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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2022 Iran MO (2nd round)
5
NT sequence
NT sequence
Source: Iran 2nd round 2022 P5
May 9, 2022
number theory
Sequence
Problem Statement
define
(
a
n
)
n
∈
N
(a_n)_{n \in \mathbb{N}}
(
a
n
)
n
∈
N
such that
a
1
=
2
a_1=2
a
1
=
2
and
a
n
+
1
=
(
1
+
1
n
)
n
×
a
n
a_{n+1}=\left(1+\frac{1}{n}\right)^n \times a_{n}
a
n
+
1
=
(
1
+
n
1
)
n
×
a
n
Prove that there exists infinite number of
n
n
n
such that
a
1
a
2
…
a
n
n
+
1
\frac{a_1a_2 \ldots a_n}{n+1}
n
+
1
a
1
a
2
…
a
n
is a square of an integer.
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