Here you will find a miscellaneous collection of problems that require some amount of insight and problem solving skills. The difficulty level varies quite a bit - and not all of them are at the level of mathematics contest questions - although certainly some are. None of these are past IB exam questions - and it is very unlikely that any Maths HL or SL exam question will look like any of these questions. There is no best description for problems like the ones I've included here - I just call them Challenge Problems.
I consider these to be true 'problems' rather than simply 'exercises'. For more on that and more on my Challenge Problems, see my blog post for 10 April 2016 - Challenge Problems (#10). Also, see my growing list of Problems of the Day - one for Maths SL and one for Maths HL (getting close to 400 total problems as of Sept 2019).
I always find it useful to have a set of more challenging problems easily available which can help in a variety of situations. In my experience these situations may include: (1) a problem to give to the students who finish an in-class assessment much faster than most of the rest of the class; (2) problems for students who need to develop their problem solving skills; (3) a problem that requires fairly basic math (for an HL or SL student) but requires some clear thinking and strategizing; (4) problems that members of a math team can use for practice; (5) for a student who is acting bored in class. Of course, challenge problems like these can be used as short assessments on their own.
So, take a look through my growing set of Challenge Problems (11 problems as of Sept 2019) - there are links to each in the left sidebar. There is a separate page for each Challenge Problem which presents the problem and also includes a link to a downloadable document containing the question and a solution (on 2nd page). Some solutions can be viewed on the page itself.