2003 IMO Shortlist S14

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries $$a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;0,&\text{if }x_i + y_j < 0.\end{cases}$$ Suppose that $B$ is an $n\times n$ matrix with entries $0$ , $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$ . Prove that $A=B$ .

设 $x_1,\ldots, x_n$ 和 $y_1,\ldots, y_n$ 为实数。设 $A = (a_{ij})_{1\leq i,j\leq n}$ 为包含条目的矩阵 $$a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;0,&\text{if }x_i + y_j < 0.\end{cases}$$ 假设 $B$ 是包含条目的 $n\times n$ 矩阵$0$ , $1$ 使得 $B$ 的每行和每列中的元素之和等于矩阵 $A$ 的相应总和。证明 $A=B$ 。