Complex Multiplication!Algebra Level 5
\(a\), \(b\), \(c\) and \(d\) are non-zero complex numbers that satisfy
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.