Bit late but I didn't notice this post. This article is a good summary+motivation for the topic; read it and then check the references that spark your interest: https://plato.stanford.edu/entries/philosophy-mathematics/>>10490034
The IMO doesn't assume a lot of technical knowledge: just some algebra, geometry and trigonometry, inequalities and combinatiorics at a basic level. You could probably understand the problems without any previous study, though their solutions may not be at all obvious (that's the whole fun in mathematics for some). My advice here would be to not worry too much and just go ahead and try your hand at the problems. All previous problems are available at the IMO's official webpage: https://www.imo-official.org/problems.aspx
It often helps to read solved problems and learn specific strategies and "tricks". For this, there's tons of free online resources (e.g. www.imomath.com
), and tons of books as well. Two such books I'd recommend are Andreescu and Enescu's "Mathematical Olympiad Treasures" and Gelca and Andreescu's "Putnam and Beyond" -- the latter is a bit more advanced since the Putnam is a competition directed at math undergraduates and not HS students. The /sci/ wiki has some additional recommendations: https://4chan-science.fandom.com/wiki/Mathematics#Problem_books>>10485862
Suppose there exist (or or whatever field/ring your course wants you to work in) such that (e.g.) . As >>10491568
noted, this is equivalent to solving the following matrix equation (verify this yourself through matrix multiplication):
If this system of equations has a solution, then you're done. If it doesn't, then you've proved (by contradiction) that no such linear combination exists.