MathDB

Integer averages implies constant

Source: 2023 Israel Olympic Revenge P4

June 15, 2023

olympic revengenumber theory

Problem Statement

Let

c

be a positive real and

a_1, a_2, \dots

be a sequence of nonnegative integers satisfying the following conditions for every positive integer

n

:
(i)

\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

a_1, a_2, \dots

is eventually constant.