2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 8 Easy

Agnes writes the following four statements on a blank piece of paper.

\cdot At least one of these statements is true.

\cdot At least two of these statements are true.

\cdot At least two of these statements are false.

\cdot 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?

  • A.

    0

  • B.

    1

  • C.

    2

  • D.

    3

  • E.

    4

Answer:B

Denote the four statements as s_1, s_2, s_3, s_4. Let T and F represent the counts of true and false statements respectively. We know T + F = 4.

Assume s_1 is false. This implies T = 0, F = 4, which would make s_4 false—a contradiction!

Since at least one of s_1, s_3 (or pick s_4) is true, at least one of s_2, s_4 (or pick s_3) is true, we have T \geq 2, confirming s_1, s_2 are true.

If s_4 were false, then T = 4, F = 0, contradicting s_4's falsity!

Hence, s_4 is true, yielding T \geq 3, F \leq 1, which implies s_3 is false, with T = 3, F = 1.

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