International Mathematical Olympiad 2021 Problem 6

Let $m \geqslant 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1, B_2, B_3, \ldots, B_m$ be subsets of $A$. Assume that for each $k = 1, 2, \ldots, m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.