2025 USAMO 第 6 题

Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.

令$m$ 和$n$ 为正整数且$m\geq n$。围成一圈排列着$m$个不同口味的纸杯蛋糕，还有$n$个喜欢纸杯蛋糕的人。每个人根据他们对纸杯蛋糕的喜爱程度，为每个纸杯蛋糕分配一个非负实数分数。假设对于每个人$P$，可以将$m$个纸杯蛋糕分成$n$组连续的纸杯蛋糕，使得每组纸杯蛋糕的$P$得分总和至少为$1$。证明可以将 $m$ 纸杯蛋糕分发给 $n$ 人，以便每个人 $P$ 收到的纸杯蛋糕的总分至少为 $1$(相对于 $P$)。