IMO Shortlist 2000 problem A2
Dodao/la:
arhiva2. travnja 2012. Let
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
be positive integers satisfying the conditions
![b > 2a](/media/m/9/4/8/948d4edf12537dcf5d64cabf644d3f3a.png)
and
![c > 2b.](/media/m/b/d/4/bd48164e4040c1c6884c50e2953f48fb.png)
Show that there exists a real number
![\lambda](/media/m/9/b/e/9be7eeb58b67ec913359062c0122ee80.png)
with the property that all the three numbers
![\lambda a, \lambda b, \lambda c](/media/m/7/a/7/7a71977c1955fbda3155e0f90978a91d.png)
have their fractional parts lying in the interval
%V0
Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$
Izvor: Međunarodna matematička olimpijada, shortlist 2000