Question 14
For real number $x$ let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x,$ and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x.$ For example, $\{3\} = 0$ and $\{4.56\} = 0.56.$ Define $f(x) = x\{x\},$ and let $N$ be the number of real-valued solutions to the equation $f(f(f(x))) = 17$ for $0 \le x \le 2020.$ Find the remainder when $N$ is divided by $1000.$