Problem 5

For each positive integer

n

, the Bank of Cape Town issues coins of denomination

\tfrac{1}{n}

. Given a finite collection of such coins (of not necessarily different denominations) with total value at most

99+\tfrac12

, prove that it is possible to split this collection into

100

or fewer groups, such that each group has total value at most

1

Step 4 of 6: Set aside the remaining tiny coins as a pile

\text{remaining coins have value }\le\tfrac1{2r+1}

Detailed analysis

All coins with denominator at least $2r+1$ form a pile; every such coin has value at most $1/(2r+1)$ .