Question 94
Let $n \ge 5$ be an odd number and $r$ an integer such that $1 \le r \le (n-1)/2$. In a sports tournament, $n$ players take part in a series of contests. Each contest involves $2r + 1$ players, and the scores obtained by the players are the numbers
$$
-r, -(r-1), \dots, -1, 0, 1, \dots, r-1, r
$$
in some order. Each possible subset of $2r + 1$ players takes part together in exactly one contest. Let the final score of player $i$ be $S_i$, for each $i = 1, 2, \dots, n$. Define $N$ to be the smallest difference between the final scores of two players, i.e.,
$$
N = \min_{i\lt j} |S_i - S_j|.
$$
Determine, with proof, the maximum possible value of $N$.
$$
-r, -(r-1), \dots, -1, 0, 1, \dots, r-1, r
$$
in some order. Each possible subset of $2r + 1$ players takes part together in exactly one contest. Let the final score of player $i$ be $S_i$, for each $i = 1, 2, \dots, n$. Define $N$ to be the smallest difference between the final scores of two players, i.e.,
$$
N = \min_{i\lt j} |S_i - S_j|.
$$
Determine, with proof, the maximum possible value of $N$.