every $x, y \in \mathbb{R}^+ \cup \{0\}$ we have
$$
f\left(\frac{x + f(x)}{2} + y\right) = 2x - f(x) + f(f(y))
$$
(note that $f$ is not necessarily strictly increasing.)
$$
f\left(\frac{x + f(x)}{2} + y\right) = 2x - f(x) + f(f(y))
$$
(note that $f$ is not necessarily strictly increasing.)