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Asian Pacific Mathematics Olympiad
›
2002
›
Problem 4
Problem 4
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive numbers such that
1
x
+
1
y
+
1
z
=
1
\frac1x+\frac1y+\frac1z=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
+
y
z
+
y
+
z
x
+
z
+
x
y
≥
x
y
z
+
x
+
y
+
z
.
Step 3 of 8: Expand the square
Previous step
Next step
L
2
=
x
+
y
+
z
+
x
y
+
y
z
+
z
x
+
2
∑
c
y
c
(
x
+
y
z
)
(
y
+
z
x
)
.
L^2=x+y+z+xy+yz+zx+2\sum_{cyc}\sqrt{(x+yz)(y+zx)}.
L
2
=
x
+
y
+
z
+
x
y
+
y
z
+
z
x
+
2
cy
c
∑
(
x
+
y
z
)
(
y
+
z
x
)
.
Detailed analysis
Expand the three square roots and group the cross terms cyclically.
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