Skip to main content
OlympiadHQ

Browse · MathNet

Print

XXIX Rioplatense Mathematical Olympiad

Argentina counting and probability

Problem

Eight teams take part in a rugby tournament in which every team plays exactly one match against each of the other seven teams. In each match, if the teams draw against each other, both of them earn 1 point; otherwise, the winner earns 2 points and the loser earns no points. At the end of the tournament, the final scores of the eight teams are all different and the score of the winning team equals the sum of the four lowest scores. Give an example of a tournament which satisfies all these conditions.
Solution
We represent the tournament as a table that in the cell contains the number of points earned by team in the match versus team . Consider the following tournament, where for every team defeats team for every .
TeamT1T2T3T4T5T6T7T8Total
T1-222222214
T20-22222212
T300-2222210
T4000-22228
T50000-2226
T600000-224
T7000000-22
T80000000-0
Despite this example is not a solution, we can observe that the four lowest scores add up to , which is close to , the score of the winning team. If we change this tournament a bit by making team and team draw against each other, we will get a tournament that does satisfy all the conditions.
TeamT1T2T3T4T5T6T7T8Total
T1-222222113
T20-22222212
T300-2222210
T4000-22228
T50000-2226
T600000-224
T7000000-22
T81000000-1
Final answer
Label teams T1 through T8. For every pair with lower index versus higher index, the lower-indexed team wins, except that the match between T1 and T8 is a draw. This yields totals 13, 12, 10, 8, 6, 4, 2, 1, which are all distinct and satisfy that the top total equals the sum of the four smallest totals.

Techniques

Games / greedy algorithmsInvariants / monovariants