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Iran MO (2nd Round)
2024 Iran MO (2nd Round)
2
Iranian sequence
Iranian sequence
Source: Iran second round 2024 p2
April 18, 2024
algebra and number theory
Problem Statement
Find all sequences
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
of positive integers such that for all integers
n
≥
3
n\geq 3
n
≥
3
we have
1
a
1
a
3
+
1
a
2
a
4
+
⋯
+
1
a
n
−
2
a
n
=
1
−
1
a
1
2
+
a
2
2
+
⋯
+
a
n
−
1
2
.
\dfrac{1}{a_1 a_3} + \dfrac{1}{a_2a_4} + \cdots + \dfrac{1}{a_{n-2}a_n}= 1 - \dfrac{1}{a_1^2+a_2^2+\cdots +a_{n-1}^2}.
a
1
a
3
1
+
a
2
a
4
1
+
⋯
+
a
n
−
2
a
n
1
=
1
−
a
1
2
+
a
2
2
+
⋯
+
a
n
−
1
2
1
.
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