AMC 12A 2025 (Problem 25)
Polynomials and each have degree and leading coefficient , and their roots are all elements of . The function has the property that there exist real numbers such that the set of all real numbers such that consists of the closed interval together with the open interval . How many functions are possible?
![Sign diagram for f(x) on the intervals [a,b] and (c,d)](/__l5e/assets-v1/46fa78ca-cb5f-433e-9d39-700f7299944f/amc12a2025p25.1.png)
By Hint 1, and should be factors of with odd multiplicity and should not be factors of .
and should be factors of , and the sum of the powers of in the numerator and denominator should be odd. The same holds for .
By Hint 2,
Case 1:
Case 2: or
Case 1:
should be outside and .
So,
should be outside and .
So,
Case 2: or
Both cases result in the same .
WLOG, .
We should select four different numbers for .
Both cases result in the same .
WLOG, .
We should select four different numbers for .
By Hints 4 and 5:
But isn't in the answer choices.
If we want to count , then in Hint 5 we should also consider , which similarly gives cases.
But isn't in the answer choices.
If we want to count , then in Hint 5 we should also consider , which similarly gives cases.