In mathematics, there are certain open questions which are the subject of current research (think of the newest papers in the top journals). I have also heard that some graduate programs ask questions such as these on their qualifying exam. How common is this practice? What kinds of questions can I expect to see on a qualifying exam? I know that a qualifying exam typically tests the knowledge gained in graduate school, but I'm not sure to what extent this knowledge is tested? How "advanced" should I expect these questions to be?