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2009 South East Mathematical Olympiad
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Inequality for six variables-China South East Olympiad 2009
Inequality for six variables-China South East Olympiad 2009
Source:
September 18, 2010
inequalities
inequalities proposed
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals such that
a
=
x
(
y
−
z
)
2
\sqrt{a}=x(y-z)^2
a
=
x
(
y
−
z
)
2
,
b
=
y
(
z
−
x
)
2
\sqrt{b}=y(z-x)^2
b
=
y
(
z
−
x
)
2
and
c
=
z
(
x
−
y
)
2
\sqrt{c}=z(x-y)^2
c
=
z
(
x
−
y
)
2
. Prove that
a
2
+
b
2
+
c
2
≥
2
(
a
b
+
b
c
+
c
a
)
a^2+b^2+c^2 \geq 2(ab+bc+ca)
a
2
+
b
2
+
c
2
≥
2
(
ab
+
b
c
+
c
a
)
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