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| Agnes writes the following four statements on a blank piece of paper.
| | #redirect [[2025 AMC 12A Problems/Problem 4]] |
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| At least one of these statements is true.
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| At least two of these statements are true.
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| At least two of these statements are false.
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| At least one of these statements is false.
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| Each statement is either true or false. How many false statements did Agnes write on the paper?
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| Answer Choices:
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| A. 0
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| B. 1
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| C. 2
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| D. 3
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| E. 4
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| Solution 1 (Process of Elimination):
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| We first number all the statements:
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| 1) At least one of these statements is true.
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| 2) At least two of these statements are true.
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| 3) At least two of these statements are false.
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| 4) At least one of these statements is false.
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| 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, which is established to be false by statement 4. As both 1 and 4 are True, statement 2 is true, meaning there are no false statements. Therefore, statement 3 is the only false statement, making the answer B.
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| -RainsJ
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