BG-MO 2012 Problem 4

Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$ , consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$ ). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$ , and for every $i = 1, 2,\ldots,n$ , exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$ .