55. 

One day, Alice met one of the brothers. He stated the following: "I am lying today, and my name is Trulala."

Which of the brothers did Alice meet?
55. The brother's statement is deliberately false (if it were true, the brother would have been lying on the day of the meeting, leading to a contradiction). Therefore, at least one of the two statements, "I am lying today," "My name is Trulala," must be false. The first statement ("I am lying today") is true, so the second statement must be false. Thus, Alice met Tralala.