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Korea National Olympiad
2023 Korea National Olympiad
7
a/b is unbounded
a/b is unbounded
Source: KMO 2023 P7
November 4, 2023
algebra
Problem Statement
Positive real sequences
{
a
n
}
\{ a_n \}
{
a
n
}
and
{
b
n
}
\{ b_n \}
{
b
n
}
satisfy the following conditions for all positive integers
n
n
n
.[*]
a
n
+
1
b
n
+
1
=
a
n
2
+
b
n
2
a_{n+1}b_{n+1}= a_n^2 + b_n^2
a
n
+
1
b
n
+
1
=
a
n
2
+
b
n
2
[*]
a
n
+
1
+
b
n
+
1
=
a
n
b
n
a_{n+1}+b_{n+1}=a_nb_n
a
n
+
1
+
b
n
+
1
=
a
n
b
n
[*]
a
n
≥
b
n
a_n \geq b_n
a
n
≥
b
n
Prove that there exists positive integer
n
n
n
such that
a
n
b
n
>
202
3
2023
.
\frac{a_n}{b_n}>2023^{2023}.
b
n
a
n
>
202
3
2023
.
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