Let omega (neq 1) be a cubic root of unity. T | vedclass

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(C) We know that $|a+b\omega+c\omega^2|^2 = (a+b\omega+c\omega^2)(a+b\omega^2+c\omega) = a^2+b^2+c^2-ab-bc-ca$. This can be rewritten as $\frac{1}{2}[

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$3$

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(C) We know that $|a+b\omega+c\omega^2|^2 = (a+b\omega+c\omega^2)(a+b\omega^2+c\omega) = a^2+b^2+c^2-ab-bc-ca$. This can be rewritten as $\frac{1}{2}[(a-b)^2+(b-c)^2+(c-a)^2]$. Since $a, b, c$ are distinct non-zero integers, the smallest possible values for these integers are chosen from the set $\{1, 2, 3\}$ or $\{1, -1, 2\}$ etc. For the set $\{1, 2, 3\}$, we have $(a-b)^2 = (1-2)^2 = 1$, $(b-c)^2 = (2-3)^2 = 1$, and $(c-a)^2 = (3-1)^2 = 4$. Thus, the minimum value is $\frac{1}{2}(1+1+4) = \frac{6}{2} = 3$.

Let $\omega (\neq 1)$ be a cubic root of unity. Then the minimum value of the set $\{|a+b\omega+c\omega^2|^2 : a, b, c \text{ are distinct non-zero integers}\}$ equals

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