Question 1
Let $\alpha$ and $\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting off from the origin $(0,0)$. Each minute she moves one unit up or one unit to the right, restricting herself to the region $|x-y|\lt 2025$, in the coordinate plane. By the time she visits a point ( $x, y$ ) she writes down the integer $\lfloor x \alpha+y \beta\rfloor$ on it. It turns out that Emerald wrote each non-negative integer exactly once. Find all the possible pairs $(\alpha, \beta)$ for which such a trip would be possible.