Given
real numbers
,
, ...,
, and
further real numbers
,
, ...,
. The entries
(with
) of an
matrix
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.](/media/m/0/a/8/0a867d5e042c04486186ff8e14ad10b6.png)
Further, let
be an
matrix whose elements are numbers from the set
satisfying the following condition: The sum of all elements of each row of
equals the sum of all elements of the corresponding row of
; the sum of all elements of each column of
equals the sum of all elements of the corresponding column of
. Show that in this case,
.
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
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
![x_2](/media/m/a/a/1/aa16f4edacb7b534405242617406658f.png)
![x_n](/media/m/3/c/5/3c57e4750d576aafa08c9ec1a939cfce.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![y_1](/media/m/9/2/e/92e328204886ca272c5a4e87b6b32005.png)
![y_2](/media/m/a/b/c/abca78e1eef61943a7b852806012be62.png)
![y_n](/media/m/2/a/0/2a08e21c3014a9761a124842ae4a687f.png)
![a_{ij}](/media/m/b/6/c/b6cda4f0bdc70b1c45ec2b74d35bcdbd.png)
![1\leq i,\;j\leq n](/media/m/a/d/3/ad35f95d90dc7ac75b7c09db1b14f63e.png)
![n\times n](/media/m/1/c/a/1caee5824fd124b98d47c32a5a96cad3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![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.](/media/m/0/a/8/0a867d5e042c04486186ff8e14ad10b6.png)
Further, let
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![n\times n](/media/m/1/c/a/1caee5824fd124b98d47c32a5a96cad3.png)
![\left\{0;\ 1\right\}](/media/m/7/f/a/7faaa5227ab0a536a18729f675037d91.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![A = B](/media/m/4/1/c/41c256139404a6e55b8b83474c31c480.png)
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