Find all strictly increasing sequences {an}∞n=0 of positive integers such that for all positive integers k,m,n an+1+an+2+⋯+an+kk+mis not an integer larger than 2020.
Problem
Source: 2020 Korean MO winter camp Test 1 P6
Tags: number theory
07.09.2020 17:09
ai=1 is a counterexample.
08.09.2020 18:27
vwu wrote: ai=1 is a counterexample. The problem is fixed!
09.09.2020 05:25
Here I am assuming that the sequence is strictly increasing (actually unbounded is enough for what follows). Take p>2020 to be a prime number. We may assume, maybe by considering the sequence after some point, that a1>p2. Also, assume for cleanness that n can be 0. By looking at the partial sums a1,a1+a2,...,a1+a2+...+ap we get two cases: First Case: p|a1+a2+...+ap. In this case we just take n=0,k=p,m=a1+a2+...+app−p>0 and get a contradiction. Second Case: p|ai+ai+1+...+ai+l for some 1≤i≤i+l≤p and l<p−1. Just take n=i−1,k=l+1,m=p−k>0 and get a contradiction. By the PGP we follow in one of the two cases above and we are done.
28.12.2022 01:07
BUMP!!! The above is incorrect since m is positive and cannot take a1+a2+...+app