How to Catch Up?

Question

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.

Answer 1

You know actually you'll probably be fine. Solving exercises is a skill you develop over time. But when you get to graduate school in math you'll discover that there are very very few other Americans in the program. And some other countries put very little emphasis on solving exercises. For example, in Italy they are obsessed with having you remember proofs of all the standard theorems in every subject. But they rarely ever collect HW or assign problem sets. The students are just expected to know the theory forwards and backwards and they'll get high marks, even if they aren't all that good at solving exercises. So they're just as scared as you are when they get here and learn they have to be able to solve problems like that cold. So don't sell yourself short, wait until you get to graduate school and you'll find you're probably as good as most everybody else at solving problems and you'll always get better at it. Don't beat yourself up over not finding elegant solutions, hindsight is always 20-20.

Answer 2

listen.

Don't feel bad you couldn't complete one problem. This is not a problem that tests knowledge but rather skill. I have a lot of passion for mathematics and spend my free time reading about it. For this particular problem, try to see why it works.

First, put all a=b=c. Then you can get an idea of when the equality conditions holds. Then try proving the same result for two variables. $1/a + 1/b > 1/sqrt(ab)$. Why is the true ? Now, you see that same inequality in action in rawer terms. Now, you go back to your general question and ask if you can start with $a+b/2 > 1/sqrt(ab)$, and write the same inequality thrice and add it to see if you end up where the question asked. In this case, we do.

In school, I liked mathematics a lot but the teachers weren't good. Also, I didn't have much exposure to things outside the syllabus. But, after entering college I found about exams like the Olympiad. Of course, I was too old to write it but I was just fascinated with questions such as these which require no knowledge of higher mathematics but still test you and are many times insolvable. It tested mathematical skill and not just mathematical knowledge. In school, we learn questions at the back of the chapter and then do them using the methods we learnt. We learn the problems on the context of the method instead of learning the methods in the contexts of the problems.

For example, here is a question. Given a point D inside a triangle ABC, we draw another triangle ADC. Can you prove that AB + BC > AD + DC ? This is a middle school problem, which becomes very difficult when taken out of its context.

I continue to study problems deeply. As people become expert performers, their problem perception improves along with their performance. What that means is if there were three problems and an expert and a novice were asked to classify them into two groups, the novice would probably use superficial characteristics like geometry, numbers, algebra. The expert would use deeper characteristics like proof by contradiction, induction, analogy, etc. An expert would have seen this problem and seen it as a problem dealing with analogies rather than inequalities.

What you need is to learn problem solving. Spend some time with some elementary books which will inspire you again. The Hungarian Problem book is great but it's hard to begin with. Here are some fun books to start with.

Mathematical Circles Mathematical Problem Solving - Alan Schoenfeld (This is more about the psychology of problem solving, than an actual Maths book.)

Answer 3

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).