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Contests
International Contests
Pan African
2016 PAMO
4
PAMO 2016 Q4
PAMO 2016 Q4
Source: PAMO 2016
April 29, 2016
Inequality
algebra
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that
x
y
z
=
1
xyz=1
x
yz
=
1
. Prove that
1
(
x
+
1
)
2
+
y
2
+
1
\frac{1}{(x+1)^2+y^2+1}
(
x
+
1
)
2
+
y
2
+
1
1
+
+
+
1
(
y
+
1
)
2
+
z
2
+
1
\frac{1}{(y+1)^2+z^2+1}
(
y
+
1
)
2
+
z
2
+
1
1
+
+
+
1
(
z
+
1
)
2
+
x
2
+
1
\frac{1}{(z+1)^2+x^2+1}
(
z
+
1
)
2
+
x
2
+
1
1
≤
\leq
≤
1
2
{\frac{1}{2}}
2
1
.
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