TITLE: How to Catch Up?
QUESTION [7 upvotes]: I am finishing up my bachelor's degree in mathematics at the University of North Florida, and I plan on going to graduate school, but I feel very behind.  One of my professor's gave us this problem:
$\frac 1a + \frac 1b + \frac 1c \ge \frac 1 { \sqrt{bc}} + \frac 1 { \sqrt{ac}} + \frac 1 { \sqrt{ab}}$
I had no idea how to solve this, and he said it was on his entrance exam for his university in Lebanon.  He showed us how to solve it using the geometric mean equation, and it was obvious after that, but why is it that we didn't immediately think of that?
This is just one example of me missing a problem he would consider elementary, and I'm wondering why that is.  Is the content and way of thinking so much different in other countries than in America?  We really don't bother with proofs or logic until undergraduate school.  I'm fine with understanding proofs, but I feel like I lack this mental database of knowledge and experience that other students have far before undergrad.
Is there a good resource for me to fill in these holes?  I want to be able to identify these similarities and common patterns that will allow me to solve problems easily, but I'm not sure what knowledge that I'm missing.

REPLY [3 votes]: I don't think the majority of high school students in any country would think that's an easy problem. 
However, there are differences between the U.S. and a number of other countries that do go some way towards explaining why something like that might be on a university entrance exam in other countries:

Some countries have separate "elite" streams for the most able students. This is particularly the case where universities have very competitive entrance exams.
Some countries are better able than others to attract competent and intelligent people to teach in elementary and secondary schools. This is partly related to the social status of teachers in society; in China, to take an extreme example, surveys have shown that the public views teachers as being on par with doctors in terms of status. Contrast this with most of Europe and the U.S., where many schools have serious discipline problems, in part because students (and their parents) don't respect teachers.
Related to the last point, teachers in the U.S., starting at the elementary level and continuing into high school, tend not to have the requisite knowledge to teach math in a way that emphasizes justification and proof rather than just algorithms. This is very clearly demonstrated in Liping Ma's comparative study of math teachers in the U.S. and in China, Knowing and Teaching Elementary Mathematics. As a result, students are inculcated with the attitude that math is a collection of techniques to solve routine problems, and this continues all the way to college, perpetuating the cycle in the next generation of teachers. The way American textbooks are written tends to exacerbate the problem, if anything. 

With regard to your question about what to do now, the answer really depends on whether you're most interested in catching up on this sort of "elite" high-school-level material, or on doing the same for undergraduate-level stuff. Seeing as you're going to graduate school, I would think the latter would be more of a priority. 
If you're curious about high-school-level problems, what Dave Renfro recommended in his comments seems like a good idea. You could also have a look at Problems in Elementary Mathematics by Lidsky or A Problem Book in Algebra by Krechmar. (These were classics for high school students preparing for the hardest university entrance exams in the Soviet Union.) 
On undergraduate material, where your focus should be depends very much on your present level of skill in calculus/analysis/algebra. If you find many of the questions on the GRE math subject test difficult, you might consider reading a book like Apostol's Calculus (Vol. 1) or Spivak's Calculus, skipping parts you know well, followed by a multivariable book like Apostol's second volume. For harder problems in undergraduate analysis, besides the standard textbooks (e.g. Rudin, Apostol, Zorich), you could consider the problem book by Makarov/Goluzina or the ones by Kaczor and Nowak. In algebra, again apart from the standard textbooks (e.g. Artin, Jacobson, Dummit/Foote, Godement), there are the problem books by Proskuryakov (linear algebra) and Faddeev/Sominski.
If you can read French, then there are a number of books with problems that are at a sort of "elite" level in undergraduate analysis/algebra (for entrance exams to the top engineering and science schools in France). The five-volume series by Ramis/Deschamps/Odoux and the four-volume one by Arnaudiès/Fraysse are the best known textbooks, but there are also problem books by Leichtnam/Schauer and another set by Francinou/Gianella (Oraux X ENS).