Question

Agnes 在一张空白纸上写下下面四个陈述。

• 这些陈述中至少有一个是真的。

• 这些陈述中至少有两个是真的。

• 这些陈述中至少有两个是假的。

• 这些陈述中至少有一个是假的。

每个陈述要么真，要么假。Agnes 在纸上写了多少个假陈述？

Agnes writes the following four statements on a blank piece of paper.

• At least one of these statements is true.

• At least two of these statements are true.

• At least two of these statements are false.

• At least one of these statements is false.

Each statement is either true or false. How many false statements did Agnes write on the paper?

$0$

$1$

$2$

$3$

$4$

答案：B

知识点：逻辑推理分类讨论

难度评级：1350

解答：

给四个陈述编号：一是至少一个真，二是至少两个真，三是至少两个假，四是至少一个假。假设第三个陈述为真，则至少有两个陈述为假。但这时第一、二、四个陈述都会是真的，因此最多只有一个假陈述，矛盾。所以第三个陈述必须为假。现在第一、二、四个陈述都为真，而且这与只有一个假陈述完全一致。所以恰好有 $1$ 个陈述是假的。因此正确答案是 B。

Number them: (1) at least one true, (2) at least two true, (3) at least two false, (4) at least one false. Suppose (3) is true. Then at least two statements are false. But then (1), (2), and (4) all read as true, which leaves at most one false statement. That's a contradiction, so (3) must be false. Now (1), (2), and (4) are all true, and each matches reality with just one false statement. So exactly $1$ statement is false. Therefore, the answer is B.

2025 AMC 10A 第 8 题

答案：B

难度评级：1350

解答：

其他年份的第 8 题