They generally require extensive mathematical knowledge and a great deal of ingenuity to solve. Additionally, much of the knowledge needed to tackle these problems is excluded from standard curriculum so you would have to self-study to a large extent to get proficient at them. I wouldn’t get too disgruntled if I were you if you can’t solve them.

Here’s what the Mathematical Assosiation of America says about the olympiad they put out every year for high school students (the USAMO): “Problems would challenge most professional mathematicians. In addition to comprehensive mathematical knowledge, success on the USAMO requires truly exceptional mathematical creativity and inventiveness.”

I would think that most of problem writers for olympiad questions must have taken and excelled at olympiads themselves when they were students.

Also, if you are looking to get better at contest problems in general check out the website. It’s a great resource. Good luck!

It`s because they are generated by people with years of experience at maths. Some that I have seen have been old matriculation questions, and some from old Oxbridge papers.
I`ve often wondered at the perversity of some of the questions too!

They are based on school programs. But they shouldn’t require some advanced(university) theory because different schools or countries have different school programs. So they look elementary. On the other hand, they are not at all elementary. The olympiad problems must not be routine problems. They are creative and make the solver think out of the box, the routine computational part being at most 50% of the problem.
To mark olympiad problems is not an easy task because there is no standard proof. One problem can have different solutions.
The disadvantage of the olymipad proofs is that they are short enough to be solved in 4 hours. But mathematics is not done in rush usually.

p.s. I guess even if they provoke the imagination, knowing different solving methods, a thouroughful knowledge of theory,
lots of practice, professors’ help, passion for math will help you being good at olympiads.

Its difficult because you need to know how to do every step in the problem correctly, and also you need to be able to do the problem faster then the person your competing against.

Math olympiad questions are about hte hardest you’ll get in math. Sometimes I get them, sometimes I don’t. Most of the times actually I don’t.

I think people who are doing research stumble upon certain ideas and concepts while working on some other problem. Then they use those ideas to formulate math questions. I might be wrong, but I think that’s how they compile the questions.

The problems require tons of steps to do. You need an advanced knowledge of math to complete them.