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Balkan MO
2011 Balkan MO
2
2
Part of
2011 Balkan MO
Problems
(1)
Unusual inequality, x+y+z=0
Source: Balkan Mathematical Olympiad 2011. Problem 2.
5/6/2011
Given real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
, show that
x
(
x
+
2
)
2
x
2
+
1
+
y
(
y
+
2
)
2
y
2
+
1
+
z
(
z
+
2
)
2
z
2
+
1
≥
0
\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0
2
x
2
+
1
x
(
x
+
2
)
+
2
y
2
+
1
y
(
y
+
2
)
+
2
z
2
+
1
z
(
z
+
2
)
≥
0
When does equality hold?
inequalities
algebra
three variable inequality