Školjka

%V0 Given $n$ real numbers $x_1$, $x_2$, ..., $x_n$, and $n$ further real numbers $y_1$, $y_2$, ..., $y_n$. The entries $a_{ij}$ (with $1\leq i,\;j\leq n$) of an $n\times n$ matrix $A$ are defined as follows: $a_{ij}=\left\{\begin{array}{c}1\text{\ \ \ \ \ \ if\ \ \ \ \ \ }x_{i}+y_{j}\geq 0;\\ 0\text{\ \ \ \ \ \ if\ \ \ \ \ \ }x_{i}+y_{j}<0.\end{array}\right.$ Further, let $B$ be an $n\times n$ matrix whose elements are numbers from the set $\left\{0;\ 1\right\}$ satisfying the following condition: The sum of all elements of each row of $B$ equals the sum of all elements of the corresponding row of $A$; the sum of all elements of each column of $B$ equals the sum of all elements of the corresponding column of $A$. Show that in this case, $A = B$. comment (This one is from the ISL 2003, but in any case, the official problems and solutions - in German - are already online, hence I take the liberty to post it here.) Darij