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2005 Silk Road
4
a(2n) \leq 2a(n)
a(2n) \leq 2a(n)
Source: SRMO 2005
April 10, 2005
inequalities
induction
algebra proposed
algebra
Problem Statement
Suppose
{
a
(
n
)
}
n
=
1
∞
\{a(n) \}_{n=1}^{\infty}
{
a
(
n
)
}
n
=
1
∞
is a sequence that:
a
(
n
)
=
a
(
a
(
n
−
1
)
)
+
a
(
n
−
a
(
n
−
1
)
)
∀
n
≥
3
a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3
a
(
n
)
=
a
(
a
(
n
−
1
))
+
a
(
n
−
a
(
n
−
1
))
∀
n
≥
3
and
a
(
1
)
=
a
(
2
)
=
1
a(1)=a(2)=1
a
(
1
)
=
a
(
2
)
=
1
. Prove that for each
n
≥
1
n \geq 1
n
≥
1
,
a
(
2
n
)
≤
2
a
(
n
)
a(2n) \leq 2a(n)
a
(
2
n
)
≤
2
a
(
n
)
.
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