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Junior Balkan MO
2012 Junior Balkan MO
1
JBMO 2012 Problem 1
JBMO 2012 Problem 1
Source: JBMO 2012
June 27, 2012
inequalities
function
JBMO
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
a
b
+
a
c
+
c
b
+
c
a
+
b
c
+
b
a
+
6
≥
2
2
(
1
−
a
a
+
1
−
b
b
+
1
−
c
c
)
.
\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).
b
a
+
c
a
+
b
c
+
a
c
+
c
b
+
a
b
+
6
≥
2
2
(
a
1
−
a
+
b
1
−
b
+
c
1
−
c
)
.
When does equality hold?
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