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Middle European Mathematical Olympiad
2014 Middle European Mathematical Olympiad
1
lowest possible value of the expression
lowest possible value of the expression
Source: Middle European Mathematical Olympiad T-1
September 21, 2014
inequalities
inequalities proposed
Problem Statement
Determine the lowest possible value of the expression
1
a
+
x
+
1
a
+
y
+
1
b
+
x
+
1
b
+
y
\frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y}
a
+
x
1
+
a
+
y
1
+
b
+
x
1
+
b
+
y
1
where
a
,
b
,
x
,
a,b,x,
a
,
b
,
x
,
and
y
y
y
are positive real numbers satisfying the inequalities
1
a
+
x
≥
1
2
\frac{1}{a+x} \ge \frac{1}{2}
a
+
x
1
≥
2
1
1
a
+
y
≥
1
2
\frac{1}{a+y} \ge \frac{1}{2}
a
+
y
1
≥
2
1
1
b
+
x
≥
1
2
\frac{1}{b+x} \ge \frac{1}{2}
b
+
x
1
≥
2
1
1
b
+
y
≥
1.
\frac{1}{b+y} \ge 1.
b
+
y
1
≥
1.
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