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Bosnia and Herzegovina EGMO TST 2018 Problem 2

Source: Bosnia and Herzegovina EGMO Team Selection Test 2018

September 19, 2018
positive integernumber theorydivides

Problem Statement

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
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