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Korea National Olympiad
2022 Korea National Olympiad
1
Chained sequences
Chained sequences
Source: KMO 2022 P1
October 29, 2022
algebra
Problem Statement
Three sequences
a
n
,
b
n
,
c
n
{a_n},{b_n},{c_n}
a
n
,
b
n
,
c
n
satisfy the following conditions.[*]
a
1
=
2
,
b
1
=
4
,
c
1
=
5
a_1=2,\,b_1=4,\,c_1=5
a
1
=
2
,
b
1
=
4
,
c
1
=
5
[*]
∀
n
,
a
n
+
1
=
b
n
+
1
c
n
,
b
n
+
1
=
c
n
+
1
a
n
,
c
n
+
1
=
a
n
+
1
b
n
\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}
∀
n
,
a
n
+
1
=
b
n
+
c
n
1
,
b
n
+
1
=
c
n
+
a
n
1
,
c
n
+
1
=
a
n
+
b
n
1
Prove that for all positive integers
n
n
n
,
m
a
x
(
a
n
,
b
n
,
c
n
)
>
2
n
+
13
max(a_n,b_n,c_n)>\sqrt{2n+13}
ma
x
(
a
n
,
b
n
,
c
n
)
>
2
n
+
13
.
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