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Find all possible values of m, n

Source: China TST 2011 - Quiz 2 - D1 - P2

May 20, 2011
calculusderivativefunctionalgebrapolynomialmodular arithmeticcombinatorics unsolved

Problem Statement

Let ℓ\ell be a positive integer, and let m,nm,n be positive integers with m≥nm\geq n, such that A1,A2,⋯ ,Am,B1,⋯ ,BmA_1,A_2,\cdots,A_m,B_1,\cdots,B_m are m+nm+n pairwise distinct subsets of the set {1,2,⋯ ,ℓ}\{1,2,\cdots,\ell\}. It is known that AiΔBjA_i\Delta B_j are pairwise distinct, 1≤i≤m,1≤j≤n1\leq i\leq m, 1\leq j\leq n, and runs over all nonempty subsets of {1,2,⋯ ,ℓ}\{1,2,\cdots,\ell\}. Find all possible values of m,nm,n.
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