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China South East Mathematical Olympiad 2021 Grade11 P5

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August 14, 2021
combinatoricsset

Problem Statement

Let A={a1,a2,⋯ ,an,b1,b2,⋯ ,bn}A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\} be a set with 2n2n distinct elements, and Bi⊆AB_i\subseteq A for any i=1,2,⋯ ,m.i=1,2,\cdots,m. If ⋃i=1mBi=A,\bigcup_{i=1}^m B_i=A, we say that the ordered m−m-tuple (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of A.A. If (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of A,A, and for any i=1,2,⋯ ,mi=1,2,\cdots,m and any j=1,2,⋯ ,n,j=1,2,\cdots,n, {aj,bj}\{a_j,b_j\} is not a subset of Bi,B_i, then we say that ordered m−m-tuple (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of AA without pairs. Define a(m,n)a(m,n) as the number of the ordered m−m-coverings of A,A, and b(m,n)b(m,n) as the number of the ordered m−m-coverings of AA without pairs. (1)(1) Calculate a(m,n)a(m,n) and b(m,n).b(m,n). (2)(2) Let m≥2,m\geq2, and there is at least one positive integer n,n, such that a(m,n)b(m,n)≤2021,\dfrac{a(m,n)}{b(m,n)}\leq2021, Determine the greatest possible values of m.m.
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