Problem 6.   Let m ≥ 2 be an integer, A be a finite set of (not necessarily positive) integers, and B₁,B₂,B₃,...,B_m be subsets of A. Assume that for each k = 1,2,...,m the sum of the elements of B_k  is m^k. Prove that A contains at least m/2 elements.

proposed by Austria