Given three n-tuples {λi}ni=1, {μi}ni=1, {vi}ni=1 of complex numbers, we introduce the problem of when there exists a pair of normal matrices A and B such that σ(A) = {λi}ni=1, σ(B) = {μi}ni=1, and σ(A+B) = {vi}ni=1, where σ(.) denote that spectrum. In the case when λk = 0, k = 2,...,n, we provide necessary and sufficient conditions for the existence of A and B. In addition, we show that the solution pair (A,B) is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when n-tuples are real.