International Mathematical Olympiad 2003 — Official contest Question 1

Let $A$ be a $101$-element subset of $S=\{1,2,\dots,10^6\}$. Prove that there exist numbers $t_1$, $t_2, \dots, t_{100}$ in $S$ such that the sets $$
A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\dots,100
$$ are pairwise disjoint.