KöMaL 2020 B5088

With respect to a given set $G$ of numbers, the positive integer $k>1$ is called interesting if there exist $k$ distinct elements in set $G$ such that the arithmetic mean of these elements also belongs to set $G$. Let $H=\{1;3;4;9;10;\ldots\}$ be the set of those numbers that can be represented as a sum of some different powers of 3. $a)$ What numbers $k>1$ are interesting with respect to the set $H$? $b)$ Let $c \notin H$ be an arbitrary positive integer. Prove that every number $k>1$ is interesting with respect to the set $H'=H \cup \{c\}$.