IMO Shortlist 2003 problem C4


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2. travnja 2012.
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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.

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(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.)

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Izvor: Međunarodna matematička olimpijada, shortlist 2003