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Final Question from IrMO 2022

Source: IrMO 2022

May 13, 2022
combinatoricsalgebra

Problem Statement

10. Let n≥5n \ge 5 be an odd number and let rr be an integer such that 1≤r≤(n−1)/21\le r \le (n-1)/2. IN a sports tournament, nn players take part in a series of contests. In each contest, 2r+12r+1 players participate, and the scores obtained by the players are the numbers −r,−(r−1),⋯ ,−1,0,1⋯ ,r−1,r-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r in some order. Each possible subset of 2r+12r+1 players takes part together in exactly one contest. let the final score of player ii be SiS_i, for each i=1,2,⋯ ,ni=1, 2,\cdots,n. Define NN to be the smallest difference between the final scores of two players, i.e., N=min⁡i<j∣Si−Sj∣.N = \min_{i<j}|S_i - S_j|. Determine, with proof, the maximum possible value of NN.
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