Additive Latin transversals

We prove that for every odd prime p, every k less than or equal to p and every two subsets A = {a(1),..., a(k)) and B = {b(1),..., b(k)) of cardinality k each of Z(p), there is a permutation pi is an element of S-k such that the sums a(i) + b(pi)(i) (in Z(p)) are pairwise distinct. This partially settles a question of Snevily. The proof is algebraic, and implies several related results as well.