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Combinatorics and Geometry

What is the reason Geometry and Combinatorics is a combined problem set? I have heard about combinatorial geometry, but the problems I've seen in that set are either combinatorics or geometry, not both, it seems.

My combinatorics is quite good, which is what brought me to level 5, but by geometry is definitely not there yet, but at about level 2 to 3, rather.

Edit, I forgot to mention: If the olympiad math sections would be split into four different sections, then they could have four problems each (while preserving the same amount of problems), which is more in line with physics and computer science.

Note by Tim Vermeulen
4 years, 1 month ago

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There are numerous considerations involved, in determining how the sections are decided.

One main consideration is the amount of knowledge that high school students would accumulate as they progress through their school systems. In general, geometry and combinatorics tend to be poorly developed concepts in several countries, with many schools only dealing with them in Grade 10-12 (if at all). For example, even the concept of sine rule and cosine rule are often treated as part of trigonometric functions which are covered under Algebra. Schools rarely develop much more application of these basic geometric concepts e.g. Heron's formula is denoted as a textbook * problem.

Another consideration is the type, and relevance of questions that we can ask. Sad to say, pure Euclidean geometry does not have much applications. There are almost no further theory that I could allude to, as opposed to concepts in Algebra, Number Theory and Combinatorics. The 'university equivalent' is Topology, but this does not go well with the high school students, who do not see the relevance to Euclidean geometry (because there is little).

The math olympiad world does love geometry problems, in part because geometry proofs require very rigorous demonstrations with little ambiguity (e.g. at IMO they deduct 1 point if you do not explain why a defined point is inside or outside of a triangle). However, in fitting with our numerical answer system, it becomes next to impossible to require exact proofs of such statements. As you pointed out in another post, there are questions which could be approached by making various assumptions which are not indicated in the question, in order to get the 'short-cut' to the answer. Furthermore, accurate diagrams could be drawn, in order to get estimates of lengths and angles in the problem, without requiring much work. This is obvious in solutions where students say "Since \(\theta = 31.4159 ^ \circ \), hence \(\cos \theta = \ldots \)".As such, these problems are much less a reflection of your problem solving ability, as opposed to your "crunching" ability.

Because of this, there are much fewer geometry problems that we can ask at the higher levels. Most of our geometry problems are capped at 200 points, and even then they have a disproportionate percentage of correct answers with little understanding.

We have considered splitting out the sections. However, even if we do this, you will not get significantly hard geometry problems, and almost certainly not comparable to the current 300 points questions. Calvin Lin Staff · 4 years, 1 month ago

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@Calvin Lin I agree that geometry and combinatorics are barely dealt with on high school. Geometry was taught starting in grade 9 (although that was little more than sine and cosine, we started proof-writing in grade 11) and combinatorics is (in my school) only part of an optional math course (in which is dealt with probability, combinatorics and just a little bit of number theory). Speaking of number theory, prime factors were only mentioned once or twice, the only thing we spent a considerable amount of time on was determining the gcd of two numbers, manually. Ugh. Algebra seems to be the only topic in olympiad maths that has been covered in high school curriculums, but not anyting like AM-GM or problems where all functions satisfying certain conditions have to be found. So, at least for me, nothing (apart from the basics of math) I learnt at high school seems to help in the olympiads, not just geometry and combinatorics.

It is a shame that geometry problems are so easy to answer on this website, without knowing how to propery solve the problem. Many users will probably be stimulated to draw an accurate diagram and just measure, because they do not know how to solve it otherwise, because they are assigned too difficult problems. So maybe splitting the geometry and combinatorics section would make sure that users will have the right level for both categories, so they can actually solve the problems without just measuring it. I never measure in order to find a solution, but rather I just don't submit anything when I don't know how to solve it, which has the same effect: the user doesn't learn anything because the problems are too difficult.

I think having to prove something in geometry (or in fact in any categorie) is often much nicer than having to give a numerical answer, but the latter is the only way in which Brilliant (in its current form) could check it. However, the problems are far from boring, as there are many underlying concepts involved that are in fact very interesting. It is a shame that users can get around those concepts by just measuring, but that does not just apply to geometry: I've seen many problems in the other categories where a computer could solve a problem immediately, or do at least most of the work. I guess you just have to trust your users, and think that if users cheat like that, all they do is lie to theirselves.

Although the title of this post suggests otherwise, I don't really understand why Algebra and Number Theory are combined as well: sure, they're both about numbers, but I think that's about it. I would say that Number Theory and Combinatorics would be more similar, as they both are about the study of natural numbers, which involves divisibility, which in turn is part of both Number Theory and Combinatorics. Tim Vermeulen · 4 years, 1 month ago

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Algebra and number theory involve the same kind of thinking, but the same doesn't really go for geometry and combinatorics. My thoughts always were that if you weren't as good in geometry, you would learn how to do more of it, faster, by viewing the solutions of that higher level because of your combinatorics. I think the people at brilliant just took two more large Olympiad categories and decided to put them together. I don't really know. Bob Krueger · 4 years, 1 month ago

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@Bob Krueger I agree that Algebra and Number Theory are somewhat similar, but they have their own key techniques. I have improved my algebra, but my number theory used to be way better than my algebra.

I don't think my geometry has improved as much as it could have: I now only solve geometry problems if they're among the easier ones. Also, most problems in Combinatorics and Geometry are about combinatorics, not about geometry. Had there been a seperate geometry section, I could try to get them all right, rather than just being lucky whenever I know how to solve one. Tim Vermeulen · 4 years, 1 month ago

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@Tim Vermeulen Right. The situation with me is just the opposite. I can hardly solve combinatorics problems(sometimes I do), but I almost solve the geometry problems whenever they are there. What I personally think is there are quite often a lot more combinatorics problems than geometry ones. Your opinion seems to be legitimate enough to be thought of. Maybe if this comment gets enough voteups, the Brilliant staff would look to it. That's my say.... Nishant Sharma · 4 years, 1 month ago

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@Nishant Sharma I'm pretty sure they will. :) Tim Vermeulen · 4 years, 1 month ago

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@Tim Vermeulen Seconded. While I was still in school (I studied in Singapore) doing Mathematical Olympiads, combinatorics was my 'pet' subject, while geometry was almost definitely my weakest!

I've got to level 5 due to combinatorics-rich problem sets, but I almost always have no idea where to start for 180+ geometry questions. Agree with the observation that the majority of the problems seem to be combinatorics ones as well. Jeremy Kong · 4 years, 1 month ago

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@Tim Vermeulen Right. That's just a prickly situation. Bob Krueger · 4 years, 1 month ago

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