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IMC 2012 Day 1, Problem 2

Source:

July 28, 2012
linear algebramatrixgeometryalgebrapolynomialinductionIMC

Problem Statement

Let nn be a fixed positive integer. Determine the smallest possible rank of an n×nn\times n matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.
Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.
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