Question

$1$ 到 $40$ 编号的四十张纸条放在帽子里。Alice 和 Bob 各不放回地抽一张，并互相隐藏自己的数字。Alice 说：“我不能判断谁的数字更大。” 然后 Bob 说：“我知道谁的数字更大。” Alice 说：“你知道？你的数字是质数吗？” Bob 回答：“是。” Alice 说：“这样的话，如果我把你的数字乘以 $100$ ，再加上我的数字，结果是一个完全平方数。” 两人抽到的数字之和是多少？

Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, “I can’t tell who has the larger number.” Then Bob says, “I know who has the larger number.” Alice says, “You do? Is your number prime?” Bob replies, “Yes.” Alice says, “In that case, if I multiply your number by $100$ and add my number, the result is a perfect square.” What is the sum of the two numbers drawn from the hat?

$27$

$37$

$47$

$57$

$67$

答案：A

知识点：逻辑推理质数完全平方数

难度评级：1950

解答：

如果 Alice 抽到 $1$ 或 $40$ ，她会知道谁的数字更大，因此她的数字不是 $1$ 或 $40$ 。

Bob 由此知道 Alice 的数字不是端点。此时只有当 Bob 的数字属于 $1,2,39$ 或 $40$ 时，他才能判断大小。又因为他的数字是质数，所以只能是 $2$ 。

现在 $100\cdot2$ 加上 Alice 的数字是介于 $201$ 和 $240$ 之间的平方数。唯一可能是 $225$ ，所以 Alice 的数字为 $25$ ，两数之和为 $2+25=27$ 。

所以正确答案是 A。

If Alice had drawn $1$ or $40,$ she would know who had the larger number. Her first statement tells Bob that Alice has neither $1$ nor $40.$

Bob can then know who has the larger number only if his number is $1,2,39,$ or $40.$ Since Bob says his number is prime, his number must be $2.$

Now $100\cdot2$ plus Alice's number is a square between $201$ and $240.$ The only square in that interval is $225,$ so Alice's number is $25.$ The sum is $2+25=27.$

Thus, the answer is A .

2021 AMC 10B Fall 第 10 题

答案：A

难度评级：1950

解答：

其他年份的第 10 题