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$-element subsets of {1, 2, . . . , n}

Source: Switzerland - 2012 Swiss MO Final Round p5

December 30, 2022
combinatoricsSubsets

Problem Statement

Let n be a natural number. Let A1,A2,...,AkA_1, A_2, . . . , A_k be distinct 33-element subsets of {1,2,...,n}\{1, 2, . . . , n\} such that ∣Ai∩Aj∣≠1|A_i \cap A_j | \ne 1 for all 1≤i,j≤k1 \le i, j \le k. Determine all nn for which there are nn such that these subsets exist.
[hide=original wording of last sentence]Bestimme alle n, fur die es n solche Teilmengen gibt.
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