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Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2021 Macedonian Team Selection Test
Problem 4
Problem 4
Part of
2021 Macedonian Team Selection Test
Problems
(1)
Cycle conditions in group of permutations determine f(2021)
Source: 2021 Macedonian Team Selection Test P4
5/30/2021
Let
S
=
{
1
,
2
,
3
,
…
2021
}
S=\{1, 2, 3, \dots 2021\}
S
=
{
1
,
2
,
3
,
…
2021
}
and
f
:
S
→
S
f:S \to S
f
:
S
→
S
be a function such that
f
(
n
)
(
n
)
=
n
f^{(n)}(n)=n
f
(
n
)
(
n
)
=
n
for each
n
∈
S
n \in S
n
∈
S
. Find all possible values for
f
(
2021
)
f(2021)
f
(
2021
)
. (Here,
f
(
n
)
(
n
)
=
f
(
f
(
f
…
f
(
⏟
n
times
n
)
)
)
…
)
)
f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))
f
(
n
)
(
n
)
=
n
times
f
(
f
(
f
…
f
(
n
)))
…
))
.)Proposed by Viktor Simjanoski
combinatorics
algebra