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Romania National Olympiad
2014 Romania National Olympiad
1
Romania National Olympiad 2014,grade 8 ,P1
Romania National Olympiad 2014,grade 8 ,P1
Source:
April 29, 2014
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
∈
(
0
,
∞
)
a,b,c\in \left( 0,\infty \right)
a
,
b
,
c
∈
(
0
,
∞
)
.Prove the inequality
a
−
b
c
a
+
2
(
b
+
c
)
+
b
−
c
a
b
+
2
(
c
+
a
)
+
c
−
a
b
c
+
2
(
a
+
b
)
≥
0.
\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.
a
+
2
(
b
+
c
)
a
−
b
c
+
b
+
2
(
c
+
a
)
b
−
c
a
+
c
+
2
(
a
+
b
)
c
−
ab
≥
0.
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