Question 13
(POL 4) Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1,a_2,\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ is defined as $(a_1b_1,a_2b_2,\ldots,a_nb_n)$. Prove that there exists a sequence $(c_1,c_2,\ldots,c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1,c_2,\ldots,c_n)$ contains at most $k^2/2^n$ sequences.