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Problem 1. Determine all positive integers $n$ with the following property: the set $\{1, 2, \dots, 2n - 1, 2n\}$ can be partitioned into two disjoint sets $\mathcal{A}$ and $\mathcal{B}$ with $n$ elements each, such that the sum of the elements of $\mathcal{A}$ divides the sum of the elements of $\mathcal{B}$.
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