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9x9 square filled with positive integers from 1 to 81

Source: Romanian District Olympiad 2006, Grade 9, Problem 2

March 11, 2006

combinatorics proposedcombinatorics

Problem Statement

9\times 9

array is filled with integers from 1 to 81. Prove that there exists

k\in\{1,2,3,\ldots, 9\}

such that the product of the elements in the row

k

is different from the product of the elements in the column

k

of the array.