2025 AMC 12A Problems/Problem 25

Polynomials $P(x)$ and $Q(x)$ each have degree $3$ and leading coefficient $1$ , and their roots are all elements of $\{1,2,3,4,5\}$ . The function $f(x) = \tfrac{P(x)}{Q(x)}$ has the property that there exist real numbers $a < b < c < d$ such that the set of all real numbers $x$ such that $f(x) \leq 0$ consists of the closed interval $[a,b]$ together with the open interval $(c,d)$ . How many functions $f(x)$ are possible?

Video Solution 1 by OmegaLearn

https://youtu.be/SPbTyq3Dz_0

2025 AMC 12A (Problems • Answer Key • Resources)
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2025 AMC 12A Problems/Problem 25

Video Solution 1 by OmegaLearn

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