SAT, 08.01.2021 01:00 dreyiamacc9695
Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
[asy]
unitsize(1.5 cm);
pair[] A;
for(int i = 1; i <= 10; ++i) {
A[i] = dir(90 - 360/10*i);
draw(dir(90 - 360/10*i)--dir(90 - 360/10*(i + 1)));
}
draw(A[10]--(A[10] + A[2] - A[3])--(A[10] + A[2] - A[3] + A[4] - A[5])--(A[10] + A[2] - A[3] + A[4] - A[5] + A[2] - A[1])--(A[10] + A[2] - A[3] + A[4] - A[5] + A[2] - A[1] + A[4] - A[3]));
draw((A[10] + A[2] - A[3])--(A[10] + A[2] - A[3] + A[10] - A[1])--(A[10] + A[2] - A[3] + A[10] - A[1] + A[8] - A[9])--(A[10] + A[2] - A[3] + A[10] - A[1] + A[8] - A[9] + A[6] - A[7]));
dot(A[10]);
[/asy]
A square, a regular pentagon, and a regular $n$-gon, all with the same side length, also completely surround a point. Find $n$.
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