AMC12 2025 B

Let $f(x)=\sin(20\pi x)$ and $g(x)=\log_{20}(x)$. Note that $g$ passes through $\left(\frac{1}{20},-1\right)$ and $(1,0)$ and $(20,1)$; these are the extrema and midpoint of $f$. We want to find the number of intersections of $f$ and $g$. Let an occurrence of sine passing under the $x$-axis a down-dip, and similarly define an up-dip. We find that the period of $f$ is $\frac{1}{10}$, so between $x=\frac{1}{20}$ and $x=1$ the number of periods is $9.5$. The first period indeed counts, so effectively we have $10$ down-dips in this interval. Each down-dip contributes $2$ to the total, so we have $20$ total intersections. From $x=1$ to $x=20$, there are $190$ periods, each of which also contributes $2$ to the total due to the up-dips. Therefore, this case contributes $380$ points to the total. But $(1,0)$ is counted in both cases, so the total is $20+380-1=\boxed{\textbf{(D) } 399}$.