出难题 is a separable verb (离合词) that literally means 'to put out a difficult problem/task'. In everyday use, it can refer to setting a challenging question (e.g. in class) or, more figuratively, to deliberately giving someone a hard time or creating obstacles for them. The splitting pattern is: 出 + measure word (道/个/etc.) + 难题 — for example, 出了一道难题 ('set a difficult problem'). At least one of your example sentences should show this separated form.