Question

七名学生按如下方式从 $1$ 数到 $1000$ ：

• Alice 说出所有数字，但跳过每连续三个数字中的中间那个。也就是说，Alice 说 $1$ ， $3$ ， $4$ ， $6$ ， $7$ ， $9, \ldots$ ， $997$ ， $999$ ， $1000$ 。

• Barbara 说出所有 Alice 没说的数字，但她也跳过自己连续三个数字中的中间那个。

• Candice 说出 Alice 和 Barbara 都没说的数字，但她也跳过自己连续三个数字中的中间那个。

• Debbie、Eliza 和 Fatima 也依次说出所有名字字母顺序在她们之前的学生都没说的数字，但各自也跳过自己连续三个数字中的中间那个。

• 最后，George 说出唯一没有别人说过的数字。

George 说的数字是多少？

Seven students count from $1$ to $1000$ as follows:

• Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1,$ $3,$ $4,$ $6,$ $7,$ $9, \ldots,$ $997,$ $999,$ $1000.$

• Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.

• Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.

• Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.

• Finally, George says the only number that no one else says.

What number does George say?

$37$

$242$

$365$

$728$

$998$

答案：C

知识点：等差数列过程模拟找规律

难度评级：2110

解答：

逐步追踪这些数列即可找出最后剩下的数。

Alice 没说的数为 $2, 5, 8, 11, 14, 17, \ldots, 998.$

Barbara 说完后剩下 $5, 14, 23, 32, 41, \ldots, 995.$

这些都是等差数列，而公差每次变为原来的 $3$ 倍。

Candice 说完后剩下 $14, 41, 68, 95, \ldots, 986.$

Debbie 说完后剩下，Eliza 说完后剩下 $41, 122, 203, \ldots, 932$ $122, 365, 608, 851.$

最后，Fatima 说完后唯一剩下的数是 $365$ ，所以 George 说的就是这个数。

所以正确答案是 C。

We can walk through all the iterations to find what is left.

Alice does not say the numbers $2, 5, 8, 11, 14, 17, \ldots, 998.$

After Barbara says her numbers, the remaining ones are $5, 14, 23, 32, 41, \ldots, 995.$

Note that both of these are arithmetic sequences where the common difference is increased by a multiple of $3.$

This pattern continues as the numbers remaining after Candice says hers are $14, 41, 68, 95, \ldots, 986.$

Then after Debbie, they are $41, 122, 203, \ldots, 932$ and after Eliza, they are $122, 365, 608, 851.$

Finally, the only number left after Fatima goes is $365,$ which is the number that George will have to say.

Thus, C is the correct answer.

2011 AMC 10A 第 23 题

答案：C

难度评级：2110

解答：

其他年份的第 23 题