MathDB

10. Let

n \ge 5

be an odd number and let

r

be an integer such that

1\le r \le (n-1)/2

. IN a sports tournament,

n

players take part in a series of contests. In each contest,

2r+1

players participate, and the scores obtained by the players are the numbers

-r, -(r-1),\cdots, -1, 0, 1 \cdots, 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

S_i

, for each

i=1, 2,\cdots,n

. Define

N

to be the smallest difference between the final scores of two players, i.e.,

N = \min_{i<j}|S_i - S_j|.

Determine, with proof, the maximum possible value of

N

10