Question

Let $\omega= e ^{ i \pi / 3}$, and $a , b , c , x , y , z$ be non-zero complex numbers such that $a +b +c=x$
$a +b \omega+c \omega^{2}=y$
$a+ b \omega^{2}+c \omega=z$
Then the value of $\frac{|x|^{2}+|y|^{2}+|z|^{2}}{|a|^{2}+|b|^{2}+|c|^{2}}$ is

Answer 1

The expression may not attain integral value for all $a , b , c$
If we consider $a = b = c$, then
$x=3 a$
$y=a\left(1+\omega+\omega^{2}\right)=a(1+i \sqrt{3})$
$z=a\left(1+\omega^{2}+\omega\right)=a(1+i \sqrt{3})$
$\therefore |x|^{2}+|y|^{2}+|z|^{2}=9|a|^{2}+4|a|^{2}+4|a|^{2}=17|a|^{2}$
$\therefore \frac{|x|^{2}+|y|^{2}+|z|^{2}}{|a|^{2}+|b|^{2}+|c|^{2}}=\frac{17}{3}$
Note: However if $\omega= e ^{ i (2 \pi / 3)}$, then the value of the expression $=3$.