2024 USAMO 第 4 题

Let $m$ and $n$ be positive integers. A circular necklace contains $m n$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$.

令 $m$ 和 $n$ 为正整数。一条圆形项链包含 $m n$ 珠子，每颗珠子都是红色或蓝色。事实证明，无论项链如何被切割成$m$块，每块$n$个连续的珠子，每块都有不同数量的红色珠子。通过证明确定有序对 $(m, n)$ 的所有可能值。