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Variant of floor function

Source: Kvant Magazine No. 9 2020 M2618

March 9, 2023

algebrafloor functionKvant

Problem Statement

For a given number

\alpha{}

let

f_\alpha

be a function defined as

f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.

Let

\alpha>1

and

\beta=1/\alpha

. Prove that for any natural

n{}

the relation

f_\beta(f_\alpha(n))=n

holds.
Proposed by I. Dorofeev

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