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National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2010 Kyrgyzstan National Olympiad
1
ineq
ineq
Source:
April 25, 2011
inequalities unsolved
inequalities
Problem Statement
Given that
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
and
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove that
a
b
a
b
+
c
+
b
c
b
c
+
a
+
c
a
c
a
+
b
⩽
3
2
\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}
ab
+
c
ab
+
b
c
+
a
b
c
+
c
a
+
b
c
a
⩽
2
3
.
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