There are three people A, B, and C. Each of them is either strictly a truth-teller or strictly a lie-teller.

A man walks up to them and asks A which he is. The man gets distracted and does not hear A’s answer. He then asks B, “What did A say?”. B says “A said he is a liar”. C then yells “Don’t trust B! He’s lying!”.

What type of person is B?
What type of person is C?

B is a liar and C is a truth-teller.

No one can say they are a liar: a truth-teller would be lying if they said they were a liar, and a liar would be telling the truth if they said they were a liar. Therefore, A could not have said he is a liar. Therefore B lied about what A said and is a liar. C must be a truth-teller because he disagrees with B.