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IMO ShortList 2003, combinatorics problem 4

Source: Problem 5 of the German pre-TST 2004, written in December 03

May 17, 2004
linear algebramatrixcombinatoricsIMO Shortlist

Problem Statement

Let x1,…,xnx_1,\ldots, x_n and y1,…,yny_1,\ldots, y_n be real numbers. Let A=(aij)1≤i,j≤nA = (a_{ij})_{1\leq i,j\leq n} be the matrix with entries aij={1,if xi+yj≥0;0,if xi+yj<0.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 BB is an n×nn\times n matrix with entries 00, 11 such that the sum of the elements in each row and each column of BB is equal to the corresponding sum for the matrix AA. Prove that A=BA=B.
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