Question 118
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,
$$
f(x+f(xy))+y=f(x)f(y)+1
$$
$$
f(x+f(xy))+y=f(x)f(y)+1
$$