2017 AMC 12A Problem 3
Below is the professionally curated solution for Problem 3 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.
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Difficulty rating: 1100
3.
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
If Lewis received an A, then he got all of the multiple choice questions right.
If Lewis received an A, then he got at least one of the multiple choice questions right.
Solution:
The promise is "all right A." An implication is equivalent only to its contrapositive: "not A not all right."
"Not all right" means at least one question was wrong, which is exactly statement B. The converse and inverse do not follow, and getting "all wrong" is a much stronger claim than the negation.
Thus, the correct answer is B.
Problem 3 in Other Years
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