Maths Olympiad Prep

Library / /9 of 34

Combinatorics Difficulty 6.2 National Olympiad Find the answer Argentina

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.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We represent the tournament as a table that in the cell (i,j)(i, j) contains the number of points earned by team ii in the match versus team jj. Consider the following tournament, where for every 1≤i≤81 \le i \le 8 team ii defeats team jj for every j>ij > i.

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 1212, which is close to 1414, the score of the winning team. If we change this tournament a bit by making team 11 and team 88 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

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.