Example 4 Some middle school students in a city participated in a mathematics invitational competition, which had a total of 6 problems. It is known that exactly 500 students answered each problem correctly, but for any two students, there is at least one problem that neither of them answered correctly. Question: What is the minimum number of middle school students in the city who participated in this mathematics invitational competition?
First, we prove that each student answered at most 4 questions correctly. In fact, according to the problem, no student answered all 6 questions correctly. If a student answered 5 questions correctly, then all other students would have answered the same question incorrectly. This contradicts the given condition that "exactly 500 students answered each question correctly"!

If a student answered 4 questions correctly, without loss of generality, let's assume they answered questions 1, 2, 3, and 4 correctly. According to the given condition, no student answered both questions 5 and 6 correctly. Therefore, 500 students answered question 5 correctly, and 500 students answered question 6 correctly. In this case, the total number of participants is \(500 + 500 + 1 = 1001\) people.

If each student answered at most 3 questions correctly, since the total number of questions answered correctly by all students is \(500 \times 6 = 3000\), the number of students must be at least \(3000 \div 3 = 1000\).

Below is a construction to show that 1000 people is possible:
100 people each answered the following sets of questions correctly: \((1,2,3), (1,3,4), (1,4,5), (1,5,6), (1,2,6), (2,4,6), (2,3,5), (2,4,5), (3,4,6), (3,5,6)\).

In summary, at least 1000 people participated in this competition.