MathLabs

Problem 1

Let AA be a 101-element subset of S={1,2,…,106}S=\{1,2,\ldots,10^6\}. Prove that there exist numbers t1,t2,…,t100t_1,t_2,\ldots,t_{100} in SS such that the sets Aj={x+tj∣x∈A}A_j=\{x+t_j\mid x\in A\}, j=1,2,…,100j=1,2,\ldots,100, are pairwise disjoint.
Step 3 of 3: Finish
Previous stepNext step
In plain words

Finish

∣T∣≥100|T|\ge100
Detailed analysis

Since 99⋅10101=999999<10699\cdot10101=999999<10^6, ∣T∣≥100|T|\ge100. Any 100 members of TT give pairwise disjoint translates.