ELMO 2011 Problem 4

Find all functions $f : \mathbb{R}^+ \mapsto \mathbb{R}^+$, where $\mathbb{R}^+$ denotes the positive reals, such that whenever $a > b > c > d > 0$ are reel numbers with $ad = bc$, $$f(a + d) + f(b - c) = f(a - d) + f(b + c).$$