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Array nxn with positive integers that add up to n^3

Source: Romanian IMO Team Selection Test TST 2003, problem 9

September 24, 2005

geometryrectanglecombinatorics proposedcombinatorics

Problem Statement

Let

n\geq 3

be a positive integer. Inside a

n\times n

array there are placed

n^2

positive numbers with sum

n^3

. Prove that we can find a square

2\times 2

of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than

3n

. Radu Gologan

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