Školjka

A coin is called a \emph{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 $99 + \frac12$, prove that it is possible to split this collection into at most $100$ groups each of total value at most $1$. \begin{flushright}\emph{(Luxembourg)}\end{flushright}