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Constructing two sets from conditions on their intersection, union and product

Source: 2021 Iberoamerican Mathematical Olympiad, P5

October 20, 2021

sets of integersIntersectionunion

Problem Statement

For a finite set

C

of integer numbers, we define

S(C)

as the sum of the elements of

C

. Find two non-empty sets

A

and

B

whose intersection is empty, whose union is the set

\{1,2,\ldots, 2021\}

and such that the product

S(A)S(B)

is a perfect square.