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Floor of the multiples of alpha

Source: Brazil National Olympiad Junior 2021 #9

February 15, 2022
floor functionalgebra

Problem Statement

Let α≥1\alpha\geq 1 be a real number. Define the set A(α)={⌊α⌋,⌊2α⌋,⌊3α⌋,… }A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} Suppose that all the positive integers that does not belong to the A(α)A(\alpha) are exactly the positive integers that have the same remainder rr in the division by 20212021 with 0≤r<20210\leq r<2021. Determine all the possible values of α\alpha.
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