CMO 1996 Problem 5

Let $r_1$ , $r_2$ , $\ldots$ , $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$ . Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$ . Determine the minimum and maximum values of $f(n)$ . Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$ .