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Contests
National and Regional Contests
China Contests
China National Olympiad
2006 China National Olympiad
5
2006cmo problem5
2006cmo problem5
Source:
January 13, 2006
induction
inequalities
inequalities unsolved
Problem Statement
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence such that:
a
1
=
1
2
a_1 = \frac{1}{2}
a
1
=
2
1
,
a
k
+
1
=
−
a
k
+
1
2
−
a
k
a_{k+1}=-a_k+\frac{1}{2-a_k}
a
k
+
1
=
−
a
k
+
2
−
a
k
1
for all
k
=
1
,
2
,
…
k = 1, 2,\ldots
k
=
1
,
2
,
…
. Prove that
(
n
2
(
a
1
+
a
2
+
⋯
+
a
n
)
−
1
)
n
≤
(
a
1
+
a
2
+
⋯
+
a
n
n
)
n
(
1
a
1
−
1
)
(
1
a
2
−
1
)
⋯
(
1
a
n
−
1
)
.
\left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right).
(
2
(
a
1
+
a
2
+
⋯
+
a
n
)
n
−
1
)
n
≤
(
n
a
1
+
a
2
+
⋯
+
a
n
)
n
(
a
1
1
−
1
)
(
a
2
1
−
1
)
⋯
(
a
n
1
−
1
)
.
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