2025 AMC 12A Problems/Problem 4

The following problem is from both the 2025 AMC 10A #8 and 2025 AMC 12A #4, so both problems redirect to this page.

Problem

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 false, 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

Solution 5 (Quick)

I tried setting that the statements regarding the true ones (I and II) were true and regarding the false ones (III and IV) were false, and it clearly doesn't work. It doesn't work because statement III was true, not false as I set it to be originally.

What if we make that statement (Statement III) true too? Well, then we have 3 T and 1 F statements, so statement IV is false, statement III is true, and statements I and II are true too.

So there is $\boxed{\text{(B) }1}$ false statement.

~Aarav22

Video Solution by Power Solve

https://youtu.be/QBn439idcPo?si=DGqtuDIJ399xE_rh&t=524

Chinese Video Solution

https://www.bilibili.com/video/BV1t72uBREof/

~metrixgo

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=dAeyV60Hu5c

Video Solution (In 1 Min)

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

Video Solution by Daily Dose of Math

https://youtu.be/gPh9w3X3QSw

~Thesmartgreekmathdude

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

2025 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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 12 Problems and Solutions

Page

Toolbox

Search