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2020 ASDAN Math Tournament
14
2020 Team #14
2020 Team #14
Source:
October 24, 2023
team test
Problem Statement
If
f
f
f
is a permutation of
S
=
{
0
,
1
,
.
.
.
,
14
}
S = \{0, 1,..., 14\}
S
=
{
0
,
1
,
...
,
14
}
, then for integers
k
≥
1
k \ge 1
k
≥
1
, define
f
k
(
x
)
=
f
(
f
.
.
.
(
f
(
x
)
)
.
.
.
)
)
⏟
k
a
p
p
l
i
c
a
t
i
o
n
s
o
f
f
f^k(x) =\underbrace{f(f...(f(x))... ))}_{k\,\,\, applications \,\,\, of \,\,\, f}
f
k
(
x
)
=
k
a
ppl
i
c
a
t
i
o
n
s
o
f
f
f
(
f
...
(
f
(
x
))
...
))
Compute the number of permutations
f
f
f
of
S
S
S
such that, for some
k
≥
1
k \ge 1
k
≥
1
,
f
k
(
x
)
=
(
x
+
5
)
m
o
d
15
f^k(x) = (x + 5) \mod \,\,\, 15
f
k
(
x
)
=
(
x
+
5
)
mod
15
for all
x
∈
S
x \in S
x
∈
S
.
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