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APMO
2005 APMO
2
cyclic, abc=8
cyclic, abc=8
Source: APMO 2005 Problem 2
March 23, 2005
inequalities
algebra
inequalities proposed
APMO
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
8
abc=8
ab
c
=
8
. Prove that
a
2
(
1
+
a
3
)
(
1
+
b
3
)
+
b
2
(
1
+
b
3
)
(
1
+
c
3
)
+
c
2
(
1
+
c
3
)
(
1
+
a
3
)
≥
4
3
\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}
(
1
+
a
3
)
(
1
+
b
3
)
a
2
+
(
1
+
b
3
)
(
1
+
c
3
)
b
2
+
(
1
+
c
3
)
(
1
+
a
3
)
c
2
≥
3
4
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