@esoterica The title reminds me of a problem I composed but never submitted to any journal: Suppose \(A_1, A_2, \ldots, A_n\) are finite sets, and each \(A_i\) has odd cardinality. Prove there must be a set \(S\subseteq \{1,2,\ldots,n\}\) such that \(\sum_{i\in S} |A_i\cap A_j|\) is odd for every \(j\in\{1,2,\ldots,n\}\).

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