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National and Regional Contests
Mexico Contests
Mexico National Olympiad
2014 Mexico National Olympiad
5
5
Part of
2014 Mexico National Olympiad
Problems
(1)
Easy inequality on a + b + c = 3
Source: Mexican Mathematical Olympiad 2014 Problem 5
11/15/2014
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove:
a
2
a
+
b
c
3
+
b
2
b
+
c
a
3
+
c
2
c
+
a
b
3
≥
3
2
\frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2}
a
+
3
b
c
a
2
+
b
+
3
c
a
b
2
+
c
+
3
ab
c
2
≥
2
3
And determine when equality holds.
inequalities
inequalities proposed
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