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Sequence of integers

Source: Polish math olympiad second round 1996

July 22, 2019
combinatoricsInteger sequence

Problem Statement

Let a1a_1, a2a_2 ,..., a99a_{99} be a sequence of digits from the set 0,...,9{0,...,9} such that if for some nn ∈ NN, an=1a_n = 1, then an+1≠2a_{n+1} \ne 2, and if an=3a_n = 3 then an+1≠4a_{n+1} \ne 4. Prove that there exist indices k,lk,l ∈ 1,...,98{1,...,98} such that ak=ala_k = a_l and ak+1=al+1a_{k+1} = a_{l+1}.
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