1 Day to 2016 Final Call

Find the largest positive integer \(k\) that satisīŦes the property: The set of positive integers can be partitioned into \(k\) subsets \(A_{1}, A_{2}, \ldots , A_{k}\) such that for all integers \(n \ge 15\) and all \(i \in \{\ 1,2, \ldots, k \}\ \), there exist two distinct elements of \(A_{i}\) whose sum is \(n\).

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