3. In a psychiatric hospital, there is a chief doctor and many lunatics. During the week, each lunatic bit someone (possibly even themselves) once a day. At the end of the week, it turned out that each of the patients had two bites, and the chief doctor had a hundred bites. How many lunatics are there in the hospital
Answer: 20 madmen.

Solution: Let there be n madmen in the hospital. Then, by the end of the week, on the one hand, 7n bites were made, and on the other hand, $2 n+100$. i.e.

$7 n=2 n+100$, from which $n=20$.