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S_1 = {1}, S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}

Source: Korean Mathematical Olympiad 1994, Final Round P5 FKMO

July 21, 2018
Integer sequenceSubsetsetnumber theory

Problem Statement

Given a set S⊂NS \subset N and a positive integer n, let S⊕{n}={s+n/s∈S}S\oplus \{n\} = \{s+n / s \in S\}. The sequence SkS_k of sets is defined inductively as follows: S1=1S_1 = {1}, Sk=(Sk−1⊕{k})∪{2k−1}S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\} for k=2,3,4,...k = 2,3,4, ... (a) Determine N−∪k=1∞SkN - \cup _{k=1}^{\infty} S_k. (b) Find all nn for which 1994∈Sn1994 \in S_n.
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