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Let \(a, b, c, d\) be real numbers, none of which are equal to \(-1\), and let w be a complex number such that \(w^3=1\) and \(w \neq 1.\) 

 

If \(\frac{1}{a + w}+\frac{1}{b + w}+\frac{1}{c + w}+\frac{1}{d + w}= \frac{2}{w}\), then find \(\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}\).  

 

Thanks for any help, I know that by factorizing the w^3=1, you can solve for w, but this does not seem to do anything to help simplify.  

 May 29, 2021
 #1
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$\omega$ is a cube root of unity so write it in exponential form everywhere. This problem is easy to guess as well.

 May 30, 2021

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