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National and Regional Contests
China Contests
XES Mathematics Olympiad
the 11th XMO
2
Hard 3 variable inequality
Hard 3 variable inequality
Source:
February 9, 2023
algebraic inequality
inequalities
Problem Statement
Suppose
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
c
=
64
abc=64
ab
c
=
64
, show that
∑
c
y
c
a
2
a
3
+
8
b
3
+
8
≥
2
3
\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}
cyc
∑
a
3
+
8
b
3
+
8
a
2
≥
3
2
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