HMMT November, 2019

Solution:

Suppose the toast has side length $s$. If we draw the three line segments from the sesame seed to the three vertices of the triangle, we partition the triangle into three smaller triangles, with areas $\frac{s}{2}$, $s$, and $2s$, so the entire piece of toast has area $\frac{7s}{2}$. Suppose the cheese has radius $r$. We similarly see that the toast has area $\frac{3rs}{2}$. Equating these, we see that $r = \frac{7}{3}$, so the area of the cheese is $\pi\left(\frac{7}{3}\right)^2 = \frac{49\pi}{9}$.