2014 IMO Shortlist N3

A coin is called a Cape Town coin if its value is $1 / n$ for some positive integer $n$. Given a collection of Cape Town coins of total value at most $99+\frac{1}{2}$, prove that it is possible to split this collection into at most 100 groups each of total value at most 1. (Luxembourg)

如果某枚硬币的价值为 $1 / n$(对于某个正整数 $n$)，则该硬币被称为开普敦硬币。给定总价值最多为 $99+\frac{1}{2}$ 的开普敦硬币收藏，证明可以将该收藏分为最多 100 组，每组总价值最多为 1。(卢森堡)