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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 55
Prove this algebraic inequality
Prove this algebraic inequality
Source: 2019 Jozsef Wildt International Math Competition
May 20, 2020
Product
Summation
inequalities
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
be
n
n
n
positive numbers such that
∑
i
=
1
n
a
i
=
n
\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}
i
=
1
∑
n
a
i
=
n
. Then
∏
i
=
1
n
−
1
(
1
+
1
a
i
)
a
i
+
1
(
1
+
1
a
n
)
a
1
≥
1
+
n
∑
i
=
1
n
a
i
\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}
i
=
1
∏
n
−
1
(
1
+
a
i
1
)
a
i
+
1
(
1
+
a
n
1
)
a
1
≥
1
+
i
=
1
∑
n
a
i
n
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