MathLabs

Problem 2

Let nn be positive and let A1,…,A2n+1A_1,\ldots,A_{2n+1} be subsets of BB. Suppose each AiA_i has exactly 2n2n elements, every two distinct AiA_i have exactly one common element, and every element of BB belongs to at least two AiA_i. For which nn can one label each element 0 or 1 so that each AiA_i has 0 on exactly nn elements?
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∣B∣=n(2n+1)|B|=n(2n+1)
Detailed analysis

Name the element belonging to A_i and A_j by (i,j). Counting incidences twice gives |B|=n(2n+1).