For each positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, the Bank of Cape Town issues coins of denomination
![\frac{1}{n}](/media/m/2/e/f/2ef3e48f5fb08d285d64b9d8a7fbb556.png)
. Given a finite collection of such coins (of not necessarily different denominations) with total value at most
![99 + \frac{1}{2}](/media/m/f/e/a/fea108a6d8e0aa0be1683b2bc7d16901.png)
, prove that it is possible to split this collection into
![100](/media/m/c/c/c/ccc0563efabf7c1a3d81b0dc63f5b627.png)
or fewer groups, such that each group has total value at most
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
.
%V0
For each 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 $1$.