KöMaL 2020 B5084

Let $n$ be a positive integer, and let $\mathcal{S}$ denote the set of $0-1-2$ sequences of length $n$. Determine which sets $\emptyset\ne A\subseteq\mathcal{S}$ have the following property: no matter how a vector $(c_1,c_2,\ldots,c_n)\in \mathcal{S}\setminus\big\{(0,0,\dots,0)\big\}$ is selected, the probabilities that the sum of products $c_1a_1+c_2a_2+\dots+c_na_n$ formed with a randomly chosen element $(a_1,a_2,\ldots,a_n)$ of set $A$ will leave a remainder of $0$, $1$, or $2$ are $1/3$ each.