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South East Mathematical Olympiad
2017 South East Mathematical Olympiad
3
China South East Mathematical Olympiad 2017 Grade11 Q3
China South East Mathematical Olympiad 2017 Grade11 Q3
Source: China Jiangxi , Jul 30, 2017
July 30, 2017
inequalities
algebra
China
BPSQ
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
+
1
>
0
a_1,a_2,\cdots,a_{n+1}>0
a
1
,
a
2
,
⋯
,
a
n
+
1
>
0
. Prove that
∑
i
−
1
n
a
i
∑
i
=
1
n
a
i
+
1
≥
∑
i
=
1
n
a
i
a
i
+
1
a
i
+
a
i
+
1
⋅
∑
i
=
1
n
(
a
i
+
a
i
+
1
)
\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})
i
−
1
∑
n
a
i
i
=
1
∑
n
a
i
+
1
≥
i
=
1
∑
n
a
i
+
a
i
+
1
a
i
a
i
+
1
⋅
i
=
1
∑
n
(
a
i
+
a
i
+
1
)
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