I wonder why topics examined in high school math contests are so different from the maths learned by those who are seriously studying a math major at a university. Firstly, contests like IMO, ARML, AMC and most of the others seem to focus on a very small area of math (graph theory, combinatorics, elementary number theory, planar geometry, etc). Topics in analysis (which include several courses like complex analysis and functional analysis), topics in mathematical physics (quantum mechanics, electromagnetism), and more advanced algebra (rings, modules) are almost completely ignored. I cannot see any obvious reasons why those important topics should be omitted -- it is true that some mathematicians specialise in small fields like combinatorics, but they are definitely not the majority. Most mathematicians are specialists in more "modern" math, like the ones I listed above (analysis, physics, algebraic and analytic number theory, etc).
You can see six or seven extra lines that must be added before one can answer the question. Such questions involve a lot of small tricks that one hardly ever needs in math research. In fact, nobody needs to know how to do this question unless they are a contest math teacher. Yes, one can really dig very deep into planar geometry, and there is beauty in it. But contest-style geometry is very "ancient", and is no longer the focus of research nowadays. Today, computers can already do such geometric proofs in a much more rigorous way than people in the contest. It doesn't seem to be a good idea to play with ancient things too much -- this IMO question is just like making a horse run faster than an airplane.
Another "ancient" technique commonly seen in contests is to construct inequalities without using calculus. It used to be part of the math course at university 100 years ago, but now it is no longer a topic that must be studied -- students simply learn those inequalities when necessary. However, such inequalities are still a large part of the contest.
So, that leads to the question: what it the reason why there are almost no high school math contests that are even close to the style of math at universities?
This might be too opinion based, but note that I am asking "why", not "what should we do", so I believe one can write a very objective answer to this question.
Also, don't tell me that is because degree level maths is too hard -- clearly that IMO question above is much harder than math at any universities. In fact, many PhD students of leading math departments cannot do such questions.
PS: I know that students learn a little bit of degree level math while training for IMO or other competitions, but their knowledge about those more advanced math is likely to be fragmented and incomplete -- they are likely to end up with false impressions about math at universities.