ISL 2006 C5

An $(n, k)$-tournament is a contest with $n$ players held in $k$ rounds such that: (i) Each player plays in each round, and every two players meet at most once. (ii) If player $A$ meets player $B$ in round $i$, player $C$ meets player $D$ in round $i$, and player $A$ meets player $C$ in round $j$, then player $B$ meets player $D$ in round $j$. Determine all pairs $(n, k)$ for which there exists an $(n, k)$-tournament. (Argentina)