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MathLinks Contest 3rd
2
0312 inequalities 3rd edition Round 1 p2
0312 inequalities 3rd edition Round 1 p2
Source:
May 9, 2021
3rd edition
inequalities
algebra
Problem Statement
Prove that for all positive reals
a
,
b
,
c
a, b, c
a
,
b
,
c
the following double inequality holds:
a
+
b
+
c
3
≥
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
8
3
≥
a
b
+
b
c
c
a
3
\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}
3
a
+
b
+
c
≥
3
8
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
3
ab
+
b
c
c
a
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