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Turkey MO (2nd round)
2011 Turkey MO (2nd round)
3
3
Part of
2011 Turkey MO (2nd round)
Problems
(1)
Hard inequality Turkey 2011
Source: Turkey NMO 2011 Problem3
12/7/2011
x
,
y
,
z
x,y,z
x
,
y
,
z
positive real numbers such that
x
y
z
=
1
xyz=1
x
yz
=
1
Prove that:
1
x
+
y
20
+
z
11
+
1
y
+
z
20
+
x
11
+
1
z
+
x
20
+
y
11
≤
1
\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1
x
+
y
20
+
z
11
1
+
y
+
z
20
+
x
11
1
+
z
+
x
20
+
y
11
1
≤
1
inequalities
inequalities unsolved