AMC10 2002 A

AMC10 2002 A · Q22

AMC10 2002 A · Q22. It mainly tests Perfect squares & cubes, Number theory misc.

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

一组编号为1至100的瓦片通过以下操作反复修改：移除所有完美平方编号的瓦片，并将剩余瓦片从1开始连续重新编号。要将瓦片数量减少到1次，需要执行该操作多少次？

(A) 10 10

(B) 11 11

(D) 19 19

(E) 20 20

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) The first application removes ten tiles, leaving 90. The second and third applications each remove nine tiles leaving 81 and 72, respectively. Following this pattern, we consecutively remove 10, 9, 9, 8, 8, ..., 2, 2, 1 tiles before we are left with only one. This requires $1 + 2(8) + 1 = 18$ applications.

（C）第一次操作移走 10 块瓷砖，剩下 90 块。第二次和第三次操作各移走 9 块瓷砖，分别剩下 81 块和 72 块。按照这个规律，我们依次移走 10，9，9，8，8，…，2，2，1 块瓷砖，直到只剩下 1 块。这需要 $1 + 2(8) + 1 = 18$ 次操作。

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