Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

Each of Alice, Bob, and Carol is either a consistent truth-teller or a consistent liar. Alice states: "At least one of Bob or Carol is a truth-teller." Bob states: "Alice and Carol are both truth-tellers." Carol states: "If Alice is a truth-teller, so too is Bob." Must they all be truth-tellers?

Solution

Solution:

Yes. If Carol were a liar, Alice would have to be a truth-teller while Bob would have to be a liar. However, Bob would then be telling the truth, a contradiction.

Thus Carol is telling the truth. Alice's statement is then true as well, and thus Bob's statement is also true. Hence, all logicians must be telling the truth.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.