CONO-SUR 2002 Problem 5

Consider the set $A = \{1, 2, ..., n\}$ . For each integer $k$ , let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$ . Determine the highest value possible of $r_k$ , where $1 \le k \le \frac{n}{2}$