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Korea Contests
Korea National Olympiad
2008 Korean National Olympiad
8
2008 KMO P8
2008 KMO P8
Source:
August 9, 2015
floor function
algebra
Problem Statement
For fixed positive integers
s
,
t
s, t
s
,
t
, define
a
n
a_n
a
n
as the following.
a
1
=
s
,
a
2
=
t
a_1 = s, a_2 = t
a
1
=
s
,
a
2
=
t
, and
∀
n
≥
1
\forall n \ge 1
∀
n
≥
1
,
a
n
+
2
=
⌊
a
n
+
(
n
+
2
)
a
n
+
1
+
2008
⌋
a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor
a
n
+
2
=
⌊
a
n
+
(
n
+
2
)
a
n
+
1
+
2008
⌋
. Prove that the solution set of
a
n
≠
n
a_n \not= n
a
n
=
n
,
n
∈
N
n \in \mathbb{N}
n
∈
N
is finite.
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