Middle European Mathematical Olympiad 2023 Problem I-1

Let $\mathbb{R}$ denote the set of all real numbers. For each pair $(\alpha, \beta)$ of nonnegative real numbers subject to $\alpha + \beta \geq 2$, determine all functions $f\colon \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x)f(y) \leq f(xy) + \alpha x + \beta y$$

for all real numbers $x$ and $y$.