How to find $f_1\circ f_2\circ\cdots f_{13}(2),$ where $f_n(x) = \frac{nx+9}{x+3}$ in a reasonable amount of time?
I solved this problem through brute forcing, but that took about an hour, and got the final answer $23/11.$ Is there a much quicker way to do this? (The recommended time is 5-10 minutes)
Express $f_n(x) = n + \frac{9-3n}{x+3}$. Then $f_3(x) = 3$, for all $x$. Thus your expression reduces to $f_1(f_2(3))$.
Edit: As cardboard_box notes in the comments, one needs to check that $f_n(x)\neq -3$ for $n\leq 13$ in the above composition (and the appropriate $x$). You can do it as described by cardboard_box.