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2019 Japan TST
9
9
Part of
2019 Japan TST
Problems
(1)
2019 Japan TST
Source: Japan
1/6/2020
Let
P
(
x
)
∈
Q
[
x
]
P(x) \in \mathbb{Q}[x]
P
(
x
)
∈
Q
[
x
]
, if
P
(
P
(
x
)
)
∈
Z
[
x
]
,
P
(
P
(
P
(
x
)
)
)
∈
Z
[
x
]
,
P(P(x)) \in \mathbb{Z}[x], P(P(P(x))) \in \mathbb{Z}[x],
P
(
P
(
x
))
∈
Z
[
x
]
,
P
(
P
(
P
(
x
)))
∈
Z
[
x
]
,
prove that
P
(
x
)
∈
Z
[
x
]
P(x) \in \mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
.
algebra
number theory