BALTIC-WAY 2025 Problem 3

Let $f: \mathbb{Q} \to \mathbb{Q}$ be a function that satisfies

$$f(x) + f(y) \geq f(x + y)$$

for all $x, y \in \mathbb{Q}$. Show that there exists an $\alpha \in \mathbb{R}$ such that $f(x) \geq \alpha x$ for all $x \in \mathbb{Q}$.