Question 156
Let $A$ be a subset of the set $S = \{1, 2, \dots, 1000000\}$ containing exactly $101$ elements. Prove that there exist numbers $t_1, t_2, \dots, t_{100}$ in $S$ such that the sets
$$
A_j = \{x + t_j \mid x \in A\} \text{ for } j = 1, 2, \dots, 100
$$
are pairwise disjoint.
$$
A_j = \{x + t_j \mid x \in A\} \text{ for } j = 1, 2, \dots, 100
$$
are pairwise disjoint.