2023 AIME I Problems/Problem 9
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Problem (Unofficial, please update when official one comes out):
is a polynomial with integer coefficients between
and
, inclusive. There is exactly one integer
such that
. How many possible values are there for the ordered triple
?
Solution
If is the only integral value that satisfies
, we can show that
is the only real value that satisfies
.
Next, we have , so therefore
We can now simplify:
Since ,
We can now apply the quadratic formula, yielding
For this to have exactly solution, we must have
, and thus
. This means that
, yielding
solutions for
. For any solution of
,
can only attain one value. And, the value of
doesn't matter. Our answer is thus
.
~mathboy100