Example 4 Some middle school students in a city participated in a mathematics invitational competition, and this competition 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 correctly. Question: What is the minimum number of middle school students from the city who participated in this mathematics invitational competition?
Solution: First, each student can answer at most 4 questions correctly.
In fact, according to the problem, it is impossible for any student to answer 6 questions correctly. If a student answers 5 questions correctly, by the problem's condition, all other students must have answered the same question incorrectly, which contradicts the condition that exactly 500 students answered each question correctly.

If a student answers 4 questions correctly, let's assume they answered questions 1, 2, 3, and 4 correctly. Then no student can answer both questions 5 and 6 correctly, otherwise it would contradict the problem's condition. Therefore, the number of students who answered question 5 and question 6 correctly are each 500, so the total number of students is at least $500 + 500 + 1 > 1000$.

If each student answers 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 is at least $3000 \div 3 = 1000$.

Below is a construction to show that 1000 people is possible. The number of people who answered the following questions correctly is 100 each:
$$
\begin{array}{l}
(1,2,3),(1,3,4),(1,4,5),(1,5,6), \\
(1,2,6),(2,4,6),(2,3,5),(2,4,5), \\
(3,4,6),(3,5,6) .
\end{array}
$$

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