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2021 Fall AMC 12B Problem 19

Problem 19 of 25HarderGeometryCombinatorics

Regular polygons with 5,5, 6,6, 7,7, and 88 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

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Solution

For two convex polygons inscribed in the same circle with no shared vertices, each side of the smaller polygon crosses the larger polygon’s boundary exactly twice, so they meet at 2min⁡(m,n)2\min(m, n) points. Summing over all pairs: (5,6),(5,7),(5,8)(5,6), (5,7), (5,8) give 1010 each; (6,7),(6,8)(6,7), (6,8) give 1212 each; (7,8)(7,8) gives 14.14. The total is 3⋅10+2⋅12+14=68.3 \cdot 10 + 2 \cdot 12 + 14 = 68. Thus, the correct answer is E.
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Tagged: regular polygon · counting intersections

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