Let be a -element subset of the set . Prove that there exist numbers , in such that the sets are pairwise disjoint.
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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.
Izvor: Međunarodna matematička olimpijada, shortlist 2003
Školjka 
be a 
-element subset of the set 
. Prove that there exist numbers 
, 
in 
such that the sets 
are pairwise disjoint.