BALTIC-WAY 1998 Problem 1

Let $\mathbb{Z}^{+}$be the set of all positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ satisfying the following conditions for all $x, y \in \mathbb{Z}^{+}$:

$$\begin{aligned} f(x, x) & =x, \\ f(x, y) & =f(y, x), \\ (x+y) f(x, y) & =y f(x, x+y) . \end{aligned}$$