Inspired by Deepanshu Gupta

Algebra Level 5

\[f(x)=x^4+ax^3+bx^2+cx+1\]

If \(f(x)\) has at least one real root, for real numbers \(a,b\) and \(c\), find the minimal value of \(a^2+b^2+c^2\). Write your answer in the form \(\frac{p}{q}\) for co-prime positive integers \(p\) and \(q\), and enter \(p+q\).

If you come to the conclusion that no minimum is attained, enter 666.


Inspiration.

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