2025 CMO 第 4 题

The *fractional distance* between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as $$\sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},$$where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$ .

两点 $(x_1,y_1)$ 和 $(x_2,y_2)$ 之间的*分数距离*定义为 $$\sqrt{ \left\| x_1 - x_2 \右\|^2 + \左\| y_1 - y_2 \right\|^2},$$其中$\left\| x \right\|$ 表示 $x$ 与其最接近的整数之间的距离。找到最大的实数 $r$ ，使得平面上存在四个点，其两两分数距离都至少为 $r$ 。