International Mathematical Olympiad 2003 Problem 1

$S$ is the set $\{1,2,3,\ldots,1000000\}$. Show that for any subset $A$ of $S$ with 101 elements we can find 100 distinct elements $x_i$ of $S$, such that the sets $\{a + x_i | a \in A\}$ are all pairwise disjoint.