US-TST 2005 Problem 1

Let $n$ be an integer greater than $1$ . For a positive integer $m$ , let $S_{m}= \{ 1,2,\ldots, mn\}$ . Suppose that there exists a $2n$ -element set $T$ such that (a) each element of $T$ is an $m$ -element subset of $S_{m}$ ; (b) each pair of elements of $T$ shares at most one common element; and (c) each element of $S_{m}$ is contained in exactly two elements of $T$ .

Determine the maximum possible value of $m$ in terms of $n$ .