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National and Regional Contests
Indonesia Contests
Indonesia TST
2010 Indonesia TST
1
Hard inequality
Hard inequality
Source:
October 23, 2010
inequalities
Problem Statement
Given
a
,
b
,
c
a,b, c
a
,
b
,
c
positive real numbers satisfying
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
1
a
b
+
b
c
+
c
a
≥
2
a
3
(
b
+
c
)
+
2
b
3
(
c
+
a
)
+
2
c
3
(
a
+
b
)
≥
a
+
b
+
c
\dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c}
ab
+
b
c
+
c
a
1
≥
3
(
b
+
c
)
2
a
+
3
(
c
+
a
)
2
b
+
3
(
a
+
b
)
2
c
≥
a
+
b
+
c
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