20. 10 people go to the bookstore to buy books. If it is known that: (1) each person bought three books; (2) any two people have at least one book in common. How many people at least bought the most popular book (the one bought by the most people)? Why?
20. Let the most popular book be bought by $k$ people, and suppose $A_{1}$ bought books $a, b, c$. Each of the other 9 people bought at least one book that is the same as $a, b, c$, so at least $\left[\frac{9-1}{3}\right]+1=3$ people (denoted as $A_{2}, A_{3}, A_{4}$) also bought the same book among $a, b, c$. Therefore, $A_{1}, A_{2}, A_{3}, A_{4}$ bought the same book, hence $k \geqslant 4$. If $k=4$, meaning the book bought by the most people is only bought by 4 people, and since 3 people bought a total of 30 books, and $4 \times 30$, it is impossible for every book to be bought by 4 people. Suppose book $a$ is only bought by 3 people, and books $b, c$ are bought by at most 4 people. Thus, apart from $A_{1}$, at most 2 more people bought book $a$, and the number of people buying books $b$ and $c$ is at most 3 each, so the total number of people $\leqslant 1+2+3+3=9$, which is a contradiction. Therefore, $k \geqslant 5$. Below is an example of a purchase method when $k=5$:
In summary, the book bought by the most people is bought by at least 5 people.