1999 AMC 12 Problem 10
Below is the professionally curated solution for Problem 10 of the 1999 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.
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Difficulty rating: 1370
10.
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.
I. The digit is
II. The digit is not
III. The digit is
IV. The digit is not
Which one of the following must necessarily be correct?
I is true.
I is false.
II is true.
III is true.
IV is false.
Solution:
Statements I and III cannot both be true, so the single false statement is one of them. Therefore statements II and IV are both true, which makes "II is true" necessarily correct.
The digit is thus or If it were then (B) and (D) are false; if it were then (A) is false; and (E) is always incorrect. Only (C) is guaranteed.
Thus, the correct answer is C.
Problem 10 in Other Years
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