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A. 712

Part of KoMaL A Problems 2017/2018

Problems(1)

Hatters gonna hat

Source: KöMaL A. 712

1/12/2018
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?
algebraSequencereal number