2005 IMO Shortlist Problems/A1
Find all monic polynomials of degree two for which there exists an integer polynomial such that is a polynomial having all coefficients .
This was also the last problem of the final round of the 2006 Polish Mathematics Olympiad.
Since the constant term of is , and and both have integral constant terms, the constant term of must be .
We note that for , (), we have
Since we must have when is the degree of and is a root thereof, this means that cannot have any roots of magnitude greater than or equal to 2.
Now, if , then we cannot have , for then one of the roots would have magnitude , and similarly, if , then we cannot have , for then one of the roots would have magnitude .
This leaves us only the possibilities . For these we have respective solutions . These are therefore the only solutions, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.