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Ireland National Math Olympiad
2004 Irish Math Olympiad
5
Inequality
Inequality
Source: 2004 IrMO Paper 1 Problem 5
December 28, 2017
inequalities
Problem Statement
Let
a
,
b
≥
0
a,b\ge 0
a
,
b
≥
0
. Prove that
2
(
a
(
a
+
b
)
3
+
b
a
2
+
b
2
)
≤
3
(
a
2
+
b
2
)
\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)
2
(
a
(
a
+
b
)
3
+
b
a
2
+
b
2
)
≤
3
(
a
2
+
b
2
)
with equality if and only if
a
=
b
a=b
a
=
b
.
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