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matrix and elements

Source: 2011-2012 china second round,problem 4

October 30, 2011
linear algebramatrixcombinatorics proposedcombinatorics

Problem Statement

Let AA be a 3×93 \times 9 matrix. All elements of AA are positive integers. We call an m×nm\times n submatrix of AA "ox" if the sum of its elements is divisible by 1010, and we call an element of AA "carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in AA.
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