2025 AMC 12A Problem 4

Question

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?

0

1

2

3

4

Let

T

be the number of true statements; the four claims say

T\ge1,

T\ge2,

T\le2,

T\le3

Find the value of

T

for which exactly

T

of the four conditions hold

Let

T

be the number of true statements. The statements assert

T \ge 1,

T \ge 2,

T \le 2,

and

T \le 3,

respectively.

Testing

T = 3

: the conditions

T\ge1,

T\ge2,

T\le3

hold (statements one, two, four) and

T\le2

fails (statement three). Exactly

3

statements are true, matching

T = 3.

For

T=0,1,2,3,4,

the respective numbers of true conditions are

2,3,4,3,2,

so no other value is self-consistent. Therefore exactly one statement is false.

Thus, the correct answer is B.

Problem 4 in other years