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2018 Pan-African Shortlist
A6
2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$
2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$
Source: 2018 Pan-African Shortlist - A6
May 6, 2019
inequalities
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
3
+
b
3
+
c
3
=
5
a
b
c
a^3 + b^3 + c^3 = 5abc
a
3
+
b
3
+
c
3
=
5
ab
c
.Show that
(
a
+
b
c
)
(
b
+
c
a
)
(
c
+
a
b
)
≥
9.
\left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9.
(
c
a
+
b
)
(
a
b
+
c
)
(
b
c
+
a
)
≥
9.
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