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South East Mathematical Olympiad
2016 South East Mathematical Olympiad
1
China South East Mathematical Olympiad 2016 Grade10 Q1
China South East Mathematical Olympiad 2016 Grade10 Q1
Source: China Nanchang
July 30, 2016
inequalities
Problem Statement
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
1
,
a
2
=
1
2
a_1=1,a_2=\frac{1}{2}
a
1
=
1
,
a
2
=
2
1
,
n
(
n
+
1
)
a
n
+
1
a
n
+
n
a
n
a
n
−
1
=
(
n
+
1
)
2
a
n
+
1
a
n
−
1
(
n
≥
2
)
.
n(n+1) a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).
n
(
n
+
1
)
a
n
+
1
a
n
+
n
a
n
a
n
−
1
=
(
n
+
1
)
2
a
n
+
1
a
n
−
1
(
n
≥
2
)
.
Prove that
2
n
+
1
<
a
n
n
<
1
n
(
n
≥
3
)
.
\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).
n
+
1
2
<
n
a
n
<
n
1
(
n
≥
3
)
.
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