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2022 IMO Shortlist
A1
Weird Sequence
Weird Sequence
Source: ISL 2022 A1
July 9, 2023
algebra
Sequence
ISL 2022
Problem Statement
Let
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
be a sequence of positive real numbers with the property that
(
a
n
+
1
)
2
+
a
n
a
n
+
2
≤
a
n
+
a
n
+
2
(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}
(
a
n
+
1
)
2
+
a
n
a
n
+
2
≤
a
n
+
a
n
+
2
for all positive integers
n
n
n
. Show that
a
2022
≤
1
a_{2022}\leq 1
a
2022
≤
1
.
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