HMMT 二月 2005 · TEAM2 赛 · 第 9 题

英文解析

9. [30] Let x and y be two distinct roots of unity. Prove that x + y is also a root of unityyif and only if is a cube root of unity.
   Solution: This is easiest to see geometrically. The vectors corresponding to x , y , andx − x − y sum to 0, so they form a triangle. In order for them all to be roots of unity,
   they must all have length one, so the triangle must be equilateral. Therefore the angley
   2 πbetween x and y is ± , that is, is a cube root of unity.
   3 x