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Sum of many sets

Source: Science ON 2021 grade VII/4

March 8, 2021
Setsequality casecombinatorics

Problem Statement

Take k∈Z≥1k\in \mathbb{Z}_{\ge 1} and the sets A1,A2,…,AkA_1,A_2,\dots, A_k consisting of x1,x2,…,xkx_1,x_2,\dots ,x_k positive integers, respectively. For any two sets AA and BB, define A+B={a+b ∣ a∈A, b∈B}A+B=\{a+b~|~a\in A,~b\in B\}.
Find the least and greatest number of elements the set A1+A2+⋯+AkA_1+A_2+\dots +A_k may have.
(Andrei Bâra)
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