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Subset of N such that inequality holds

Source: Chinese Mathematical Olympiad 2010 Problem 4

November 28, 2010
inequalitiesfloor functionalgebra unsolvedalgebra

Problem Statement

Let m,n≥1m,n\ge 1 and a1<a2<…<ana_1 < a_2 < \ldots < a_n be integers. Prove that there exists a subset TT of N\mathbb{N} such that ∣T∣≤1+an−a12n+1|T| \leq 1+ \frac{a_n-a_1}{2n+1} and for every i∈{1,2,…,m}i \in \{1,2,\ldots , m\}, there exists t∈Tt \in T and s∈[−n,n]s \in [-n,n], such that ai=t+sa_i=t+s.
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