MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 1
2014 Taiwan TST Round 1
1
Simple cube root inequality [Taiwan 2014 Quizzes]
Simple cube root inequality [Taiwan 2014 Quizzes]
Source:
July 18, 2014
inequalities
Hi
Problem Statement
Prove that for positive reals
a
a
a
,
b
b
b
,
c
c
c
we have
3
(
a
+
b
+
c
)
≥
8
a
b
c
3
+
a
3
+
b
3
+
c
3
3
3
.
3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}.
3
(
a
+
b
+
c
)
≥
8
3
ab
c
+
3
3
a
3
+
b
3
+
c
3
.
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