2021 IMO 第 6 题

Let $m \geq 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_{1}, B_{2}, \ldots , B_{m}$ subsets of $A$ . Suppose that, for every $k = 1, 2, \ldots , m$ , the sum of the elements of $B_{k}$ is $m^{k}$ . Prove that $A$ contains at least $\frac{m}{2}$ elements.

令 $m \geq 2$ 为整数， $A$ 为有限整数集(不一定为正数)，以及 $B_{1}, B_{2}, \ldots , B_{m}$ $A$ 的子集。假设对于每个 $k = 1, 2, \ldots , m$ ， $B_{k}$ 的元素之和为 $m^{k}$ 。证明$A$至少包含$\frac{m}{2}$元素。