I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff just so that I can win a few math competitions, and just become a better math student overall.

I'm starting off with inequalities, which seem very prevalent in olympiad math. They seem quite daunting to me, but I guess if I start step by step, I'll improve with a lot of practice.

To start off learning inequalities, I've begun with the AM-GM Inequality, learning it, practising & perfecting it. In addition to this, I check out some external sources like the Cauchy-Schwarz Master Class, which was a book suggested to me by another user on this website. I skimmed through it very quickly, and it turned out to be something I was looking for. It has a section on AM-GM Inequality, which is what I'm starting with, which is nice. I'll be proceeding to Cauchy-Schwarz straight after, I guess.

I have trouble kind of understanding what it is some people do with AM-GM though. I also see many authors and people use AM-GM to solve a particularly daunting inequality which I could NEVER solve. When they explain their logic, and how they may use the AM-GM inequality to solve the problem, hey, it makes sense, it actually is logical and I can kind of appreciate how they did it all. But I don't understand where they get all these ideas and solutions from, I feel like I'm supposed to memorize whatever number of cases there are for AM-GM, and to somehow apply them when I'm faced with another question involving it. I don't know how to become necessarily proficient at it.

Since I'm super new to this, the only thing I'm really doing is reading the Cauchy-Schwarz Master Class, practicing a few of these AM-GM problems on AoPS forums, and looking up YouTube tutorials regarding AM-GM whenever I can.

If someone could help me out with learning AM-GM further, that would be very helpful.

Also, if you guys don't mind-- as a separate super noob question, I must ask,

The author of the Cauchy-Schwarz Master Class takes the inequality,

$xy < (x^2 / 2) + (y^2 / 2)$

Which makes perfect sense to me, and then he says:

Let's replace x and y with their square roots, and then go on from there, which I don't really get.

Would we then get,

$\sqrt{xy} < (x / 2) + (y / 2)$?

They then end up with,

$4\sqrt{xy}< 2x + 2y$,

I see the author relates that inequality to the areas & perimeters of squares and rectangles, which I kind of understand. Could we use that fact to then apply the AM-GM inequality to optimization problems for rectangles/squares?

I don't completely understand how the author got $4sqrt(xy) < 2x + 2y$, I also don't understand how that inequality relates to anything. If I were to guess, I guess I'd say 4 times the square root of the area of the rectangle, will always be less than the perimeter of the square?

I'm in 9th grade and a COMPLETE beginner when it comes to this, so help would be really greatly appreciated. I just need someone to clear up things for me.


  • $\begingroup$ He/she multiplied both sides of the inequality by 4. $\endgroup$ – Hayden Jul 30 '14 at 17:15
  • 3
    $\begingroup$ Can you ask the question up front, rather than burying it in your biography? People need to quickly know if they can be any help to you, and hiding the question doesn't let them do that. $\endgroup$ – Thomas Andrews Jul 30 '14 at 17:21

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