IMO Shortlist 2005 problem A1
Dodao/la:
arhiva2. travnja 2012. Find all pairs of integers
![a,b](/media/m/7/d/8/7d8bdace47e602448e6040957d8cf923.png)
for which there exists a polynomial
![P(x) \in \mathbb{Z}[X]](/media/m/c/8/8/c88440085030c32a41066167316e5adc.png)
such that product
![(x^2+ax+b)\cdot P(x)](/media/m/2/a/2/2a2a013d8ddfad91261dfecebd7034f6.png)
is a polynomial of a form
![x^n+c_{n-1}x^{n-1}+...+c_1x+c_0](/media/m/1/5/e/15e17b51e49a4d64a430bea025ec74ba.png)
where each of
![c_0,c_1,...,c_{n-1}](/media/m/d/4/b/d4bd1502d72b11d6d1cb6645ec056e1a.png)
is equal to
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
or
![-1](/media/m/6/1/c/61cf05f5d8d6a4f0d373e7452cde9c3c.png)
.
%V0
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form $$x^n+c_{n-1}x^{n-1}+...+c_1x+c_0$$ where each of $c_0,c_1,...,c_{n-1}$ is equal to $1$ or $-1$.
Izvor: Međunarodna matematička olimpijada, shortlist 2005