AMC 10 Step by Step
2021 AMC 10AProblem 18P16-20~4 minPrint

Let ff be a function defined on the set of positive rational numbers with the property that f(ab)=f(a)+f(b)f(a\cdot b)=f(a)+f(b) for all positive rational numbers aa and bb . Furthermore, suppose that ff also has the property that f(p)=pf(p)=p for every prime number pp . For which of the following numbers xx is f(x)<0f(x)<0 ?

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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.