2006 AMC 12A 第 12 题

Question

若干个相扣的环，每个厚

1

cm，挂在一个钉子上。最上面的环外径为

20

cm。其余每个环的外径都比上方的环小

1

cm。最下面的环外径为

3

cm。从最上面环的顶部到最下面环的底部的距离是多少 cm？

A number of linked rings, each

1

cm thick, are hanging on a peg. The top ring has an outside diameter of

20

cm. The outside diameter of each of the other rings is

1

cm less than that of the ring above it. The bottom ring has an outside diameter of

3

cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

171

173

182

188

210

答案：B

知识点：等差数列求和

难度评级：1370

小提示：

最上面的环跨越

20

cm；每个下面的环增加其外径减

2

的长度，因为有重叠。

The top ring spans

20

cm; each lower ring adds its outside diameter minus

2

for the overlap

大提示：

总长

= 20 + (17 + 16 + \cdots + 1)

。

Total

= 20 + (17 + 16 + \cdots + 1)

解答：

最上面的环跨越

20

cm。下面每个环与上方的环重叠

2

cm（两倍的

1

-cm 厚度），所以它增加的长度为外径减

2

。

下面各环外径为

19, 18, \ldots, 3

，贡献

17, 16, \ldots, 1

。因此总距离为

\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173 \text{ cm} \end{gathered}\text{。}

因此，正确答案是 B。

The top ring spans

20

cm. Each ring below overlaps the ring above by

2

cm (twice the

1

-cm thickness), so it adds its outside diameter minus

2.

The lower rings have outside diameters

19, 18, \ldots, 3,

contributing

17, 16, \ldots, 1.

Thus the total distance is

\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173 \text{ cm}. \end{gathered}

Thus, the correct answer is B.

2006 AMC 12A 第 12 题

答案：B

难度评级：1370

小提示：

大提示：

解答：

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