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1997 IMO Problems/Problem 4

Revision as of 15:16, 6 October 2023 by Tomasdiaz (talk | contribs) (Created page with "==Problem== An <math>n \times n</math> matrix whose entries come from the set <math>S={1,2,...,2n-1}</math> is called a <math>\textit{silver}</math> matrix if, for each <math...")
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Problem

An $n \times n$ matrix whose entries come from the set $S={1,2,...,2n-1}$ is called a $\textit{silver}$ matrix if, for each $i=1,2,...,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no $\textit{silver}$ matrix for $n=1997$;

(b) $\textit{silver}$ matrices exist for infinitely many values of $n$.

Solution

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