jmc

The stated property of $f(x)$ can be written as an equation which holds for all $x$: $$f(x-20)=f(x).$$We are looking for the smallest positive $a$ such that the equation $$f\left(\frac{x-a}{5}\right)=f\left(\frac{x}{5}\right)$$holds for all $x$. Rewriting this equation as $$f\left(\frac{x}{5}-\frac{a}{5}\right)=f\left(\frac{x}{5}\right),$$we see that it is implied by the known property of $f(x)$ if $\frac{a}{5}$ is equal to $20$ (or a multiple of $20$), or in other words, if $a$ is equal to $100$ (or a multiple of $100$). So, the smallest positive $a$ for which we know that this property holds is $a=\boxed{100}$.