MathDB

P4

Part of 2023 Israel Olympic Revenge

Problems(1)

Integer averages implies constant

Source: 2023 Israel Olympic Revenge P4

6/15/2023
Let cc be a positive real and a1,a2,…a_1, a_2, \dots be a sequence of nonnegative integers satisfying the following conditions for every positive integer nn:
(i)2a1+2a2+⋯+2ann\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n} is an integer;
(ii)\textbullet 2^{a_n}\leq cn.
Prove that the sequence a1,a2,…a_1, a_2, \dots is eventually constant.
olympic revengenumber theory