MEMO 2014 ekipno problem 1
Dodao/la:
arhiva24. rujna 2014. 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{,}](/media/m/9/9/d/99d35622e30515e95f323fb393fd8448.png)
where
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
, and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
are positive real numbers satisfying the inequalities
%V0
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 $$
\frac{1}{a + x} \geq \frac12, \quad
\frac{1}{a + y} \geq \frac12, \quad
\frac{1}{b + x} \geq \frac12, \quad \text{and}
\ \frac{1}{b + y} \geq 1 \text{.}
$$
Izvor: Srednjoeuropska matematička olimpijada 2014, ekipno natjecanje, problem 1