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a [ b n] = b [ a n ] , floor function equation

Source: 2019 Austrian Federal Competition For Advanced Students, Part 1 p4

March 4, 2020
floor functionalgebrafunction

Problem Statement

Find all pairs (a,b)(a, b) of real numbers such that a⋅⌊b⋅n⌋=b⋅⌊a⋅n⌋a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor applies to all positive integersn n. (For a real number x,⌊x⌋x, \lfloor x\rfloor denotes the largest integer that is less than or equal to xx.)
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