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1,2,...,100 in a a 10 x 10 chessboard , half negatives, 5 negatives in each row

Source: 1994 ITAMO p6

January 30, 2020
combinatoricsChessboard

Problem Statement

The squares of a 10×1010 \times 10 chessboard are labelled with 1,2,...,1001,2,...,100 in the usual way: the ii-th row contains the numbers 10i−9,10i−8,...,10i10i -9,10i - 8,...,10i in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.
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