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Contests
National and Regional Contests
China Contests
China Northern MO
2006 China Northern MO
8
Sequence Inequality
Sequence Inequality
Source: China North MO
August 14, 2006
inequalities
inequalities unsolved
Problem Statement
Given a sequence
{
a
n
}
\{ a_{n}\}
{
a
n
}
such that
a
n
+
1
=
a
n
+
1
2006
a
n
2
a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}
a
n
+
1
=
a
n
+
2006
1
a
n
2
,
n
∈
N
n \in N
n
∈
N
,
a
0
=
1
2
a_{0}=\frac{1}{2}
a
0
=
2
1
. Prove that
1
−
1
2008
<
a
2006
<
1
1-\frac{1}{2008}< a_{2006}< 1
1
−
2008
1
<
a
2006
<
1
.
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