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International Contests
APMO
2002 APMO
4
4
Part of
2002 APMO
Problems
(1)
Sum 1/x = 1
Source: APMO 2002
4/8/2006
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive numbers such that
1
x
+
1
y
+
1
z
=
1.
{1\over x}+{1\over y}+{1\over z}=1.
x
1
+
y
1
+
z
1
=
1.
Show that
x
+
y
z
+
y
+
z
x
+
z
+
x
y
≥
x
y
z
+
x
+
y
+
z
\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}
x
+
yz
+
y
+
z
x
+
z
+
x
y
≥
x
yz
+
x
+
y
+
z
inequalities