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IMO ShortList 1998, combinatorics theory problem 1

Source: IMO ShortList 1998, combinatorics theory problem 1

October 22, 2004
algorithmmaxtrixroundingIMO Shortlist

Problem Statement

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number xx in the array can be changed into either ⌈x⌉\lceil x\rceil or ⌊x⌋\lfloor x\rfloor so that the row-sums and column-sums remain unchanged. (Note that ⌈x⌉\lceil x\rceil is the least integer greater than or equal to xx, while ⌊x⌋\lfloor x\rfloor is the greatest integer less than or equal to xx.)
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