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8
Korea Third Round (FKMO) 2012 #1
Korea Third Round (FKMO) 2012 #1
Source: FKMO 2012
March 25, 2012
inequalities unsolved
inequalities
Problem Statement
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers. Prove that
2
x
2
+
x
y
(
y
+
z
x
+
z
)
2
+
2
y
2
+
y
z
(
z
+
x
y
+
x
)
2
+
2
z
2
+
z
x
(
x
+
y
z
+
y
)
2
≥
1
\frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1
(
y
+
z
x
+
z
)
2
2
x
2
+
x
y
+
(
z
+
x
y
+
x
)
2
2
y
2
+
yz
+
(
x
+
yz
+
y
)
2
2
z
2
+
z
x
≥
1
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