Maths Olympiad Prep

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Combinatorics Difficulty 4.7 AIME Find the answer Italy

Problem:

Marco, Fabrizio and Giovanni, three mathematicians, challenge a group of four physicists to a foosball tournament. They play one match for every possible combination of two mathematicians (one on offense, one on defense) against two physicists (one on offense, one on defense). Each match has the same duration, and in total the tournament lasts a full 24 hours (without breaks). How long does Marco play on defense?
Note that, for example, there will be two different matches of Marco and Fabrizio against a certain forward and a certain defender among the physicists: one with Marco on offense and Fabrizio on defense, one the other way around.

Pick one

Solution

Solution:

The answer is (D). By symmetry, each of Marco, Fabrizio and Giovanni must spend the same amount of time on defense for the mathematicians' team, which therefore turns out to be one third of the 24 hours of the tournament, that is, 8 hours. Alternatively, we can observe that there will be 32433 \cdot 2 \cdot 4 \cdot 3 matches in total (3 ways to choose the mathematicians' forward, 2 for the defender, 4 for the physicists' forward and 3 for the defender); the matches in which Marco is defender are 2432 \cdot 4 \cdot 3, and thus the hours he spends on defense are 243324324=243=8\frac{2 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 4 \cdot 3} 24=\frac{24}{3}=8.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.