HMMT 二月 2005 · TEAM1 赛 · 第 15 题

英文解析

15. [35] Suppose S tiles N . Show that S is symmetric; that is, if − S = {− s , . . . , − s } ,
    n 0
    show that S ∼ − S .
    Solution: Assume without loss of generality that the minimum element of S is 0. Bythe previous problem, S tiles the set { 1 , 2 , . . . , k } for some positive integer k . Then let
    ∑
    P ( x ) be the polynomial x . To say that the set { 1 , 2 , . . . , k } , or equivalently theisni =0
    set { 0 , 1 , . . . , k − 1 } , is tiled by S is to say that there is some polynomial Q ( x ) withk − 1 kcoefficients 0 or 1 such that P ( x ) Q ( x ) = 1 + x + · · · + x = ( x − 1) / ( x − 1). It followsthat all the roots of P ( x ) are roots of unity, but P (1) 6 = 0. By question 11 above, thisimplies that P ( x ) is symmetric. Therefore, s + s = s + s = · · · = s + s , so S
    0 n 1 n − 1 n 0
    is symmetric. 6