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elements of AB form finite arithmetic progression, A or B has at most 3 elements

Source: Ukraine TST 2015 p7

May 2, 2020

arithmetic sequenceSetsProductalgebra

Problem Statement

Let

A

and

B

be two sets of real numbers. Suppose that the elements of the set

AB = \{ab: a\in A, b\in B\}

form a finite arithmetic progression. Prove that one of these sets contains no more than three elements