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Problems
Contests
International Contests
Baltic Way
2007 Baltic Way
4
Cauchy-Schwarz prone
Cauchy-Schwarz prone
Source: Baltic Way 2007
November 30, 2010
inequalities proposed
inequalities
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers, and let
S
=
a
1
+
a
2
+
…
+
a
n
S=a_1+a_2 +\ldots +a_n
S
=
a
1
+
a
2
+
…
+
a
n
. Prove that
(
2
S
+
n
)
(
2
S
+
a
1
a
2
+
a
2
a
3
+
…
+
a
n
a
1
)
≥
9
(
a
1
a
2
+
a
2
a
3
+
…
+
a
n
a
1
)
2
(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2
(
2
S
+
n
)
(
2
S
+
a
1
a
2
+
a
2
a
3
+
…
+
a
n
a
1
)
≥
9
(
a
1
a
2
+
a
2
a
3
+
…
+
a
n
a
1
)
2
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