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Russia Contests
Saint Petersburg Mathematical Olympiad
2023 Saint Petersburg Mathematical Olympiad
5
Iteration of a weird function
Iteration of a weird function
Source: St. Petersburg 2023 11.5
August 12, 2023
function
algebra
Problem Statement
Let
a
>
1
a>1
a
>
1
be a positive integer and let
f
(
n
)
=
n
+
[
a
{
n
2
}
]
f(n)=n+[a\{n\sqrt{2}\}]
f
(
n
)
=
n
+
[
a
{
n
2
}]
. Show that there exists a positive integer
n
n
n
, such that
f
(
f
(
n
)
)
=
f
(
n
)
f(f(n))=f(n)
f
(
f
(
n
))
=
f
(
n
)
, but
f
(
n
)
≠
n
f(n) \neq n
f
(
n
)
=
n
.
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