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1 nxn square divided into n^2 unit squares, real number in each unti square

Source: 2011 Romania JBMO TST 2.2

June 1, 2020

combinatoricssquare

Problem Statement

We consider an

n \times n

(

n \in N, n \ge 2

) square divided into

n^2

unit squares. Determine all the values of

k \in N

for which we can write a real number in each of the unit squares such that the sum of the

n^2

numbers is a positive number, while the sum of the numbers from the unit squares of any

k \times k

square is a negative number.