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Every element is in exactly two of the pairs.

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September 14, 2010
combinatorics proposedcombinatorics

Problem Statement

Given that nn elements a1,a2,…,ana_1, a_2,\dots, a_n are organized into nn pairs P1,P2,…,PnP_1, P_2, \dots, P_n in such a way that two pairs Pi,PjP_i, P_j share exactly one element when (ai,aj)(a_i, a_j) is one of the pairs, prove that every element is in exactly two of the pairs.
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