2023 WSMO Accuracy Round Problems/Problem 8

Problem

Let $f(x)=x^3-3x^2+4x+5$ have complex roots $a,b,c$ . Then, the value of $\frac{1}{a^2+b^2}+\frac{1}{a^2+c^2}+\frac{1}{b^2+c^2}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n$

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2023 WSMO Accuracy Round Problems/Problem 8

Problem

Solution