Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be a
![101](/media/m/2/d/d/2dd895ad02fd3a5d479b776d16593108.png)
-element subset of the set
![S=\{1,2,\ldots,1000000\}](/media/m/5/7/9/5799f9470967a80fc0b3502803f78fdd.png)
. Prove that there exist numbers
![t_1](/media/m/d/3/3/d332512f256ff456647ffb3819f44831.png)
,
![t_2, \ldots, t_{100}](/media/m/8/5/a/85a6c39c6c18c6cfcca54ac08b671818.png)
in
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
such that the sets
![A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100](/media/m/7/2/d/72dc820df19abec10f84e7bf8545a350.png)
are pairwise disjoint.
%V0
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.