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Korea National Olympiad
2014 Korea National Olympiad
3
3 variable inequality, hard
3 variable inequality, hard
Source: Korea National 2014 #7
January 21, 2015
inequalities
inequalities proposed
Problem Statement
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be the real numbers that satisfies the following.
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
8
,
x
3
+
y
3
+
z
3
=
1
(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
8
,
x
3
+
y
3
+
z
3
=
1
Find the minimum value of
x
4
+
y
4
+
z
4
x^4+y^4+z^4
x
4
+
y
4
+
z
4
.
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