I'm currently attempting to consolidate my failing foundation in maths from ground zero upwards. I want to be able to do olympiad problems but the books that I'm learning from don't teach me how to solve the problems brought up in famous olympiads like the IMO even though it's from the basic level of algebra or integers. It's frustrating to me because I'm unsure of whether the way I've been learning is inadequate or rather the books I'm learning from don't address the techniques and proofs used for that level since not many people care to learn maths to such heights.
Furthermore, as result of this I've tried consulting proofs but find that I don't understand it because of the confusing symbols which I neither know how they are called or read aloud. If I knew what the symbols mean and how the statement is read it's easy for me to understand its meaning. An example would be from this book called Elementary Number Theory by Underwood Dudley on page 3:
Lemma 2. Ifd l al, dla!, .. . ,dla", then dl(clal + c2a2 + . . . +c"a,,) for any integers Cl, C2 , ••• , CII' Proof. From the definition, there are integers qI> q2,' .. ,qll such that al = dql, a2 = dq2' . .. , a" = dq". Thus clal + cza: + ... + cna" = d(Clql + C2Q! + ... + C.,.Q,,) , and from the definition again, dlcla] + CZa2 + ... + c"a".
if your purpose is to be able to solve Olympiad problems, then you should look for books on problem-solving, there are a lot of books that fall under that category, you should pay attention to the books that are specifically written for the purpose of math competitions, I can think of Terence Tao's book "Solving mathematical problems", there are others as well, now if you want to learn number theory for the sole purpose of solving Olympiad problems, then I would say that you are in the wrong direction, number theory is fascinating for its own sake, if you are interested in number theory, then don't worry about Olympiad problems so much, for number theory books I can recommend "Number theory through inquiry" and "Nuggets of number theory, a visual approach", both of those books are suitable for an introduction to proof based mathematics and explains the topic very well.