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Romania National Olympiad
1997 Romania National Olympiad
2
Nice but seems difficult
Nice but seems difficult
Source: Romania 1997
March 31, 2006
inequalities
inequalities proposed
Problem Statement
I found this inequality in "Topics in Inequalities" (I 85) For all positive reals
x
,
y
,
z
x,y,z
x
,
y
,
z
with
x
y
z
=
1
xyz=1
x
yz
=
1
prove:
x
9
+
y
9
x
6
+
x
3
y
3
+
y
6
+
y
9
+
z
9
y
6
+
y
3
z
3
+
z
6
+
z
9
+
x
9
z
6
+
z
3
x
3
+
x
6
≥
2
\frac{x^9+y^9}{x^6+x^3y^3+y^6}+\frac{y^9+z^9}{y^6+y^3z^3+z^6}+\frac{z^9+x^9}{z^6+z^3x^3+x^6}\geq 2
x
6
+
x
3
y
3
+
y
6
x
9
+
y
9
+
y
6
+
y
3
z
3
+
z
6
y
9
+
z
9
+
z
6
+
z
3
x
3
+
x
6
z
9
+
x
9
≥
2
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