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Problem 1104

AIME late

Combinatorics Difficulty 5.0 Prove it Berkeley Math Circle: Monthly Contest 6 · United States

On Predictable Island, everyone is either a liar (who always lies) or a truth-teller (who always tells the truth). You encounter Aerith who says, "Bob and I are both liars." What are they actually?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

If Aerith was a truth-teller, she would be lying, because it wouldn't be the case that they're both liars. Thus, she must be a liar. Therefore, the statement must be false, which means Bob has to be a truth-teller.

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