2014 IMO 第 5 题

For every positive integer $n$ , the Bank of Cape Town issues coins of denomination $\frac{1}{n}$ . Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99 + \frac{1}{2}$ , prove that it is possible to split this collection into 100 or fewer groups, such that each group has total value at most 1.

对于每个正整数 $n$ ，开普敦银行发行面额为 $\frac{1}{n}$ 的硬币。给定此类硬币(不一定是不同面额)的有限集合，其总价值最多为 $99 + \frac{1}{2}$ ，证明可以将该集合分成 100 个或更少的组，使得每个组的总价值最多为 1。