BALKAN 2013 Problem 3

Let $S$ be the set of positive real numbers. Find all functions $f: S^{3} \rightarrow S$ such that, for all positive real numbers $x, y, z$ and $k$, the following three conditions are satisfied: (i) $x f(x, y, z)=z f(z, y, x)$; (ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$; (iii) $f(1, k, k+1)=k+1$. (United Kingdom)