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Subpoint a) is canonic; roots of unity that don’t resemble an equilateral triang

Source: Romanian District Olympiad 2009, Grade X, Problem 4

October 8, 2018
complex numbersgeometryabsolute value

Problem Statement

a) Let z1,z2,z3 z_1,z_2,z_3 be three complex numbers of same absolute value, and 0=z1+z2+z3. 0=z_1+z_2+z_3. Show that these represent the affixes of an equilateral triangle.
b) Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.
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