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Turkey Contests
Turkey MO (2nd round)
2015 Turkey MO (2nd round)
2
3-variable inequality
3-variable inequality
Source: Turkey National Olympiad 2015 P2
December 14, 2015
inequalities
inequalities proposed
Problem Statement
x
x
x
,
y
y
y
and
z
z
z
are real numbers where the sum of any two among them is not
1
1
1
. Show that,
(
x
2
+
y
)
(
x
+
y
2
)
(
x
+
y
−
1
)
2
+
(
y
2
+
z
)
(
y
+
z
2
)
(
y
+
z
−
1
)
2
+
(
z
2
+
x
)
(
z
+
x
2
)
(
z
+
x
−
1
)
2
≥
2
(
x
+
y
+
z
)
−
3
4
\dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}
(
x
+
y
−
1
)
2
(
x
2
+
y
)
(
x
+
y
2
)
+
(
y
+
z
−
1
)
2
(
y
2
+
z
)
(
y
+
z
2
)
+
(
z
+
x
−
1
)
2
(
z
2
+
x
)
(
z
+
x
2
)
≥
2
(
x
+
y
+
z
)
−
4
3
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers satisfying the equality case.
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