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JBMO ShortLists
2019 JBMO Shortlist
A2
Simple inequality
Simple inequality
Source: Shortlist BMO 2018, A1
May 3, 2019
Balkan
algebra
BPSQ
inequalities proposed
inequalities
Hi
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
2
3
.
abc = \frac {2} {3}.
ab
c
=
3
2
.
Prove that:
a
b
a
+
b
+
b
c
b
+
c
+
c
a
c
+
a
⩾
a
+
b
+
c
a
3
+
b
3
+
c
3
.
\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.
a
+
b
ab
+
b
+
c
b
c
+
c
+
a
c
a
⩾
a
3
+
b
3
+
c
3
a
+
b
+
c
.
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