IMO Shortlist 1971 problem 11


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2. travnja 2012.
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The matrix
A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots  \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}
satisfies the inequality \sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M for each choice of numbers x_i equal to \pm 1. Show that
|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.
Izvor: Međunarodna matematička olimpijada, shortlist 1971