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Balkan MO
2014 Balkan MO
1
1
Part of
2014 Balkan MO
Problems
(1)
xy+yz+xz=3xyz
Source: Balkan Mathematics Olympiad 2014 - Problem-1
5/4/2014
Let
x
,
y
x,y
x
,
y
and
z
z
z
be positive real numbers such that
x
y
+
y
z
+
x
z
=
3
x
y
z
xy+yz+xz=3xyz
x
y
+
yz
+
x
z
=
3
x
yz
. Prove that
x
2
y
+
y
2
z
+
z
2
x
≥
2
(
x
+
y
+
z
)
−
3
x^2y+y^2z+z^2x \ge 2(x+y+z)-3
x
2
y
+
y
2
z
+
z
2
x
≥
2
(
x
+
y
+
z
)
−
3
and determine when equality holds.UK - David Monk
algebra
inequalities
Balkan Mathematics Olympiad