Question

В соревнованиях по бегу и прыжкам в длину приняло участие в общей сложности 55 человек. Известно, что в обоих видах спорта соревновалось 12 человек. Докажите, что в одном из видов спорта соревновалось не менее 34 человек. ____ n(A объединение В) = 55, n (А пересечение В) = 12 n(А объединение В) = n(A)+n(В)-n(А пересечение В) n(А)+n(B)=67... А дальше что? Или я вообще не в том направлении двигаюсь?

Answer 1

Если подумать логически то тебя просто просят найти остаток людей из общего числа. 55-12*2=31>22, так как 12+22=34. Вот в принципе и всё.

Answer 2

Problem Analysis

We are given that a total of 55 people participated in a competition involving running and long jump. It is also known that 12 people competed in both sports. We need to prove that in at least one of the sports, there were at least 34 participants.

Solution

To solve this problem, we can use the principle of inclusion-exclusion. Let's denote the number of participants in running as A and the number of participants in long jump as B.

We are given the following information: - n(A ∪ B) = 55 (the total number of participants) - n(A ∩ B) = 12 (the number of participants who competed in both sports)

Using the principle of inclusion-exclusion, we can calculate the number of participants in either running or long jump as follows:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Substituting the given values:

55 = n(A) + n(B) - 12

Simplifying the equation:

n(A) + n(B) = 67

Now, we need to prove that in at least one of the sports, there were at least 34 participants. Let's assume that there were fewer than 34 participants in both sports. This means:

n(A) < 34 and n(B) < 34

Adding these two inequalities:

n(A) + n(B) < 68

But we know that n(A) + n(B) = 67, which contradicts our assumption. Therefore, our assumption is incorrect, and it must be true that in at least one of the sports, there were at least 34 participants.

Therefore, we have proved that in one of the sports, there were at least 34 participants.

Conclusion

In the competition involving running and long jump, a total of 55 people participated. It is proven that in at least one of the sports, there were at least 34 participants.