Competition problem — Combinatorics

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Combinatorics Difficulty 5.3 AIME, harder Prove it United States

Problem:

There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:

```
Alan: "All of us are truth-tellers."
Bob: "No, only Alan and I are truth-tellers."
Casey: "You are both liars."
Dan: "If Casey is a truth-teller, then Eric is too."
Eric: "An odd number of us are liars."
```
Who are the liars?

Solution

Solution:

Alan, Bob, Dan, and Eric are liars.

Alan and Bob each claim that both of them are telling the truth, but they disagree on the others. Therefore, they must both be liars, and Casey must be a truth-teller. If Dan is a truth-teller, then so is Eric, but then there would only be two truth-tellers, contradicting Eric's claim. Therefore, Dan is a liar, and so is Eric.

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