2023 AIME I Problems/Problem 9
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
I believe this solution is wrong. The answer is . ~r00tsOfUnity
??? I checked with like 5 people ~mathboy100