I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare for it, I'm doing questions on Brilliant.org, Art of Problem Solving forums, as well as frequently checking out the questions and answers on this website since they provide pretty valuable insight into many problems. As a further plan, I do intend on trying to step into extremely high-level math competitions like the IMO in junior or senior year.
I do have a few questions I'd like to ask.
Firstly, could someone point me to useful and valuable resources I could use to prepare, study, etc? Any books, e-books, problems, perhaps any specific authors I should check out?
Next, this question might make me seem very ignorant, but hopefully I'll get some insight into this:
In Canada, in the 9th grade, we've completed the basics of pre-algebra-- things like exponents, scientific notation, rational equations, simplifying radicals, performing various operations with radicals, that type of thing. The pretty easy stuff.
I then check out the questions on math contests and olympiads, and, of course, they're much harder. I notice a certain emphasis on inequality questions, and finding all numbers that satisfy the equation. Questions like these:
For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \ldots, n$. For example, $S(5) = 60$. How many positive integers $n$ with $1 < n < 100$ have $S(n) = S(n + 4)$ ?
Where I have to find all positive integers of $n$, those types of questions confuse me. I realize that yes, I could just use trial and error, but that seems far too inefficient. I realize these types of questions are all over math contests and olympiads, and as such I should prepare for them.
How can I prepare for these types of questions? I was suggested to take a look into the beginning chapters of number theory, and to check out the wiki article on the Art of Problem Solving website involving inequalities, but some of the concepts listed there are far beyond my knowledge and understanding.
Do rules such as the AM-GM rule for inequalities, and number theory intertwine when it comes to inequality questions such as the one listed above?
Thank you in advance. I appreciate it.