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Question

Let ω1 be a cube root of unity. For distinct non-zero integers a,b, and c, the minimum value of z=|a+bω+cω2|2 is

A
0
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B
1
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C
3
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D
4
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Solution

The correct option is C 3
|a+bω+cω2|2
=(a+bω+cω2)(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a+bω+cω2)
=(a+bω+cω2)(a+bω2+cω)
=a2+b2+c2abbcca
=12[(ab)2+(bc)2+(ca)2]

minz occurs when a,b,c{1,2,3}
zmin=12(12+12+22)=3

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