HMMT 二月 2005 · TEAM2 赛 · 第 14 题

英文解析

14. [30] Suppose that S tiles the set of all integer cubes. Prove that S has only oneelement.
    Solution: Let the difference between the smallest and largest element of S be a . Then
    3 3
    the set equivalent to S that contains b can only contain integers between b − a and
    3 3
    b + a , inclusive. But for sufficiently large b , b is the only cube in this range, so S canonly have one element.