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2025 AMC 10A Problems/Problem 8: Difference between revisions

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Now, if Statement <imath>III</imath> is true, then both Statement <imath>III</imath> and Statement <imath>IV</imath> is false, contradicting the fact that it is true. Statement <imath>III</imath> is hence false and Statement <imath>IV</imath> tells the truth since Statement <imath>III</imath> lied so indeed, there are at least one lie. There are a total of 3 truths and 1 lie, making the answer <imath>\boxed{\text{(B) }1}</imath>. ~hxve
Now, if Statement <imath>III</imath> is true, then both Statement <imath>III</imath> and Statement <imath>IV</imath> is false, contradicting the fact that it is true. Statement <imath>III</imath> is hence false and Statement <imath>IV</imath> tells the truth since Statement <imath>III</imath> lied so indeed, there are at least one lie. There are a total of 3 truths and 1 lie, making the answer <imath>\boxed{\text{(B) }1}</imath>. ~hxve
==Video Solution==
== Video Solution (In 1 Min) ==
https://youtu.be/gWSZeCKrOfU
https://youtu.be/uv3uIMwIkrg?si=XCbsXL7ikMawCGyM ~ Pi Academy
 
~MK


==See Also==
==See Also==

Revision as of 18:28, 6 November 2025

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.

$\bullet$ 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 if it were true, 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 $\boxed{\text{(B) }1}$. -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

Solution 3

Let's say there are $T$ true statements. We know that $T$ can be any integer from $0$ to $4$. We denote $A$ as $T \geq 1$, Statement $B$ as $T \geq 2$, Statement $C$ as $T \leq 2$, and Statement $D$ as $T \leq 3$.

If $T=0$, then $C$ and $D$ are met, so there are $2$ true statements, which is a contradiction.

If $T=1$, then $A,C,D$ are met, so there are $3$ true statements, which is a contradiction.

If $T=2$, then $A,B,C,D$ are met, so there are $4$ true statements, which is a contradiction.

If $T=3$, then $A,B,D$ are met, so there are $3$ true statements, which is consistent with our assumption that $T=3$.

If $T=4$, then $A,B$ are met, so there are $2$ true statements, which is a contradiction.

Only $T=3$ was consistent, so there are $3$ true statements and $4-3=\boxed{1}$ false statement. (In particular, Statement C is the false statement).

~lprado

Solution 4

Suppose Statement $I$ is false, then none of the Statements are true, which contradicts the fact that a false Statement $III$ or $IV$ is telling the truth. Therefore, Statement $I$ is true and assume Statement $II$ is false.

Statement $II$ thus implies that only Statement $I$ was the truth, and the rest, false. But then, there are 3 false statements but then Statement $III$ and Statement $IV$ are telling the truth. So Statement $II$ is also true.

Now, if Statement $III$ is true, then both Statement $III$ and Statement $IV$ is false, contradicting the fact that it is true. Statement $III$ is hence false and Statement $IV$ tells the truth since Statement $III$ lied so indeed, there are at least one lie. There are a total of 3 truths and 1 lie, making the answer $\boxed{\text{(B) }1}$. ~hxve

Video Solution (In 1 Min)

https://youtu.be/uv3uIMwIkrg?si=XCbsXL7ikMawCGyM ~ Pi Academy

See Also

2025 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

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