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Putnam 2000 B5

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September 6, 2011
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Problem Statement

Let S0S_0 be a finite set of positive integers. We define finite sets S1,S2,⋯S_1, S_2, \cdots of positive integers as follows: the integer aa in Sn+1S_{n+1} if and only if exactly one of a−1a-1 or aa is in SnS_n. Show that there exist infinitely many integers NN for which SN=S0∪{N+a:a∈S0}S_N = S_0 \cup \{ N + a: a \in S_0 \}.
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