AMC10 2025 A
AMC10 2025 A · Q8
AMC10 2025 A · Q8. It mainly tests Logic puzzles.
Agnes writes the following four statements on a blank piece of paper.
$\bullet$ At least one of these statements is true.
$\bullet$ At least two of these statements are true.
$\bullet$ At least two of these statements are false.
$\bullet$ 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?
阿格尼斯在一张白纸上写下了以下四个陈述。
$\bullet$ 这些陈述中至少有一个是真命题。
$\bullet$ 这些陈述中至少有两个是真命题。
$\bullet$ 这些陈述中至少有两个是假命题。
$\bullet$ 这些陈述中至少有一个是假命题。
每个陈述要么真要么假。阿格尼斯写了多少个假陈述？
(A)
0
0
(B)
1
1
(C)
2
2
(D)
3
3
(E)
4
4
Answer
Correct choice: (B)
正确答案：（B）
Solution
We first number all the statements:
1) At least one of these statements is true.
2) At least two of these statements are true.
3) At least two of these statements are false.
4) At least one of these statements is false.
We can immediately see that statement 4 must be true, as it would contradict itself if it were false. Similarly, statement 1 must be true, as all the other statements must be false if it were false, which is contradictory because statement 4 is true. Since both 1 and 4 are true, statement 2 has to be true. Therefore, statement 3 is the only false statement, making the answer $\boxed{\text{(B) }1}$.
我们首先给所有陈述编号：
1) 这些陈述中至少有一个是真命题。
2) 这些陈述中至少有两个是真命题。
3) 这些陈述中至少有两个是假命题。
4) 这些陈述中至少有一个是假命题。
我们立即可以看到陈述4必须为真，因为如果它是假的就会自相矛盾。类似地，陈述1必须为真，因为如果它是假的，那么其他所有陈述必须为假，这与陈述4为真相矛盾。由于1和4都是真的，陈述2必须为真。因此，陈述3是唯一的假陈述，答案是$\boxed{\text{(B) }1}$。
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