19. For $0 \leq x \leq 1$ and positive integer $n$, let $f_{0}(x)=|1-2 x|$ and $f_{n}(x)=f_{0}\left(f_{n-1}(x)\right)$. How many solutions are there to the equation $f_{10}(x)=x$ in the range $0 \leq x \leq 1$ ?
對於 $0 \leq x \leq 1$ 和正整數 $n$, 設 $f_{0}(x)=|1-2 x|$ 和 $f_{n}(x)=f_{0}\left(f_{n-1}(x)\right)$ 。那麼方程 $f_{10}(x)=x$ 在 $0 \leq x \leq 1$ 區間內有多少個解?
19. 2048
19. In an iterative manner, we can work out the graphs of $f_{0}, f_{1}$ and $f_{2}$ (together with the graph of $y=x$ in dotted line) in the range $0 \leq x \leq 1$, which explain everything:

Based on the pattern, it is easy to see (and work out an inductive proof) that the graph of $f_{10}$ consist of $2^{10}=1024$ copies of ' $\mathrm{V}$ ', and hence $1024 \times 2=2048$ intersections with the line $y=x$. This gives 2048 as the answer.