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4
Turkey NMO 2012 Problem 4
Turkey NMO 2012 Problem 4
Source: Turkey NMO 2012
November 26, 2012
inequalities
inequalities proposed
Problem Statement
For all positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
, show that
x
(
2
x
−
y
)
y
(
2
z
+
x
)
+
y
(
2
y
−
z
)
z
(
2
x
+
y
)
+
z
(
2
z
−
x
)
x
(
2
y
+
z
)
≥
1
\frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1
y
(
2
z
+
x
)
x
(
2
x
−
y
)
+
z
(
2
x
+
y
)
y
(
2
y
−
z
)
+
x
(
2
y
+
z
)
z
(
2
z
−
x
)
≥
1
is true.
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