2021 AMC Fall 10B Problem 18

1. Set up. Put the common center at $(0, 0)$ and let the bottom sheet have corners at $(\pm 3, \pm 3)$. Its corner $(3, 3)$ is at $45^\circ$ from the positive $x$-axis. The other two sheets are the same square turned $30^\circ$ and $60^\circ$, so all twelve corners are at the same distance from the center, spaced every $30^\circ$: at $15^\circ, 45^\circ, 75^\circ, \ldots$
2. Where are the other 12 corners of the outline? Between two neighboring square corners, the outline dips inward to a point where two sheets' edges cross. The whole picture looks the same after a $30^\circ$ turn, and it is also symmetric across the line halfway between two neighboring square corners. So each crossing point sits exactly halfway, on the lines at $0^\circ, 30^\circ, 60^\circ, \ldots$
3. Find one crossing point. Take the corner $C = (3, 3)$ at $45^\circ$. Walking clockwise from $C$ along the bottom sheet's right edge $x = 3$, you reach the crossing point $X$ on the $30^\circ$ line. On the $30^\circ$ line, height $=$ distance across $\times \frac{1}{\sqrt3}$ (a $30$-$60$-$90$ triangle). With distance across $3$, the height is $\frac{3}{\sqrt3} = \sqrt3$. So $X = (3, \sqrt3)$.
4. One thin triangle. Triangle $OCX$ has corners $O = (0,0)$, $C = (3,3)$, $X = (3, \sqrt3)$. Use $CX$ as the base: it lies on the line $x = 3$ and has length $3 - \sqrt3$. The height is the distance from $O$ to that line, which is $3$. Area:
$$\frac12 \times 3 \times (3 - \sqrt3) = \frac{9 - 3\sqrt3}{2}.$$
5. All 24 triangles. Going around the center, the outline is made of $24$ copies of this triangle (corner to crossing point, crossing point to corner, and so on). Total area:
$$24 \times \frac{9 - 3\sqrt3}{2} = 108 - 36\sqrt3.$$
6. So $a = 108$, $b = 36$, $c = 3$, and $a + b + c = 147$.

The answer is $\boxed{\textbf{(E)}\ 147}$.

When identical shapes are stacked around a common center and turned by equal steps, the result repeats all the way around. Find the smallest slice that repeats, compute its area, and multiply. The key fact is that the outline's corners alternate between square corners and crossing points at equal angle steps, which tells you exactly which line each crossing point lies on.

A quick size check helps catch slips. The shape contains one whole square, area $36$, and it fits inside the circle through all the corners, which has radius $3\sqrt2$ and area $18\pi \approx 56.5$. The answer $108 - 36\sqrt3 \approx 108 - 62.4 = 45.6$ lies between these, as it must.