2025 AMC 10A Problems/Problem 8
- The following problem is from both the 2025 AMC 10A #8 and 2025 AMC 12A #4, so both problems redirect to this page.
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?
Solution 1
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, which is established to be false by statement 4. As both 1 and 4 are True, statement 2 is true. Therefore, statement 3 is the only false statement, as there are three true statements, making the answer 
.
-Rainjs
Solution 2
Statements 
and 
are true, while statement 
is false. Hence, there are 
true statements and false statement. This result can be checked by examining the statements individually again.
Statements 
and 
will be true because there are 
true statements. Statement is also true because there is 
false statement. Finally, statement is false because there are 
false statements.
~Tacos_are_yummy_1
See Also
| 2025 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by 2024 AMC 10B Problems |
Followed by 2025 AMC 10B Problems | |
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| All AMC 10 Problems and Solutions | ||
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