1956 AMC 12 第 5 题

Question

将一枚五美分硬币放在桌上。围绕它最多可以放置多少枚五美分硬币，使每一枚都与中间的硬币及另外两枚硬币相切？

A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:

4

5

6

8

12

答案：C

知识点：circle packing central angles regular hexagon

难度评级：1070

小提示：

将中间硬币的圆心与两枚相邻硬币的圆心连接

Join the center of the middle nickel to the centers of two neighboring nickels

大提示：

三个两两相切的等圆，其圆心构成等边三角形

Those three mutually tangent equal circles make an equilateral triangle of centers

解答：

中间硬币与任意两枚相邻硬币的圆心两两相距两个半径，所以它们构成等边三角形。因此，相邻两枚外圈硬币的圆心在中间硬币圆心处所成的角为

60^\circ

。恰有

\frac{360^\circ}{60^\circ}=6

枚硬币可以围在它周围。

因此，正确答案是 C。

The centers of the middle nickel and any two neighboring nickels are pairwise two radii apart, so they form an equilateral triangle. Therefore consecutive outer centers subtend

60^\circ

at the center of the middle nickel. Exactly

\frac{360^\circ}{60^\circ}=6

nickels fit around it.

Thus, the correct answer is C.

1956 AMC 12 第 5 题

答案：C

难度评级：1070

小提示：

大提示：

解答：

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