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National and Regional Contests
Korea Contests
Korea National Olympiad
2009 Korea National Olympiad
2
Easy 3 variables inequality
Easy 3 variables inequality
Source: Korea National Olympiad 2009 Problem 2
September 9, 2012
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
a
3
c
(
a
2
+
b
c
)
+
b
3
a
(
b
2
+
c
a
)
+
c
3
b
(
c
2
+
a
b
)
≥
3
2
.
\frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} .
c
(
a
2
+
b
c
)
a
3
+
a
(
b
2
+
c
a
)
b
3
+
b
(
c
2
+
ab
)
c
3
≥
2
3
.
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