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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2023 Iran MO (2nd Round)
P5
P5
Part of
2023 Iran MO (2nd Round)
Problems
(1)
2023 Iran MO 2nd round P5
Source: 2023 Iran MO 2nd round
5/17/2023
5. We call
(
P
n
)
n
∈
N
(P_n)_{n\in \mathbb{N}}
(
P
n
)
n
∈
N
an arithmetic sequence with common difference
Q
(
x
)
Q(x)
Q
(
x
)
if
∀
n
:
P
n
+
1
=
P
n
+
Q
\forall n: P_{n+1} = P_n + Q
∀
n
:
P
n
+
1
=
P
n
+
Q
\newline
We have an arithmetic sequence with a common difference
Q
(
x
)
Q(x)
Q
(
x
)
and the first term
P
(
x
)
P(x)
P
(
x
)
such that
P
,
Q
P,Q
P
,
Q
are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that:
\newline
a)
P
(
x
)
P(x)
P
(
x
)
is divisible by
Q
(
x
)
Q(x)
Q
(
x
)
\newline
b)
deg
(
P
(
x
)
Q
(
x
)
)
=
1
\text{deg}(\frac{P(x)}{Q(x)}) = 1
deg
(
Q
(
x
)
P
(
x
)
)
=
1
algebra
polynomial