34. What is the smallest possible integer value of $n$ such that the following statement is always true?
In any group of $2 n-10$ persons, there are always at least 10 persons who have the same birthdays.
(For this question, you may assume that there are exactly 365 different possible birthdays.)
34. Answer: 1648
By the pigeonhole principle in any group of $365 \times 9+1=3286$ persons, there must be at least 10 persons who share the same birthday.
Hence solving $2 n-10 \geq 3286$ gives $n \geq 1648$. Thus the smallest possible $n$ is 1648 since $2 \times 1647-10=3284<365 \times 9$, and it is possible for each of the 365 different birthdays to be shared by at most 9 persons.