AMC 10 Step by Step
2019 AMC 10AProblem 24P21-25~7 minPrint

Let pp , qq , and rr be the distinct roots of the polynomial x322x2+80x67x^3 - 22x^2 + 80x - 67 . It is given that there exist real numbers AA , BB , and CC such that 1s322s2+80s67=Asp+Bsq+Csr\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r} for all s∉{p,q,r}s\not\in\{p,q,r\} . What is 1A+1B+1C\tfrac1A+\tfrac1B+\tfrac1C ?

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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.