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Vietnam Team Selection Test
2024 Vietnam Team Selection Test
6
6
Part of
2024 Vietnam Team Selection Test
Problems
(1)
Old idea in a new context
Source: Vietnam TST 2024 P6
3/27/2024
Let
P
(
x
)
∈
Z
[
x
]
P(x) \in \mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
be a polynomial. Determine all polynomials
Q
(
x
)
∈
Z
[
x
]
Q(x) \in \mathbb{Z}[x]
Q
(
x
)
∈
Z
[
x
]
, such that for every positive integer
n
n
n
, there exists a polynomial
R
n
(
x
)
∈
Z
[
x
]
R_n(x) \in \mathbb{Z}[x]
R
n
(
x
)
∈
Z
[
x
]
satisfies
Q
(
x
)
2
n
−
1
=
R
n
(
x
)
(
P
(
x
)
2
n
−
1
)
.
Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).
Q
(
x
)
2
n
−
1
=
R
n
(
x
)
(
P
(
x
)
2
n
−
1
)
.
algebra
polynomial