Question

1) В 5-этажном доме живут друзья Лера, Света, Ира, Рома, Сеня и Марат. Докажи, что хотя бы двое друзей живут на одном этаже. 2) Подумай, сколько должно быть друзей, чтобы на одном этаже жили хотя бы трое из них. При ответе на этот вопрос ученики предложили такие ответы Лена: "Друзей должно быть 7" Олег: "Друзей должно быть 11" Объясни, какой из ответов ты считаешь верным. 3) Придумай свою похожую задачу.

Answer 1

 

1)

этажей 5

друзей 6

6/5=1 (1 ост)

то есть 6й из друзей попадет на один этаж с кем то из 5

 

2) 11. прав олег. 2 друга на каждом этаже +1 на любом из этажей

то есть 5*2+1 = 11

 

3) сколько друзей должно быть в доме, что бы на одном из этажей жило 4 друга? (ответ 16) 

Answer 2

1) Proof that at least two friends live on the same floor in a 5-story building:

To prove that at least two friends live on the same floor in a 5-story building, we can use the Pigeonhole Principle. The Pigeonhole Principle states that if there are more "pigeons" than "pigeonholes," then at least one "pigeonhole" must contain more than one "pigeon."

In this case, if we have 6 friends (Lera, Svetlana, Ira, Roma, Senya, and Marat) and 5 floors, we can consider each friend as a "pigeon" and each floor as a "pigeonhole." Since there are more "pigeons" (6 friends) than "pigeonholes" (5 floors), by the Pigeonhole Principle, at least two friends must live on the same floor.

Therefore, at least two friends live on the same floor in the 5-story building.

2) Determining the minimum number of friends for at least three of them to live on the same floor:

To determine the minimum number of friends required for at least three of them to live on the same floor, we can use the Pigeonhole Principle again. This time, we want to find the minimum number of friends needed to guarantee that at least three of them live on the same floor.

Using the Pigeonhole Principle, we can calculate the minimum number of friends required as follows: - If there are 7 friends, then at least 3 of them must live on the same floor. - If there are 11 friends, then at least 3 of them must live on the same floor.

Based on the Pigeonhole Principle, the minimum number of friends required for at least three of them to live on the same floor is 7.

Therefore, the answer provided by Lena, "Dрузей должно быть 7," is the correct one.

3) Creating a similar problem:

Let's consider a scenario where there are 4 friends (A, B, C, D) and 3 floors in a building. The task is to determine the minimum number of friends required to guarantee that at least two of them live on the same floor.

Using the Pigeonhole Principle, we can calculate the minimum number of friends required as follows: - If there are 5 friends, then at least 2 of them must live on the same floor.

Therefore, the question would be: "What is the minimum number of friends needed to guarantee that at least two of them live on the same floor in a 3-story building?"

This problem is similar to the previous one but with different numbers, providing a variation of the Pigeonhole Principle application.