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Iran Contests
Iran MO (3rd Round)
2024 Iran MO (3rd Round)
6
6
Part of
2024 Iran MO (3rd Round)
Problems
(1)
Polynomial and sequence
Source: Iran 2024 3rd round p6
8/25/2024
Sequence of positive integers
{
x
k
}
k
≥
1
\{x_k\}_{k\geq 1}
{
x
k
}
k
≥
1
is given such that
x
1
=
1
x_1=1
x
1
=
1
and for all
n
≥
1
n\geq 1
n
≥
1
we have
x
n
+
1
2
+
P
(
n
)
=
x
n
x
n
+
2
x_{n+1}^2+P(n)=x_n x_{n+2}
x
n
+
1
2
+
P
(
n
)
=
x
n
x
n
+
2
where
P
(
x
)
P(x)
P
(
x
)
is a polynomial with non-negative integer coefficients. Prove that
P
(
x
)
P(x)
P
(
x
)
is the constant polynomial.
algebra