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1 solution
if a+b+c=0, then 0 = a^3 + b^3 + c^3 + 3(ab+ac+bc) – 3 abc, or a^3 + b^3 + c^3 = 3 abc. Consider ,a^5 + b^5 + c^5 = (a^3 + b^3 + c^3)( a^2 + b^2 + c^2) – (a^2 b^2 + a^2 c^2 + b^2 c^2)(a+b+c) + abc(ab+ac+bc) And 0 = ( a^2 + b^2 + c^2) + 2(ab+ac+bc), or ( a^2 + b^2 + c^2) = - 2(ab+ac+bc), So (a^5+b^5+c^5)/(ab+ac+bc) = [(a^3 + b^3 + c^3)( a^2 + b^2 + c^2) + abc(ab+ac+bc)]/ (ab+ac+bc)] = - 5 abc.