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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1992 Iran MO (2nd round)
2
Sequence Inequality
Sequence Inequality
Source:
July 15, 2010
inequalities
induction
inequalities unsolved
Problem Statement
In the sequence
{
a
n
}
n
=
0
∞
\{a_n\}_{n=0}^{\infty}
{
a
n
}
n
=
0
∞
we have
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
2
a_1=2
a
1
=
2
and
a
n
+
1
=
a
n
+
a
n
−
1
1
+
a
n
−
1
2
∀
n
≥
1
a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1
a
n
+
1
=
a
n
+
1
+
a
n
−
1
2
a
n
−
1
∀
n
≥
1
Prove that
52
<
a
1371
<
65
52 < a_{1371} < 65
52
<
a
1371
<
65
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