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 each problem was solved correctly by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. How many middle school students from the city participated in this mathematics invitational competition at least?
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.