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National and Regional Contests
Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
2
3 variable inequality
3 variable inequality
Source: Korea National 2013 #2
November 10, 2013
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
>
0
a, b, c>0
a
,
b
,
c
>
0
such that
a
b
+
b
c
+
c
a
=
3
ab+bc+ca=3
ab
+
b
c
+
c
a
=
3
. Prove that
∑
c
y
c
(
a
+
b
)
3
(
2
(
a
+
b
)
(
a
2
+
b
2
)
)
1
3
≥
12
\sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }
cyc
∑
(
2
(
a
+
b
)
(
a
2
+
b
2
))
3
1
(
a
+
b
)
3
≥
12
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