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Ireland National Math Olympiad
2012 Irish Math Olympiad
3
Inequality
Inequality
Source: 2012 IrMO Paper 2 Problem 3
February 16, 2018
inequalities
Problem Statement
Suppose
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers. Prove that
(
a
b
+
b
c
+
c
a
+
1
)
2
≥
(
2
a
+
b
+
c
)
(
2
a
+
1
b
+
1
c
)
\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2\ge (2a+b+c) \left(\frac{2}{a}+\frac{1}{b}+\frac{1}{c}\right)
(
b
a
+
c
b
+
a
c
+
1
)
2
≥
(
2
a
+
b
+
c
)
(
a
2
+
b
1
+
c
1
)
with equality if and only if
a
=
b
=
c
a=b=c
a
=
b
=
c
.
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