25. Let $S=\{1,2,3, \ldots, 48,49\}$. What is the maximum value of $n$ such that it is possible to select $n$ numbers from $S$ and arrange them in a circle in such a way that the product of any two sdjacent numbers in the circle is less than 100?
25. Answer: 18
First note that the product of any two different 2-digit numbers is greater than 100. Thus if a 2-digit number is chosen, then the two numbers adjacent to it in the circle must be single-digit numbers. Note that at most nine single-digit numbers can be chosen from $S$, and no matter how these nine numbers $1,2, \ldots, 9$ are arranged in the circle, there is at most one 2-digit number in between them. Hence it follows that $n \leq 18$. Now the following arrangement
$$
1,49,2,33,3,24,4,19,5,16,6,14,7,12,8,11,9,10,1
$$
shows that $n \geq 18$. Consequently we conclude that the maximum value of $n$ is 18 .