IMO Shortlist 2004 problem N4
Dodao/la:
arhiva2. travnja 2012. Let
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be a fixed integer greater than 1, and let
![{m=4k^2-5}](/media/m/3/1/5/315f4b9c6f49b0beca51469adea674eb.png)
. Show that there exist positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
such that the sequence
![(x_n)](/media/m/1/c/9/1c91ef86967bca399379e8bcfcef22b3.png)
defined by
![x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots](/media/m/5/7/7/5779cd993da300e7cd61b66ddf680dae.png)
has all of its terms relatively prime to
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
.
%V0
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by $$x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots$$ has all of its terms relatively prime to $m$.
Izvor: Međunarodna matematička olimpijada, shortlist 2004