1971 IMO Problems/Problem 6

Problem

Let $A = (a_{ij})(i, j = 1, 2, \cdots, n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij} = 0$ , the sum of the elements in the $i$ th row and the $j$ th column is $\geq n$ . Prove that the sum of all the elements of the matrix is $\geq n^2 / 2$ .

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1971 IMO Problems/Problem 6

Problem