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Equations with the closest integer function

Source: KoMaL A. 849

April 12, 2023

number theoryfunction

Problem Statement

For real number

r

let

f(r)

denote the integer that is the closest to

r

(if the fractional part of

r

is

1/2

, let

f(r)

be

r-1/2

). Let

a>b>c

rational numbers such that for all integers

n

the following is true:

f(na)+f(nb)+f(nc)=n

. What can be the values of

a

,

b

and

c

?
Submitted by Gábor Damásdi, Budapest

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