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Mathematical Excellence Olympiad
2017 IMEO
4
IMEO 2017 Problem 4
IMEO 2017 Problem 4
Source: IMEO
September 28, 2017
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
a
3
1
+
b
c
+
b
3
1
+
a
c
+
c
3
1
+
a
b
≥
2
\sqrt{\frac{a^3}{1+bc}}+\sqrt{\frac{b^3}{1+ac}}+\sqrt{\frac{c^3}{1+ab}}\geq 2
1
+
b
c
a
3
+
1
+
a
c
b
3
+
1
+
ab
c
3
≥
2
Are there any triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
, for which the equality holds? Proposed by Konstantinos Metaxas.
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