Complex Multiplication!

Algebra Level 5

\(a\), \(b\), \(c\) and \(d\) are non-zero complex numbers that satisfy

\[a+b+c+d=a^3+b^3+c^3+d^3=0\]

The sum of all possible values of \((a+b)(a+c)(a+d)\) can be expressed as \(p+qi\) where \(p\) and \(q\) are real numbers and \(i\) is the imaginary unit that satisfies \(i^2=-1\). What is \(p+q\)?


This problem is inspired by a problem that appeared in ITT - \(1994\).


This problem is from the set "Olympiads and Contests Around the World -1". You can see the rest of the problems here.

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