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2006 Korea National Olympiad
6
3-Vari Ineq
3-Vari Ineq
Source: 2006 Korea National Olympid #6
March 18, 2018
inequalities
algebra
Problem Statement
Prove that for any positive real numbers
x
,
y
x,y
x
,
y
and
z
,
z,
z
,
x
y
z
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
≤
(
1
+
2
(
x
y
+
y
z
+
z
x
)
3
)
3
xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3
x
yz
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
≤
(
1
+
3
2
(
x
y
+
yz
+
z
x
)
)
3
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