22nd CanMO 1990 Problem 1 A competition is played amongst n > 1 players over d days. Each day one player gets a score of 1, another a score of 2, and so on up to n. At the end of the competition each player has a total score of 26. Find all possible values for (n, d).
(n, d) = (3, 13), (12, 4) or (25, 2) Solution Each day n(n+1)/2 points are scored, so dn(n+1)/2 over the whole competition. This equals 26n, so d(n+1) = 52. So d = 1, 2, 4, 13 or 52. But d = 1 does not work, because one player scores 1 not 26. d = 2 (n = 25) does work: one player scores 1, 25 on successive days, the next 2, 24, the third 3, 23 and so on. d = 4 (n = 12) also works. For example, the players' scores could be: day 1 1 2 3 4 5 6 7 8 9 10 11 12 day 2 12 11 10 9 8 7 6 5 4 3 2 1 day 3 1 2 3 4 5 6 7 8 9 10 11 12 day 4 12 11 10 9 8 7 6 5 4 3 2 1 d = 13 (n = 3) also works (in the first three days each players scores 6, then in each 5 successive pairs of days he scores 4): day 1 2 3 4 5 6 7 8 9 10 11 12 13 player 1 1 2 3 1 3 1 3 1 3 1 3 1 3 player 2 2 3 1 3 1 3 1 3 1 3 1 3 1 player 3 3 1 2 2 2 2 2 2 2 2 2 2 2 d = 52 does not work, because there are no players! 22nd CanMO 1990 © John Scholes jscholes@kalva.demon.co.uk 3 Aug 2002 Last corrected/updated 7 Mar 04