[ Inequality problems. Case analysis ]

Once, during a moment of rest, the musketeer friends - Athos, Porthos, Aramis, and d'Artagnan - decided to test their strength in a tug-of-war. Porthos and d'Artagnan easily overpowered Athos and Aramis. But when Porthos teamed up with Athos, they managed to win against Aramis and d'Artagnan, but not as easily. When Porthos and Aramis were pitted against Athos and d'Artagnan, neither pair could overpower the other. Can you determine how the musketeers rank in strength?
Note: Four musketeers can be divided into pairs in three different ways: (1,2)-(3,4); (1,3)-(2,4); (1,4)-(2,3). Here, the numbers denote the strength of each musketeer based on their position.

## Solution

Four musketeers can be divided into pairs in three different ways. If we number the musketeers in descending order of strength, the pairs will look like this: (1,2)-(3,4), (1,3)-(2,4), (1,4)-(2,3). It is clear that in the first two cases, the first pairs will definitely win (very easily in the first case, and more difficultly in the second case), while in the third case, the outcome of the duel can be any (in our case, it was a draw). Therefore, the first and second places were taken by d'Artagnan and Porthos, the first and third places by Porthos and Athos, and the second and fourth places by d'Artagnan and Aramis. From this, it follows that the musketeers are ranked in strength as follows: Porthos, d'Artagnan, Athos, Aramis.