Question

17 участников команды комета набрали 125 баллов. Докажите, что какие-то двое из них набрали равное количество баллов

Answer 1

Если они все заработали разное количество очков, от 0 до 16, то в сумме будет
0+1+2+3+...+16 = 136.
А у нас всего 125 очков, значит, как минимум кто-то получил меньше очков,
то есть какие-то два человека получили одинаковое количество очков.

Answer 2

Problem Statement

We are given that a team called "Kometa" consists of 17 participants, and the total number of points they scored is 125. We need to prove that there are at least two participants who scored the same number of points.

Proof by Pigeonhole Principle

To prove that at least two participants scored the same number of points, we can use the Pigeonhole Principle. The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.

In this case, the "pigeons" are the participants of the team "Kometa," and the "pigeonholes" are the possible scores they can achieve. Since there are 17 participants and the total number of points scored is 125, we can assume that each participant can score a maximum of 125 points. Therefore, we have 125 possible scores or pigeonholes.

According to the Pigeonhole Principle, if we have more pigeons (17 participants) than pigeonholes (125 possible scores), then at least one pigeonhole must contain more than one pigeon. In other words, at least two participants must have scored the same number of points.

Hence, we can conclude that there are at least two participants from the team "Kometa" who scored the same number of points.

Conclusion

Using the Pigeonhole Principle, we have proven that in a team of 17 participants called "Kometa," where the total number of points scored is 125, there are at least two participants who scored the same number of points.

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