Problem 7.7. All 25 students in class 7A participated in a quiz consisting of three rounds. In each round, each participant scored a certain number of points. It is known that in each round, as well as in the total of all three rounds, all participants scored a different number of points.

Student Kolya from 7A was third in the first round of the quiz, fourth in the second, and fifth in the third. What is the lowest place Kolya could have taken among all classmates in the total points for all three rounds of the quiz?
Answer: 10.

Solution. In the first round, 2 classmates overtook Kolya, in the second - 3, in the third - 4. Then, in the sum of all three rounds, he could be overtaken by no more than $2+3+4=9$ classmates, i.e., in the sum of the three rounds, he could not end up lower than 10th place.

Now let's provide an example of how Kolya could end up exactly in 10th place. Suppose

- in each of the rounds, Kolya scored 100 points;
- in the first round, Andrey and Boris scored 1000 and 2000 points respectively and took the first two places
- in the second round, Vlad, Gennady, and Denis scored 10000, 20000, 30000 points respectively and took the first three places
- in the third round, Masha, Svetlana, Tanya, and Katya scored 100000, 200000, 300000, 400000 points respectively and took the first four places;
- in each of the rounds, all other participants after Kolya were ordered alphabetically and scored a number of points equal to their position in the ranking from the end.

It is easy to see that all the conditions of the problem are satisfied.