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Permutation of Positive Integers with Bounded Difference

Source: Germany VAIMO 2018 P2

April 15, 2020
algebra

Problem Statement

A positive integer dd and a permutation of positive integers a1,a2,a3,…a_1,a_2,a_3,\dots is given such that for all indices i≥10100i\geq 10^{100}, ∣ai+1−ai∣≤2d|a_{i+1}-a_{i}|\leq 2d holds. Prove that there exists infinity many indices jj such that ∣aj−j∣<d|a_j -j|< d.
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