Question 7
Let $A$ be a 101-element subset of the set $S = \{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_{1}, t_{2}, \ldots, t_{100}$ in $S$ such that the sets
$$ A_{j}=\{x+t_{j}\mid x\in A\},\qquad j=1,2,\ldots,100 $$
are pairwise disjoint.
$$ A_{j}=\{x+t_{j}\mid x\in A\},\qquad j=1,2,\ldots,100 $$
are pairwise disjoint.