2009 AMC 8 第 9 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
9.
在一个等边三角形的一边上作一个正方形。在正方形的一条不相邻的边上作一个正五边形，如图所示。在五边形的一条不相邻的边上作一个正六边形。用同样方式继续作正多边形，直到作出正八边形。所得图形的外边界有多少条边？
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a regular hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
小提示：
只数每个多边形留在外边界上的边
Count only the outside sides of each polygon.
大提示：
中间的多边形各共享两条边，两端的多边形各共享一条边
Middle polygons share two sides; the two end polygons share one side each.
解答：
三角形和八边形在链的两端，所以各有一条边成为共享边而不在外边界上。正方形、五边形、六边形和七边形在中间，所以各有两条边成为共享边。
所得图形有 条边。
所以正确答案是 B。
The triangle and octagon are at the ends of the chain, so each loses one side to a shared edge. The square, pentagon, hexagon, and heptagon are in the middle, so each loses two sides to shared edges.
The resulting polygon has sides.
Thus, B is the correct answer.
其他年份的第 9 题
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