2021 IMO Shortlist A6

Let $A$ be a finite set of (not necessarily positive) integers, and let $m \geq 2$ be an integer. Assume that there exist non-empty subsets $B_{1}, B_{2}, B_{3}, \ldots, B_{m}$ of $A$ whose elements add up to the sums $m^{1}, m^{2}, m^{3}, \ldots, m^{m}$, respectively. Prove that $A$ contains at least $m / 2$ elements.

令 $A$ 为有限(不一定为正)整数集，并令 $m \geq 2$ 为整数。假设$A$存在非空子集$B_{1}、B_{2}、B_{3}、\ldots、B_{m}$，其元素之和分别为$m^{1}、m^{2}、m^{3}、\ldots、m^{m}$。证明 $A$ 至少包含 $m / 2$ 个元素。