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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1996 Moldova Team Selection Test
11
11
Part of
1996 Moldova Team Selection Test
Problems
(1)
the function $f^{n-2}$ is constant, but $f^{n-3}$ is not
Source: Moldova TST 1996
8/8/2023
Let
A
A{}
A
be a set with
n
n{}
n
(
n
≥
3
)
(n\geq3)
(
n
≥
3
)
elements. Iterations
f
2
,
f
2
,
…
f^2,f^2,\ldots
f
2
,
f
2
,
…
of the function
f
:
A
→
A
f:A\rightarrow A
f
:
A
→
A
are defined as
f
2
(
x
)
=
f
(
f
(
x
)
)
,
f
i
+
1
=
f
(
f
i
(
x
)
)
,
∀
i
≥
2
f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2
f
2
(
x
)
=
f
(
f
(
x
))
,
f
i
+
1
=
f
(
f
i
(
x
))
,
∀
i
≥
2
. Find the number of functions
f
:
A
→
A
f:A\rightarrow A
f
:
A
→
A
with the property: the function
f
n
−
2
f^{n-2}
f
n
−
2
is constant, but
f
n
−
3
f^{n-3}
f
n
−
3
is not.
function