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在北京举办的2013年全国数学竞赛命题工作研讨会上,来自复旦大学附属中学的李朝晖老师给出下面一道题.记S是长度为7的0/1数字串(共2~7=128个)组成的集合,T为其一个16元子集,满足T中任意两个元素的“距离”不小于3(距离是指对应数位出现不同数字的次数).证明:若0000000∈T,则11∈T.证明略.原解答证明了:若这样的集合T存在,则1111111∈T.
At the 2013 National Mathematics Contest Proposition Seminar held in Beijing, Teacher Li Zhaohui from the Fudan Middle School Affiliated Secondary School gave the following question. Note that S is a 0/1 digit string of length 7 (2~7=128 total). The set of components, T is a 16-element subset that satisfies the “distance” of any two elements in T that is not less than 3 (the distance is the number of times the corresponding digit appears a different number). Proof: If 0000000∈T, then 11 ∈ T. proves slightly. The original solution proves: If such a set T exists, 1111111 ∈ T.