The expression may not attain integral value for all a, b, c. If we consider a = b = c, then x = 3a y = a(1+ω+ω2) = a(1+i√3) z = a(1+ω2+ω) = a(1+i√3) Therefore, |x|2+|y|2+|z|2 = 9|a|2+4|a|2+4|a|2 = 17|a|2 Hence,
|x|2+|y|2+|z|2
|a|2+|b|2+|c|2
=
17
13
Note: However, if ω = ei(2
π
3
), then the value of the expression is 3. Answer is not a single digit integer.