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

Answer: 3

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