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National and Regional Contests
Serbia Contests
Serbia JBMO TST
2017 Serbia JBMO TST
2
Inequality
Inequality
Source: Serbia JBMO TST 2017
May 22, 2017
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers.Prove that
(
x
y
2
+
y
z
2
+
z
x
2
)
(
x
2
y
+
y
2
z
+
z
2
x
)
(
x
y
+
y
z
+
z
x
)
≥
3
(
x
+
y
+
z
)
2
(
x
y
z
)
2
.
(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)(xy+yz+zx)\geq 3(x+y+z)^2(xyz)^2.
(
x
y
2
+
y
z
2
+
z
x
2
)
(
x
2
y
+
y
2
z
+
z
2
x
)
(
x
y
+
yz
+
z
x
)
≥
3
(
x
+
y
+
z
)
2
(
x
yz
)
2
.
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