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?

$0$

$1$

$2$

$3$

$4$

Answer: B

Concepts:logical deduction casework

Difficulty rating: 1200

Solution:

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.

2025 AMC 12A Problem 4

Answer: B

Difficulty rating: 1200

Solution:

Problem 4 in Other Years