IMO Shortlist 2011 problem N6
Dodao/la:
arhiva23. lipnja 2013. Let
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
and
![Q(x)](/media/m/2/2/9/229c25a1e13a21d9e725e8b48e6a685f.png)
be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
and
![Q(x).](/media/m/0/2/2/022f5d48486cfb77a0388f4dfd8aab36.png)
Suppose that for every positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
the integers
![P(n)](/media/m/f/6/8/f680d9bbded52866a9815281ae83c905.png)
and
![Q(n)](/media/m/9/6/b/96b59eb878ad13910e80248ad982871d.png)
are positive, and
![2^{Q(n)}-1](/media/m/7/a/4/7a470f7e94067bc14711b84d0cccbe04.png)
divides
![3^{P(n)}-1.](/media/m/8/e/a/8ea5ad1ad3f3397f6cfc596dc1cba843.png)
Prove that
![Q(x)](/media/m/2/2/9/229c25a1e13a21d9e725e8b48e6a685f.png)
is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine
%V0
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine
Izvor: Međunarodna matematička olimpijada, shortlist 2011