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Integer sequence and divisibility

Source: 2020 Korean MO winter camp Test 1 P6

September 7, 2020
number theory

Problem Statement

Find all strictly increasing sequences {an}n=0∞\{a_n\}_{n=0}^\infty of positive integers such that for all positive integers k,m,nk,m,n an+1+an+2+⋯+an+kk+m\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m} is not an integer larger than 20202020.
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