4. The quiz participants were asked four questions: 90 participants answered the first question correctly, 50 answered the second, 40 answered the third, and 20 answered the fourth, and no one was able to answer more than two questions correctly. What is the minimum number of participants in the quiz under these conditions?
Answer: 100.

Solution. The total number of answers is 200. Since no one answered more than two questions, the minimum possible number of participants in the quiz is 100, and in this case, each participant in the quiz must answer exactly 2 questions correctly. We will provide an example of a quiz where the described situation is realized. Number the participants from 1 to 100. Let participants numbered from 1 to 90 answer the first question, from 91 to 100 and from 1 to 40 answer the second question, from 41 to 80 answer the third question, and from 81 to 100 answer the fourth question.