SCHWEITZER 2011 Problem 5

Let n, k be positive integers. Let $f_a(x) := ||x - a||^{2n}$ , where the vectors $x = (x_1, ..., x_k) , a\in R^k$ , and ||·|| is the Euclidean norm. Let the vector space $Q_{n, k}$ be generated by the functions $f_a$ ( $a\in R^k$ ). What is the largest integer N for which $Q_{n, k}$ contains all polynomials of $x_1, ..., x_k$ whose total degree is at most N?