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Part of 2018 Bosnia and Herzegovina EGMO TST

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Bosnia and Herzegovina EGMO TST 2018 Problem 2

Source: Bosnia and Herzegovina EGMO Team Selection Test 2018

9/19/2018
Prove that for every pair of positive integers (m,n)(m,n), bigger than 22, there exists positive integer kk and numbers a0,a1,...,aka_0,a_1,...,a_k, which are bigger than 22, such that a0=ma_0=m, a1=na_1=n and for all i=0,1,...,k−1i=0,1,...,k-1 holds ai+ai+1∣aiai+1+1 a_i+a_{i+1} \mid a_ia_{i+1}+1
positive integernumber theorydivides