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2009 Irish Math Olympiad
5
Three Variables Inequality
Three Variables Inequality
Source:
September 24, 2010
inequalities
inequalities open
Problem Statement
Hello.Suppose
a
a
a
,
b
b
b
,
c
c
c
are real numbers such that
a
+
b
+
c
=
0
a+b+c = 0
a
+
b
+
c
=
0
and
a
2
+
b
2
+
c
2
=
1
a^{2}+b^{2}+c^{2} = 1
a
2
+
b
2
+
c
2
=
1
. Prove that
a
2
b
2
c
2
≤
1
54
a^{2}b^{2}c^{2}\leq \frac{1}{54}
a
2
b
2
c
2
≤
54
1
and determine the cases of equality.
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