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Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2005 Iran MO (3rd Round)
1
a,b,c
a,b,c
Source: Iran 2005
August 27, 2005
inequalities
3-variable inequality
cyclic inequality
cauchy schwarz
Problem Statement
Suppose
a
,
b
,
c
∈
R
+
a,b,c\in \mathbb R^+
a
,
b
,
c
∈
R
+
. Prove that :
(
a
b
+
b
c
+
c
a
)
2
≥
(
a
+
b
+
c
)
(
1
a
+
1
b
+
1
c
)
\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)
(
b
a
+
c
b
+
a
c
)
2
≥
(
a
+
b
+
c
)
(
a
1
+
b
1
+
c
1
)
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