In this post I am going to talk mainly about how to prepare for Math Olympiad. I am going to discuss two different approaches. At the bottom of this post are two videos I created with more information on me, Math Olympiad prep, and my online tutoring services.
The first method is to start off with solving Olympiad-type questions. You can, for example, find past IMO exams here and see how well you can do. There are many other websites and books containing Olympiad-type problems and solutions. In this method I'd suggest you start with old exams. They tend to be much easier. In the process, you will find out which subjects you need to work on. Try to spend at least 2 hours on each problem before looking up their solutions. You will most probably not be able to solve many of the problems and you will find some subjects in the solutions that you might not be familiar with. Then you can work on those subjects and theorems. This method is good if you feel confident that you know the subjects in Math Olympiad fairly well or you are at least fairly comfortable with the materials. Otherwise this method won't work for you. Many solutions to Olympiad problems have similar ideas. So the more problems you solve the more experienced you get.
The second method is to try to learn all subjects well before starting to solve any problems. This method is a lot more time consuming and I'd suggest it to 10th or 11th grade students who have enough time to prepare for Olympiad. The advantage of this method is that what you learn not only helps you with Olympiad preparation, but it is also helpful for you in the future as it gives you a better understanding and insight of Math. Also this method ensures you that you have covered all the materials needed for Math Olympiad. In contrast the first method makes you a problem solver which is very good and may be enough to be successful in Math Olympiad. However it may not give you a good mathematician or may not give you a good understanding and insight of Math.
These two general approaches remain valid for Putnam preparation or any similar exams.
In the end, if you want to be successful you need to do A LOT of problems. Even if you find a solution to a problem, look at the solutions and try to understand all different solutions in case there are more than one solutions. Each solution gives you a new method. Have all these methods in mind as they are going to help you some day in solving a problem.
Try to keep a few latest official exams from IMO, USAMO or similar exams untouched. When you are a few weeks away from you official exam give yourself 9 hours and solve these exams separately. This would help you understand how you are doing. However, keep in mind that you won't necessarily perform the same in the actual exam.
I enjoy working with students to help them prepare for advanced mathematics competitions. The first video below presents my approach towards mentoring students. The second video discusses more my credentials and then demonstrates how I work with students online using Tutor Hero's professional technology for 1-on-1 tutoring. Click here if you'd like to sign-up for a tutoring session with me.