2025 APMO 第 2 题

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 $|x - y|< 2025$ , 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.

Answer: $(\alpha ,\beta)$ such that $\alpha +\beta = 2$

设 $\alpha$ 和 $\beta$ 为正实数。 Emerald 从原点 $(0,0)$ 开始在坐标平面中进行一次旅行。她每分钟向上移动一个单位或向右移动一个单位，将自己限制在坐标平面中的区域 $|x - y|< 2025$ 内。当她访问点 $(x,y)$ 时，她在其上写下整数 $\lfloor x\alpha +y\beta \rfloor$。事实证明，Emerald 将每个非负整数只写了一次。找到所有可能的对 $(\alpha ,\beta)$ ，这样的旅行是可能的。

答案：$(\alpha ,\beta)$ 使得 $\alpha +\beta = 2$