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Difference between revisions of "1971 IMO Problems/Problem 6"

(Created page with "Let A = (aij), where i, j = 1, 2, ... , n, be a square matrix with all aij non-negative integers. For each i, j such that aij = 0, the sum of the elements in the ith row and t...")
 
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Let A = (aij), where i, j = 1, 2, ... , n, be a square matrix with all aij non-negative integers. For each i, j such that aij = 0, the sum of the elements in the ith row and the jth column is at least n. Prove that the sum of all the elements in the matrix is at least n2/2.
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==Problem==
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Let <math>A = (a_{ij})(i, j = 1, 2, \cdots, n)</math> be a square matrix whose elements are non-negative integers. Suppose that whenever an element <math>a_{ij} = 0</math>, the sum of the elements in the <math>i</math>th row and the <math>j</math>th column is <math>\geq n</math>. Prove that the sum of all the elements of the matrix is <math>\geq n^2 / 2</math>.

Revision as of 14:10, 29 January 2021

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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