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A3
x^2y/(x+2y)+y^2z/(y+2z)+z^2x/(z+2x)<(x+y+z)^2/8 , in positive real
x^2y/(x+2y)+y^2z/(y+2z)+z^2x/(z+2x)<(x+y+z)^2/8 , in positive real
Source: INAMO Shortlist 2014 A3
May 21, 2019
algebra
inequalities
three variable inequality
Problem Statement
Prove for each positive real number
x
,
y
,
z
x, y, z
x
,
y
,
z
,
x
2
y
x
+
2
y
+
y
2
z
y
+
2
z
+
z
2
x
z
+
2
x
<
(
x
+
y
+
z
)
2
8
\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}
x
+
2
y
x
2
y
+
y
+
2
z
y
2
z
+
z
+
2
x
z
2
x
<
8
(
x
+
y
+
z
)
2
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