<s>[INST] Prove $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}=2$ if $\frac{1}{a+w}+\frac{1}{b+w}+\frac{1}{c+w}+\frac{1}{d+w}=\frac{2}{w}$ 
If $\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}+\frac{1}{d+\omega}=\frac{2}{\omega}$, where $a,b,c,d\in\mathcal{R}$ and $\omega$ is a non-real cube root of unity, then prove that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}=2$

As it was asked as a multiple choice question(this expression was one option to pick) I really do not see an easy way to prove this.
$$
\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}+\frac{1}{d+\omega}=\frac{2}{\omega}\\
\frac{1}{a+\omega^2}+\frac{1}{b+\omega^2}+\frac{1}{c+\omega^2}+\frac{1}{d+\omega^2}=\frac{2}{\omega^2}\\
\frac{1}{a\omega^2+\omega}+\frac{1}{b\omega^2+\omega}+\frac{1}{c\omega^2+\omega}+\frac{1}{d\omega^2+\omega}=2\\
\frac{1}{a\omega+1}+\frac{1}{b\omega+1}+\frac{1}{c\omega+1}+\frac{1}{d\omega+1}=2\\
$$
Is it no coincidence that replacing $\omega$ with other two cube root of unity satisfy the equation ?
 [/INST] Note that $(1+a_1)(1+a_2)...(1+a_n)= 1 + SP + DP + TP + ... nP$, where:


*

*SP stands for "single products": $a_1 + a_2 + ... + a_n$,

*DP for "double products" $a_1a_2 + a_1a_3 + ... $,

*TP for "triple products" $a_1a_2a_3 + ...$ and so on until nP which is $a_1a_2... a_n$
And, for the same reasons, $(\omega +a_1)(\omega+a_2)...(\omega+a_n)= \omega^n + \omega^{n-1}SP + \omega^{n-2}DP + \omega^{n-3}TP + ... nP$.
Then, you just want to sum everything:
$$0 = \dfrac{1}{a+\omega} + \dfrac{1}{b+\omega} +\dfrac{1}{c+\omega} +\dfrac{1}{d+\omega}  - \dfrac{2}{w} = \dfrac{m\omega - n}{(a+\omega)(b+\omega)(c+\omega)(d+\omega)\omega},$$
where $m= 2 - abc - abd - acd - bcd$ and $n = 2abcd - a-b-c-d$ are real numbers. Please note that the numerator was reduced by using $\omega^3=1$ and $\omega^4=\omega$ when needed, and the double products cancel due to the "-2" term.
If $m\neq 0$, then $\omega = \dfrac{n}{m} \in \mathbb{R}$ which contradicts the hypotesis. Thus, $m=0$, and it follows that $n=0$ too.
In particular, $$0 = m-n = 2 - abc - abd - acd - bcd - (2abcd - a-b-c-d)$$ Then, $$ 0 = \dfrac{2 - abc - abd - acd - bcd - (2abcd - a-b-c-d)}{(1+a)(1+b)(1+c)(1+d)} $$ $$= \dfrac{1}{1+a} + \dfrac{1}{1+b} + \dfrac{1}{1+c} + \dfrac{1}{1+d} - 2, $$ so we finish.

PS. It would be nice to see proofs using other methods less rustic than this :)
</s>