2010 AMC 12A 第 12 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
12.
在一片神奇的沼泽中，有两种会说话的两栖动物：蟾蜍总是说真话，青蛙总是说假话。Brian、Chris、LeRoy 和 Mike 四只两栖动物一起住在这片沼泽中，并作出以下陈述。
Brian：“Mike 和我是不同物种。”
Chris：“LeRoy 是青蛙。”
LeRoy：“Chris 是青蛙。”
Mike：“我们四个中至少有两个是蟾蜍。”
这四只两栖动物中有多少只是青蛙？
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: “Mike and I are different species.”
Chris: “LeRoy is a frog.”
LeRoy: “Chris is a frog.”
Mike: “Of the four of us, at least two are toads.”
How many of these four amphibians are frogs?
小提示：
Chris 和 LeRoy 不可能是同一物种。
Chris and LeRoy cannot have the same species
大提示：
如果 Brian 是蟾蜍，Mike 的陈述会产生矛盾。
If Brian were a toad, Mike’s statement would create a contradiction
解答：
Chris 和 LeRoy 必定属于不同物种：如果 Chris 是蟾蜍，他的话说明 LeRoy 是青蛙；如果 Chris 是青蛙，他这句假话说明 LeRoy 是蟾蜍。因此他们两个中恰好有一只是蟾蜍。
如果 Brian 是蟾蜍，他说的是真话，于是 Mike 是青蛙。如果 Brian 是青蛙，他这句假话意味着他和 Mike 是同一物种，此时 Mike 仍然是青蛙。所以 Mike 一定是青蛙。
因此 Mike 的陈述是假话，也就是说蟾蜍不足两只。而 Chris 和 LeRoy 中已经恰好有一只蟾蜍，所以 Brian 只能是青蛙。于是四只中有一只蟾蜍和 只青蛙。
所以正确答案是 D。
Chris and LeRoy are of opposite species: if Chris is a toad, his statement makes LeRoy a frog; if Chris is a frog, his false statement makes LeRoy a toad. Thus exactly one of them is a toad.
If Brian is a toad, his true statement makes Mike a frog. If Brian is a frog, his false statement says that he and Mike are the same species, again making Mike a frog. Therefore Mike is always a frog.
Mike’s statement is therefore false, so there are fewer than two toads. Chris and LeRoy already supply exactly one toad, forcing Brian to be a frog. Hence there is one toad and frogs.
Thus, D is the correct answer.
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