12. It is known that an exam consists of 6 questions, each of which is solved by exactly 100 people, and for any two participants, there is at least one question that neither of them solved. How many people participated in the exam at a minimum?
12. The minimum number of people required is 200. On one hand, number 200 people from 1 to 200, where those numbered $1 \sim 50$ solve questions $1, 2, 3$, those numbered $51 \sim 100$ solve questions $1, 4, 6$, those numbered $101 \sim 150$ solve questions $2, 4, 5$, and those numbered $151 \sim 200$ solve questions $3, 5, 6$. Thus, each question is solved by exactly 100 people, and any two people have exactly one question that neither of them solved. On the other hand, if the number of people $n$ is less than 200, then by $3n < 600 = 6 \times 100$, there must be a person $A$ who has solved at least 4 questions. Suppose $A$ has solved questions $1, 2, 3, 4$. If $A$ also solved question 5, then the person who solved question 6 would have no question that both $A$ and this person did not solve, leading to a contradiction. Therefore, $A$ must have solved exactly questions $1, 2, 3, 4$. Since $n < 200$, there must be another person $B$ among the remaining people who solved questions 5 and 6 (because the last 2 questions were solved a total of 200 times), and for $A$ and $B$, there is no question that both did not solve, leading to a contradiction. Hence, it must be that $n \geqslant 200$. In summary, the minimum number of people who participated in the exam is 200.