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Written in a unique way - ISL 1978

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September 20, 2010
arithmetic sequencealgebranumber theoryfactorizationpartitionIMO Shortlist

Problem Statement

For every integer d≥1d \geq 1, let MdM_d be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference dd, having at least two terms and consisting of positive integers. Let A=M1A = M_1, B=M2∖{2},C=M3B = M_2 \setminus \{2 \}, C = M_3. Prove that every c∈Cc \in C may be written in a unique way as c=abc = ab with a∈A,b∈B.a \in A, b \in B.
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