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Baltic Way
2003 Baltic Way
3
Cyclic inequality with xyz=1
Cyclic inequality with xyz=1
Source: Baltic Way 2003 , problem 3
October 7, 2005
inequalities
inequalities proposed
Problem Statement
Let
x
x
x
,
y
y
y
and
z
z
z
be positive real numbers such that
x
y
z
=
1
xyz = 1
x
yz
=
1
. Prove that
(
1
+
x
)
(
1
+
y
)
(
1
+
z
)
≥
2
(
1
+
x
z
3
+
y
x
3
+
z
y
3
)
.
\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).
(
1
+
x
)
(
1
+
y
)
(
1
+
z
)
≥
2
(
1
+
3
z
x
+
3
x
y
+
3
y
z
)
.
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