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Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.4 AIME Prove it United States

Problem:

Let $x$ , $y$ , and $z$ be three roots of unity. Prove that $x + y + z$ is also a root of unity if and only if $x + y = 0$ , $y + z = 0$ , or $z + x = 0$ .

Solution

Solution:

Again, we consider the geometric picture. Arrange the vectors $x$ , $y$ , $z$ , and $-x-y-z$ so as to form a quadrilateral. If they are all roots of unity, they form a quadrilateral all of whose side lengths are $1$ . If the quadrilateral is degenerate, then two of the vectors sum to $0$ , which implies the result. But even if it is not degenerate, the quadrilateral must be a rhombus, and since opposite sides of a rhombus are parallel, this again implies that two of the four roots of unity sum to $0$ .

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