2023 AIME I Problems/Problem 9
Find the number of cubic polynomials , where
,
, and
are integers in
, such that there is a unique integer
with
Solution
It can be easily noticed that is independent of the condition
, and can thus safely take all
possible values between
and
.
There are two possible ways for to be the only integer satisfying
:
has a double root at
or a double root at
.
Case 1: has a double root at
:
In this case, , or
. Thus
ranges from
to
. One of these values,
corresponds to a triple root at
, which means
. Thus there are
possible values of
. (It can be verified that
is an integer).
Case 2: has a double root at
:
See the above solution. This yields possible combinations of
and
.
Thus, in total we have combinations of
.
-Alex_Z