Problem 29 · 2022 Math Kangaroo
Stretch
Logic & Word Problems
caseworksum-constraint
Eight teams take part in a football tournament where each team plays each other team exactly once. In each game the winner gets 3 points and the loser no points. In case of a draw both teams get 1 point. In the end all teams together have 61 points. What is the maximum number of points that the team with the most points could have gained?
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Answer: D — 17
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Hint 1 of 2
A decided game adds 3 points to the total; a draw adds only 2, so the total tells you the draws.
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Hint 2 of 2
Maximise one team's points while keeping the overall total at 61.
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Approach: count decided games from the total, then load wins onto one team
- There are \(\binom{8}{2}=28\) games; a decided game gives out 3 points and a draw gives out 2.
- If \(d\) games are drawn then total points \(=3(28-d)+2d=84-d=61\), so \(d=23\) and only \(5\) games are decided.
- A team can win only a decided game, so it wins at most all \(5\); giving it those 5 wins plus drawing its other 2 games gives \(5\cdot 3+2=17\) points, answer D.
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