AMC 10A/12A 2025 (Problem 8/4)
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?
• 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?
Use casework based on the second statement.
Case 1: The second statement is true.
Then the first statement must also be true.
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.
Then the first statement must also be true.
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.
Case 1-1: The third statement is true.
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.
Then, we can have at most one false statement which contradicts the truth of the third statement.
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.
Then, we can have at most one false statement which contradicts the truth of the third statement.
Case 1-2: The third statement is false.
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.
Therefore, the fourth statement must be true.
So, we found one valid case.
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.
Therefore, the fourth statement must be true.
So, we found one valid case.
Case 2: The second statement is false.
Then the fourth statement is true, and the first statement is also true.
However, both possibilities for the third statement contradict themselves.
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.
Then the fourth statement is true, and the first statement is also true.
However, both possibilities for the third statement contradict themselves.
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.
(B)
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