2005 AIME II Problems/Problem 13
Problem
Let be a polynomial with integer coefficients that satisfies
and
Given that
has two distinct integer solutions
and
find the product
Solution
Define the polynomial . By the givens,
,
,
and
. Note that for any polynomial
with integer coefficients and any integers
we have
divides
. So
divides
, and so
must be one of the eight numbers
and so
must be one of the numbers
or
. Similarly,
must divide
, so
must be one of the eight numbers
or
. Thus,
must be either 19 or 22. Since
obeys the same conditions and
and
are different, one of them is 19 and the other is 22 and their product is
.