MEMO 2011 pojedinačno problem 4
Dodao/la:
arhiva28. travnja 2012. Let
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
and
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
, with
![k > m](/media/m/8/2/9/82941fe68d7b58d0b938f52d2825d675.png)
, be positive integers such that the number
![km(k^2 - m^2)](/media/m/9/c/8/9c843893fc5f871f1bd25022611ce988.png)
is divisible by
![k^3 - m^3](/media/m/c/8/6/c86731ce4d1e34fae4d9faa5b4fbc447.png)
. Prove that
![(k - m)^3 > 3km](/media/m/8/b/a/8ba238510d7ec09d3da05b62d33b9a72.png)
.
%V0
Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.
Izvor: Srednjoeuropska matematička olimpijada 2011, pojedinačno natjecanje, problem 4