IMO 1997 Problem 4

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

(a) there is no silver matrix for $n = 1997$ ;

(b) silver matrices exist for infinitely many values of $n$ .