2022 SSMO Accuracy Round Problems/Problem 4

Problem

A monic polynomial $f$ has real roots $r,s,t.$ A monic polynomial $g$ has roots $r^3,s^3,t^3.$ Given that the minimum possible value of $\frac{g(1)}{f(1)}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

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2022 SSMO Accuracy Round Problems/Problem 4

Problem

Solution