IMO 1997-4

[Phlembac Adib Hasan]

An (n x n) matrix with entries from $\{1,2,...,2n-1\}$ will be called silver metrix if for each $1\leq i\leq n$, the union of $i^{th}$ row and column contains $2n-1$ different integers.Show that
$(1)$ There is no silver matrix for $n=1997$.
$(2)$ There exists at least one silver matrix for infinitely many integers.

[*Mahi*]

Hint:

$(1)$ There is no silver matrix for $n$ odd.
$(2)$ Construct a silver matrix from $n \times n$ to $2n \times 2n$.

[sm.joty]

Hint:

$(1)$ There is no silver matrix for $n$ odd.
$(2)$ Construct a silver matrix from $n \cross n$ to $2n \cross 2n$.

Mahi there is something wrong with your LATEX. Please edit.

[*Mahi*]

Edited now, thanks!