Trouble with inequalities 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.
 A: What I'd personally recommend
AoPS book series followed by
Polynomials by E.J. Barbeau (I loved this book!)
and then practice AMC 8-10-12 exams, AIME exams, USAJMO exams, USAMO exams, MOP/IMO exams
as well as other math contests such as national qualifying tests for other countries and ARML.
Brilliant and AOPS Alcumus are also excellent places to warm up. 
Just get into the habit of taking every problem (especially @ the AIME and higher level) and attempting to generalize it as much as possible and solve it until you basically can't any longer. At that point a site like AoPSForums or Stackexchange is good for posting and garnering more help.
Everyone can practice, and that alone will take you very far. But going that extra mile to try to understand every subtle detail going on can definitely give you a strong competitive edge over a lot of other contestants. Speed is also extremely helpful for passing through lower level competitions such as the AMC 12. If you're familiar with python scripting you can make yourself simple terminal programs that basically shoot at you very simple problems such as "multiply these two numbers" or "compute 10 choose k" which you can practice on to up your speed at simple arithmetic.
Furthermore the terminal program can more complex by asking you to solve quadratic equations, trickier word problems, systems of congruences, etc... It's a great training ground for upping your speed and with practice learning how to handle extremely complex and large expressions mostly in your head.
Between generalizing, terminal practice, and doing the standard practice set you are pretty much good to go IMO. (No pun intended ;))
A: I, too, have a hard time with Olympiad/Putnam (like a college version of Olmpiad math) style questions, but have gotten much better with time.  The first thing I would suggest is to be patient - problem solving is hard and is about improving your fundamental ability to think.
In particular I would suggest you adopt the following "strategies" that I have found useful:
(1) Unless you immediately know how to solve the problem, instead of trying something right off the bat, make observations.
(2) In a similar vein, if you don't know what to do try simple cases.
Note that trying simple cases is one of the best strategies in complicated problems, as it will often lead you to make conjectures that will solve the problem, and at worst will give you a better understanding of what is going on.
A: As someone who's also striving for the IMO: Learning how to google will help you a lot. Despite what's usually advertised, math contests' syllabi can be very extensive so you'll often find yourself googling some new technique you heard about in a solution. Also, the following two books are almost unanimously agreed to be must-reads: "The Art and Craft of Problem Solving" by Paul Zeitz (read this first) and "Problem Solving Strategies" by Arthur Engel (more direct to the point but lots of problems with solutions).
That said: About inequalities, it's good to know AM-GM and Cauchy-Schwarz inequalities by heart. A quick search along the lines "basic olympiad inequalities" will give you lots of study material and practice problems. After that, read the sections on inequalities from the two books I mentioned. If you want to study inequalities more in-depth then the book Semiclassical mentioned will be of great help. I also found this book to be particularly helpful: "Inequalities, A Mathematical Olympiad Approach" (lots of problems and solutions!).
About the problem you posted: Testing for each and every natural number $1<n<100$ is far from being the ideal solution, but testing for the first few numbers is not only permitted but encouraged. If you have no idea on what to tackle a problem, testing a bunch of small cases will give you an idea on how the solution looks like, this will often help you generalize the result using mathematical induction on any other means.
Yes, you should read the first few chapters of a number theory book. This isn't taught in high school but it is one of the four major topics in nearly every math olympiad (the other three being geometry, combinatorics, and algebra). 
There's no standard way to studying for olympiads so you should go with what works best for you. Still, the first paragraph is the best advice I can give you, other than the obvious one: Practice, practice, practice, practice... Good luck and have fun. (:
