Question

一组编号为 $1$ 到 $100$ 的瓷砖反复进行如下操作：移除所有编号为完全平方数的瓷砖，然后将剩余瓷砖从 $1$ 开始重新连续编号。需要进行多少次操作，才能把瓷砖数量减少到一？

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?

$10$

$11$

$18$

$19$

$20$

答案：C

知识点：完全平方数递推找规律

难度评级：1790

解答：

从 $n^2$ 块瓷砖开始，一次操作会移除 $n$ 个完全平方编号，剩 $n^2-n$ 块。下一次操作会移除 $n-1$ 个完全平方编号，剩 $n^2-n-(n-1)=(n-1)^2$ 块。

因此每两次操作会把 $n^2$ 减到 $(n-1)^2$ 。从 $10^2=100$ 减到 $1^2=1$ ，需要 $2(10-1)=18$ 次操作。

所以正确答案是 C。

Starting from $n^2$ tiles, one operation removes the $n$ perfect squares, leaving $n^2-n.$ The next operation removes $n-1$ perfect squares, leaving $n^2-n-(n-1)=(n-1)^2.$

So every two operations reduce $n^2$ to $(n-1)^2.$ Going from $10^2=100$ down to $1^2=1$ takes $2(10-1)=18$ operations.

Thus, the correct answer is C.

2002 AMC 10A 第 22 题

答案：C

难度评级：1790

解答：

其他年份的第 22 题