Middle European Mathematical Olympiad 2018 Problem I-1

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers and let $\alpha \in \mathbb{Q}^+$. Determine all functions $f\colon \mathbb{Q}^{+}\to (\alpha , + \infty)$ satisfying

$$f \left(\frac {x + y}{\alpha}\right) = \frac {f (x) + f (y)}{\alpha}, \quad \text {for all} \, x, y \in \mathbb {Q} ^ {+}.$$