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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2023 Iran MO (2nd Round)
P2
P2
Part of
2023 Iran MO (2nd Round)
Problems
(1)
2023 Iran MO 2nd round P2
Source: 2023 Iran MO 2nd round
5/17/2023
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
]
number theory