MathDB
Problems
Contests
International Contests
Middle European Mathematical Olympiad
2013 Middle European Mathematical Olympiad
2
Four variables
Four variables
Source: Middle European Mathematical Olympiad 2013 T-2
May 17, 2014
inequalities proposed
inequalities
Problem Statement
Let
x
,
y
,
z
,
w
x, y, z, w
x
,
y
,
z
,
w
be nonzero real numbers such that
x
+
y
≠
0
x+y \ne 0
x
+
y
=
0
,
z
+
w
≠
0
z+w \ne 0
z
+
w
=
0
, and
x
y
+
z
w
≥
0
xy+zw \ge 0
x
y
+
z
w
≥
0
. Prove that
(
x
+
y
z
+
w
+
z
+
w
x
+
y
)
−
1
+
1
2
≥
(
x
z
+
z
x
)
−
1
+
(
y
w
+
w
y
)
−
1
\left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}
(
z
+
w
x
+
y
+
x
+
y
z
+
w
)
−
1
+
2
1
≥
(
z
x
+
x
z
)
−
1
+
(
w
y
+
y
w
)
−
1
Back to Problems
View on AoPS