Problem

Source: INAMO 2023 P2 (OSN 2023)

Tags: functional equation, algebra, floor function, Floor, Indonesia



Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$: \[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]Note: $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.