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KoMaL A Problems
KoMaL A Problems 2022/2023
A. 849
A. 849
Part of
KoMaL A Problems 2022/2023
Problems
(1)
Equations with the closest integer function
Source: KoMaL A. 849
4/12/2023
For real number
r
r
r
let
f
(
r
)
f(r)
f
(
r
)
denote the integer that is the closest to
r
r
r
(if the fractional part of
r
r
r
is
1
/
2
1/2
1/2
, let
f
(
r
)
f(r)
f
(
r
)
be
r
−
1
/
2
r-1/2
r
−
1/2
). Let
a
>
b
>
c
a>b>c
a
>
b
>
c
rational numbers such that for all integers
n
n
n
the following is true:
f
(
n
a
)
+
f
(
n
b
)
+
f
(
n
c
)
=
n
f(na)+f(nb)+f(nc)=n
f
(
na
)
+
f
(
nb
)
+
f
(
n
c
)
=
n
. What can be the values of
a
a
a
,
b
b
b
and
c
c
c
?Submitted by Gábor Damásdi, Budapest
number theory
function