> I have no idea what you are saying. Sorry not to be clear. My view of efforts in grades 8-12 or so to have students do well in math competitions, e.g., the

> I have no idea what you are saying. Sorry not to be clear. My view of efforts in grades 8-12 or so to have students do well in math competitions, e.g., the Math Olympiad, is that those efforts are not promising for helping those students do well with math or anything else later in life. Instead, mostly the efforts are just wasteful for all concerned. The main reason for the waste is that the people, usually the high school math teachers, running the programs don't know enough math to give good direction to the students that might want to do especially well in math, well enough that 30 years later we would commonly see significant positive effects from that direction and the resulting efforts of the students. So, the direction that is commonly given is to have the students study relatively useless material, say, recreational math, some puzzle problems, every tricky thing they can see in Pascal's triangle (the binomial coefficients, that is, the coefficients of (x + y)^n with one row for each value of positive integer n), and there are a lot of such tricky things. But, right, the binomial coefficients have some nice properties, some of which are good to know, and an early volume of D. Knuth's The Art of Computer Programming has some good coverage. But, net, mostly to heck with the triangle and, if insist, then read the relevant sections of Knuth -- but even there Knuth has a good reason for presenting that material, and what's important is the applications Knuth makes, so, really, just read on in Knuth. So, instead, students who do want to do better in math, and I did, should just proceed along the usual topics, texts, etc. for a ugrad math major. In particular, as a first goal, ASAP get a good college calculus text, get the basic prerequisites, and dig in. By the way, the time I looked at the AP calculus materials I didn't like them and concluded that the people who wrote the materials didn't understand calculus very well. Instead, just get a good college calculus text. For the rest I mentioned, about have to have a relatively applied ugrad math major to understand: So, would need to know some linear programming, network flows, optimization, integer programming, something about the question of P versus NP, and more. For Lagrangian relaxation, that's a bit tricky and specialized and not commonly well known even among college math profs. I did give an outline in https://news.ycombinator.com/item?id=8919311 https://news.ycombinator.com/item?id=8919311 but it's not really easy reading. But it's a cute topic with some a good intuitive views, a bit amazing in some ways, and at times shockingly powerful. E.g., the guy who first did the many worlds interpretation of quantum mechanics, a Ph.D. physics student of J. Wheeler at Princeton, an expert in relativity theory, was H. Everett, and at one time he had a nice applied math shop near DC doing resource allocation for the US DoD using a first, simple version of Lagrangian relaxation. What I outlined in https://news.ycombinator.com/item?id=8919311 https://news.ycombinator.com/item?id=8919311 is more general and powerful and makes some cute use of some convexity -- that there is convexity there is also surprising but true and powerful and, with the right setup, easy to prove.