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Excellent step lengths

Source: Finnish Mathematics Competition 2009, Final Round - P5

November 16, 2011
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

We say that the set of step lengths D⊂Z+={1,2,…}D\subset \mathbb{Z}_+=\{1,2,\ldots\} is excellent if it has the following property: If we split the set of integers into two subsets AA and Z∖A\mathbb{Z}\setminus{A}, at least other set contains element a−d,a,a+da-d,a,a+d (i.e. {a−d,a,a+d}⊂A\{a-d,a,a+d\} \subset A or {a−d,a,a+d}∈Z∖A\{a-d,a,a+d\}\in \mathbb{Z}\setminus A from some integer a∈Z,d∈Da\in \mathbb{Z},d\in D.) For example the set of one element {1}\{1\} is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set {1,2,3,4}\{1,2,3,4\} is excellent but it has no proper subset which is excellent.
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