2005 IMO Shortlist Problems/A1
Problem
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.
Solution
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.