HMMT 二月 2005 · TEAM2 赛 · 第 10 题

英文解析

10. [30] Let x , y , and z be three roots of unity. Prove that x + y + z is also a root of unityif and only if x + y = 0, y + z = 0, or z + x = 0.
    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 aquadrilateral all of whose side lengths are 1. If the quadrilateral is degenerate, then twoof the vectors sum to 0, which implies the result. But even if it is not degenerate, thequadrilateral 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.
    Early Re-tile-ment [150]
    Let S = { s , . . . , s } be a finite set of integers, and define S + k = { s + k, . . . , s + k } .
    0 n 0 n
    We say that S and T are equivalent , written S ∼ T , if T = S + k for some k . Given a
    (possibly infinite) set of integers A , we say that S tiles A if A can be partitioned into subsetsequivalent to S . Such a partition is called a tiling of A by S . 3