MathDB

9

Part of 2021 Brazil National Olympiad

Problems(1)

Floor of the multiples of alpha

Source: Brazil National Olympiad Junior 2021 #9

2/15/2022
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.
floor functionalgebra