smc

1. Draw the graph of $f_0(x)$ by dividing the domain into three parts. 2. Apply the recursive rule a few times to find the pattern. Note: $f_n(x)=|f_{n-1}(x)|-10$ is used to enlarge the difference, but the reasoning is the same. 3. Extrapolate to $f_{100}$. Notice that the summits start $100$ away from $0$ and get $1$ closer each iteration, so they reach $0$ exactly at $f_{100}$. $f_{100}(x)$ reaches $0$ at $x=-300$, then zigzags between $0$ and $-1$, hitting $0$ at every even $x$, before leaving $0$ at $x=300$. This means that $f_{100}(x)=0$ at all even $x$ where $-300\le x\le 300$. This is a $601$-integer odd-size range with even numbers at the endpoints, so just over half of the integers are even, or $\frac{601+1}{2}=\boxed{301}$. (Revised by Flamedragon & Jason,C & emerald_block)