IMO Shortlist 2008 problem A5
Dodao/la:
arhiva2. travnja 2012. Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
,
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
be positive real numbers such that
![abcd = 1](/media/m/a/b/0/ab0e0a49c710eea48f1cbb230263f275.png)
and
![a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}](/media/m/a/c/a/aca7b818cb81060fb235f9bb87e93cd1.png)
. Prove that
Proposed by Pavel Novotný, Slovakia
%V0
Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd = 1$ and $a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$. Prove that
$$a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}$$
Proposed by Pavel Novotný, Slovakia
Izvor: Međunarodna matematička olimpijada, shortlist 2008