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> IMO problems are simple, but only in the sense that they don’t require any advanced math — even calculus is considered beyond the scope of the competition. I

• gowld 6y ago
  They are elementary. They don't require advanced theory, but they require an extremely large breadth of mathematical knowledge or creative.
  • SketchySeaBeast 6y ago
    "and each set of three points determines a non-degenerate triangle with rational area" <- that's not language I knew in elementary school. Just being able to read what the question wants is a skill that isn't available to most elementary students.
    • saagarjha 6y ago
      "Elementary" in the sense of "introductory", not in the sense of "taught in elementary school".
      • SketchySeaBeast 6y ago
        I understand that, but elementary quantum physics is still grossly past the level of the students and required more advanced theory just to get to that elementary level understanding of the subject.
        • saagarjha 6y ago
          Right, but this is elementary mathematics that is specifically chosen so that university-level mathematics is not required. (And rarely helps.)
    • mciancia 6y ago
      Well, maybe not elementary school, but in highschool you should encounter those, especially second part. Non-degenerate in that case is fancy word for saying that those point cannot form a line (or a single point) - just that triangle is actually a triangle. This is pretty common term when it comes to math Olympiad so in would be extremely unlikely that someone can get to imo without encountering that in the past during training
  • blueicecubes 6y ago
    "'Elementary' means that very little is required to know ahead of time in order to understand it, except to have an infinite amount of intelligence."
• dmurray 6y ago
  In practice, yeah, "an average person" wouldn't know where to begin. And a brilliant high schooler who hasn't studied maths beyond the high-school curriculum won't ace the test. Though he could get a medal, particularly if the geometry and combinatorics questions are friendly. But the conceit of the IMO has always been that the problems are drawn from a very limited selection of fields, which are/were commonly taught in high schools around the world. No need for complex numbers, calculus, abstract algebra, or trigonometry (though any of these may end up used in an alternative solution).
  • im3w1l 6y ago
    Can't say I've seen inequality-problems like IMO has anywhere else.
    • singhrac 6y ago
      They show up in a few other areas, like probabilistic analysis of algorithms or sketching (e.g. Chernoff bounds and definitely the Cauchy-Schwartz ineq.). But you're right, nothing heavyweight like Muirheads has many applications.
    • dmurray 6y ago
      But a surprising number of them can be solved with the insight only that x^2 >= 0. (Both Cauchy-Schwartz and AM/GM reduce to that if you squint hard enough). And that's accessible to any maths student.
• saagarjha 6y ago
  IMO problems are designed such that an average (ok, maybe slightly-above average) person can follow the solution, but it takes skill to come up with it in the first place. Just like calculus is standard curriculum for hundreds of thousands of high school students around the world, but we still think Isaac Newton and Gottfried Leibniz were genii for coming up with it.
• lacker 6y ago
  Yeah, they do often require a lot of math. It's just a restricted set of topics. The IMO doesn't do any calculus or anything beyond calculus like real analysis, topology, etc. It does have a lot of geometry problems that use geometry that typically isn't in the American curriculum, and it has topics like number theory that aren't really in a standard high school curriculum either. But the topics are restricted so that they are pretty close to understandable by someone who has just done well in a standard high school mathematics curriculum. Here's an example. For the problem they include in the article: Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. This is a whole lot easier if you're familiar with Pick's Theorem, which gives you a formula for the area of a polygon whose corners are integer lattice points. In particular, any three integer lattice points define a triangle with rational area. So all you have to do is find n points that aren't colinear with each other and have irrational distance, and you'll solve the problem. Technically, you could do this without Pick's Theorem, because you'll be able to find an equivalent formula for the area of these triangles. But it's much easier if you do know Pick's Theorem, and that's the sort of thing that you typically don't pick up in standard high school math, but you would pick up if you were practicing a bunch of different Olympiad-type problems.
  • logicallee 6y ago
    Ha, Pick's theorem. Such hopelessly advanced problems (I'll give an example shortly) are well beyond what a cluster of ten thousand GPU's can solve today if trained for a thousand days on the types of math you've mentioned and then given a hundred hours to compute the answer to an infinitely, staggeringly and mind-bendingly intricate problem such as this one: "Starting at 2, how many times must you double the number for the result to exceed 10?". (Or a similarly trivial one.) Yeah. Really. Try it. They don't understand language at all. Just try to interact with any of the assistant or computational knowledge engines. Here's what wolfram alpha gives you for that query: Query: Starting at 2, how many times must you double the number for the result to exceed 10? Computed response: 1 | adjective | (especially of eyes) bulging or protruding as with fear 2 | adjective | appropriate to the beginning or start of an event 3 | noun | a turn to be a starter (in a game at the beginning)