113. 

Regarding A and B, it is known that each of them is either a knight or a liar. A states: "If B is a knight, then I am a liar." Who is A and who is B?
113. A must be a knight, and B must be a liar.

First, let's prove that only a knight can make a statement of the form "If $P$, then I am a liar." Let's recall that a true statement follows from any statement. Therefore, if the statement "I am a liar" is true, then the entire statement "If $P$, then I am a liar" is also true. But if I am a liar, then no true statement could have come from me. Therefore, by making the statement "If $P$, then I am a liar," I must be a knight.

Thus, A must be a knight. Consequently, it is also true that if B is a knight, then A is a liar (because A insists on the truth of this statement). Therefore, B cannot be a knight, for if B were a knight, A would have to be a liar, which he is not. Therefore, B is a liar.