Question 101
Seven Skiers, Numbered from $1$ to $7$, Participated in a race. They started one by one, and each skier passed the distance It Happened That Each Skier Participated In An Overtaking And the one who is finished (that List of Ranks is published) finished the race). Prove that all the conditions being satisified, not more than two possible lists might appear. Seven skiers with $1, 2, \ldots, 7$ numbers left the start in turn and covered the distance - each with its own constant speed. It turned out that each skier participated exactly twice in overtaking. (Everybody involved in each overtaking involves exactly two skiers - the one who overtakes, and the one who is overtaking.) At the end of the race, a protocol consisting of skiing numbers in the order of finishing must be drawn up. Prove that in a race with the described properties, no more than two different protocols can be obtained.