2025 AMC 10A Problems/Problem 8

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?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

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 $\boxed{\text{(B) }1}$ . -Rainjs

Solution 2

Statements $I,II,$ and $IV$ are true, while statement $III$ is false. Hence, there are $3$ true statements and $\boxed{\text{(B) }1}$ false statement. This result can be checked by examining the statements individually again.

Statements $I$ and $II$ will be true because there are $3\ge2$ true statements. Statement $IV$ is also true because there is $1\ge1$ false statement. Finally, statement $III$ is false because there are $1\ngeq2$ false statements.

~Tacos_are_yummy_1

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2025 AMC 10A Problems/Problem 8

Solution 1

Solution 2