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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2023 Iran MO (2nd Round)
P2
2023 Iran MO 2nd round P2
2023 Iran MO 2nd round P2
Source: 2023 Iran MO 2nd round
May 17, 2023
number theory
Problem Statement
2. Prove that for any
2
≤
n
∈
N
2\le n \in \mathbb{N}
2
≤
n
∈
N
there exists positive integers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
such that
∀
i
≠
j
:
gcd
(
a
i
,
a
j
)
=
1
\forall i\neq j: \text{gcd}(a_i,a_j) = 1
∀
i
=
j
:
gcd
(
a
i
,
a
j
)
=
1
and
∀
i
:
a
i
≥
1402
\forall i: a_i \ge 1402
∀
i
:
a
i
≥
1402
and the given relation holds.
[
a
1
a
2
]
+
[
a
2
a
3
]
+
⋯
+
[
a
n
a
1
]
=
[
a
2
a
1
]
+
[
a
3
a
2
]
+
⋯
+
[
a
1
a
n
]
[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]
[
a
2
a
1
]
+
[
a
3
a
2
]
+
⋯
+
[
a
1
a
n
]
=
[
a
1
a
2
]
+
[
a
2
a
3
]
+
⋯
+
[
a
n
a
1
]
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