Question

Три человека выписали по 100 различных слов. После этого слова, встречающиеся не менее двух раз, вычеркнули. В результате у одного осталось 45 слов, у другого – 68, а у третьего – 54. Докажите, что по крайней мере одно слово выписали все трое

Answer 1

100-45=55

100-68=32

100-54=46

Для того что бы все слова были выписаны только по два раза, сумма вычеркнутых слов не должна превышать 100. Соответственно: 55+32+46=133.

 

 

 

 

 

 

Answer 2

Problem Analysis

We are given that three people wrote down 100 different words. After that, any word that appeared at least twice was crossed out. We are also given that one person was left with 45 words, another with 68 words, and the third with 54 words. We need to prove that at least one word was written down by all three people.

Proof by Contradiction

To prove that at least one word was written down by all three people, we will assume the opposite, i.e., that no word was written down by all three people. Let's denote the number of words written down by the first person as A, the number of words written down by the second person as B, and the number of words written down by the third person as C.

If no word was written down by all three people, it means that any word that was written down by one person was not written down by the other two people. Therefore, the total number of unique words written down by the three people would be A + B + C.

According to the given information, A = 45, B = 68, and C = 54. Therefore, the total number of unique words written down by the three people would be 45 + 68 + 54 = 167.

However, we are given that each person wrote down 100 different words. Therefore, the total number of unique words written down by the three people cannot exceed 100.

This leads to a contradiction: 167 > 100. Therefore, our assumption that no word was written down by all three people must be false.

Hence, we can conclude that at least one word was written down by all three people.

Note: The given search results did not provide any relevant information for this specific problem. The proof provided is based on logical reasoning and does not require external sources.