IMO Shortlist 1969 problem 28
Dodao/la:
arhiva2. travnja 2012. ![(GBR 5)](/media/m/a/7/3/a73a0a443b75d0729ccd9d0b080c552f.png)
Let us define
![u_0 = 0, u_1 = 1](/media/m/9/6/a/96a8b97f5b6db823b7c9e278b0a418a8.png)
and for
![n\ge 0, u_{n+2} = au_{n+1}+bu_n, a](/media/m/d/a/9/da94de25b90172fbfcbd5f8b89b22e55.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
being positive integers. Express
![u_n](/media/m/4/e/f/4ef22a367647acd2e57faa44940d07c1.png)
as a polynomial in
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b.](/media/m/9/0/a/90a5f50b15448e78bc5971529300acc5.png)
Prove the result. Given that
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
is prime, prove that
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
divides
%V0
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
Izvor: Međunarodna matematička olimpijada, shortlist 1969