Question

Let $\omega \ne 1$ be a cube root of unity. For distinct non-zero integers $a,b,$ and $c,$ the minimum value of $z=|a+b\omega +c{\omega }^2{|}^2$ is

• A. $0$
• B. $1$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$|a+b\omega +c{\omega }^2{|}^2$
$=(a+b\omega +c{\omega }^2)\left(\overline{a+b\omega +c{\omega }^2}\right)$
$=(a+b\omega +c{\omega }^2)(a+b{\omega }^2+c\omega )$
$=a^2+b^2+c^2-ab-bc-ca$
$=\frac{1}{2}[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$

$\therefore minz$ occurs when $a,b,c\in \{1,2,3\}$
$z_{min}=\frac{1}{2}(1^2+1^2+2^2)=3$