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2025 AMC 12A Problem 4

Problem 4 of 25EasierLogicProblem-Solving Techniques

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?

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Solution

Let TT be the number of true statements. The statements assert T≥1,T \ge 1, T≥2,T \ge 2, T≤2,T \le 2, and T≤3,T \le 3, respectively. Testing T=3T = 3: the conditions T≥1,T\ge1, T≥2,T\ge2, T≤3T\le3 hold (statements one, two, four) and T≤2T\le2 fails (statement three). Exactly 33 statements are true, matching T=3.T = 3. For T=0,1,2,3,4,T=0,1,2,3,4, the respective numbers of true conditions are 2,3,4,3,2,2,3,4,3,2, so no other value is self-consistent. Therefore exactly one statement is false. Thus, the correct answer is B.
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