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Hatters gonna hat

Source: KöMaL A. 712

January 12, 2018
algebraSequencereal number

Problem Statement

We say that a strictly increasing positive real sequence a1,a2,⋯a_1,a_2,\cdots is an elf sequence if for any c>0c>0 we can find an NN such that an<cna_n<cn for n=N,N+1,⋯n=N,N+1,\cdots. Furthermore, we say that ana_n is a hat if an−i+an+i<2ana_{n-i}+a_{n+i}<2a_n for 1≤i≤n−1\displaystyle 1\le i\le n-1. Is it true that every elf sequence has infinitely many hats?
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