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2009 Cuba MO
3
min x^2 + y^2 + z^2 if x^3 + y^3 + z^3 -3xyz = 1
min x^2 + y^2 + z^2 if x^3 + y^3 + z^3 -3xyz = 1
Source: 2009 Cuba 2.3
August 27, 2024
algebra
inequalities
Problem Statement
Determine the smallest value of
x
2
+
y
2
+
z
2
x^2 + y^2 + z^2
x
2
+
y
2
+
z
2
, where
x
,
y
,
z
x, y, z
x
,
y
,
z
are real numbers, so that
x
3
+
y
3
+
z
3
ā
3
x
y
z
=
1.
x^3 + y^3 + z^3 -3xyz = 1.
x
3
+
y
3
+
z
3
ā
3
x
yz
=
1.
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