Discussion about Landau's "Theoretical Minimum" has already been posted here. Unfortunately I couldn't find much about some examples of questions he gave to students. There are three questions in the post I linked and also in the linked arXiv article.
What I am mostly interested in are math exercises.
There is a history of Alexei Abrikosov who after presenting himself as a 3rd year student to Landau was asked if he had a pen and then was given an integral to solve. He managed to solve it and Landau was very happy :).
There are many other such anecdotes but they all lack exercise formulation. Maybe you know what this integral could be or at least of what type it was?