Skip to main content

2017 AMC 12A Problem 3

Problem 3 of 25Easier

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?

Answer choices

Show solution

Solution

The promise is “all right \Rightarrow A.” An implication is equivalent only to its contrapositive: “not A \Rightarrow 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.

More practice

Concepts: logical deduction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.