HMMT Spring, 2021

Solution:

$X_{n}$ is the set of points within $n$ units of some point in $X_{0}$. The diagram above shows $X_{0}$, $X_{1}$, $X_{2}$, and $X_{3}$. As seen above it can be verified that $X_{n}$ is the union of
- $X_{0}$,
- three rectangles of height $n$ with the sides of $X_{0}$ as bases, and
- three sectors of radius $n$ centered at the vertices and joining the rectangles

Therefore the total area of $X_{n}$ is
$$\left[X_{0}\right] + n \cdot \operatorname{perimeter}\left(X_{0}\right) + n^{2} \pi$$
Since $X_{n-1}$ is contained entirely within $X_{n}$, the area within $X_{n}$ but not within $X_{n-1}$ is
$$\operatorname{perimeter}\left(X_{0}\right) + (2n-1) \pi$$
Since $X_{0}$ is a $(3,4,5)$ triangle, and $n=21$, this is $12 + 41\pi$.