MEMO 2014 ekipno problem 1


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Sept. 24, 2014
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Determine the lowest possible value of the expression 
    \frac{1}{a + x} + \frac{1}{a + y} + \frac{1}{b + x} + \frac{1}{b + y} \text{,}
where a, b, x, and y are positive real numbers satisfying the inequalities 
    \frac{1}{a + x} \geq \frac12, \quad
    \frac{1}{a + y} \geq \frac12, \quad
    \frac{1}{b + x} \geq \frac12, \quad \text{and}
    \ \frac{1}{b + y} \geq 1 \text{.}
Source: Srednjoeuropska matematička olimpijada 2014, ekipno natjecanje, problem 1