HMMT 二月 2005 · TEAM2 赛 · 第 11 题
HMMT February 2005 — TEAM2 Round — Problem 11
题目详情
- [20] Find all sets S with minimum element 1 that tile A = { 1 , . . . , 12 } .
解析
- [20] Find all sets S with minimum element 1 that tile A = { 1 , . . . , 12 } . Solution: This can be done by brute force. Alternatively, note that if P ( x ) and Q ( x ) 2 12 are polynomials with coefficients either 0 or 1 with P ( x ) Q ( x ) = x + x + · · · + x , then the set consisting of the exponents of nonzero terms in P tiles A . Either way, we find that S is one of the following: { 1 } , { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 1 , 7 } , { 1 , 2 , 3 } , { 1 , 3 , 5 } , { 1 , 5 , 9 } , { 1 , 2 , 3 , 4 } , { 1 , 2 , 7 , 8 } , { 1 , 4 , 7 , 10 } , { 1 , 2 , 3 , 4 , 5 , 6 } , { 1 , 2 , 3 , 7 , 8 , 9 } , { 1 , 2 , 5 , 6 , 9 , 10 } , { 1 , 3 , 5 , 7 , 9 , 11 } , or A itself.