By Nakul Iyer ’20

What is competition math? Each year, the Phillips Academy Math Team participates in several regional and national competitions, such as the AMC (American Math Competition) series, NEML (New England Math League), and ARML (American Regions Mathematics League), to name a few. How do students in the math team view competition math, and how do they prepare for these tests? I interviewed four students, Sebastian Zhu ‘20, Max Tao ‘20, Justin Chang ‘19, and Wendy Wu ‘20 independently to find the answers to a few questions.

Nakul: First off, how would you describe competition math? What are the differences and similarities between competition math and school math?

Sebastian: ​Parts of competition math can be similar to school math. The main difference is that competition math will use topics and apply them in very creative and deep ways, and school math really just tests you on your basic knowledge. School math might test you on some trigonometric fact, maybe that sin is opposite over adjacent. It might say “What’s the sine of this angle?” and give you the two relevant sides, and you just say “​this​ over ​this​”, and that’s it. In competition math, however, you’re not going to get the explicit statement of trig in the problem. It’s going to be some geometry problem, and you’re going to have to use your trig rules to figure it out, without them having to tell you exactly what to do. So, competition math is a lot more creative than school math and it offers you a lot more ways to be creative with your problem solving and to apply everything you know to be able to solve a problem instead of just doing what the teacher tells you and reciting from memory.

Max: ​I think the main difference between competition math and school math is that competition math requires a certain sort of creativity to be able to apply a few basic topics to solve a wide variety of problems. School math tends to be much more straightforward. You know exactly what’s on the test and the application of the topics you learned is more direct. In that way, I think the math at PA is much more similar to competition math than the math at any of my previous schools. Much like competition math, PA math requires a very good grasp of the topics and often a creative approach to solving problems.

Wendy: ​The way of competition math is more problem-solving focused, I would say. The goal isn’t really to learn a concept; it’s to solve your problem. Even with these differing goal, the way of thinking in competition math applies very well to school math, in my opinion. But the aims are completely different.

Nakul: What math problems do you tend to like the most?

Sebastian: ​I like creative problems in which the solution isn’t immediately obvious, but as you work through it you start to see a solution path, and you start to see techniques that might work to solve the problem. Once you try those, some of them might work, some of them might not work, and you figure out exactly which ones you need to do. Eventually, when you solve it, it feels really good to be able to solve these kinds of problems in which they seem really hard but if you work through it, you’ll be able to solve it.

Justin: ​Actually, Barycentric Geometry has really appealed to me lately. Barycentric Geometry is a coordinate system except the origin is the centroid of a triangle and the axes would be from any point in a triangle you define, P, to each of the vertices, so there will be three axes. It’s the fastest way to do 3d graphics, so any first person shooter game or any animated movie will probably have used Barycentric Geometry extensively.

Max: ​I think many people associate difficult math problems with crazy symbols and bad notation, and some problems are certainly like that, but I don’t tend to like those types of problems. I tend to like problems with simple statements, and I find that those problems often require much more in-depth analyses and have much more elegant solutions (sometimes) ie. 2018 JMO 6.

Wendy:​ One that, either if you had never encountered it before, you can just stop and think for a moment and get a solution and/or if you give up, you’ll look at the solution and think “Wow!”. There are a lot of problems that fall into these two categories. Not something that’s ridiculously straightforward.

Nakul: ​How do you approach a problem in a intense, timed environment?

Sebastian: ​Most math competitions won’t leave you enough time to at least contemplate each problem and spend a little bit of time trying to see if it’s worth approaching or not. Some of the easier ones—I use easier in a relative way—like the AMC 10 or AMC 12, top math students will be able to get through every single question, at least attempt it and usually solve it. They usually don’t have a problem with their time management; they just go through it, just like the countless practices they’ve done. Usually, math competitors know their limits, what they can and can’t do. Sometimes, they do a little bit better than what they expect, and that could be a fluke or it could mean that they’re getting better. If you reach a problem you don’t know, don’t think too much about it. Approach Problem 25 the same as you would approach Problem 1.

Max: ​For competitions like the AMC 10 or AMC 12 where there are a lot of easier problems but the time limit is very tight, if I am unable to see a solution after 15-30 seconds, I skip the problem. For AMC style problems, and all math problems in general, it often helps to come back to a problem after you’ve taken a break, even one as short as 2-3 minutes.

Wendy:​ I would kind of know what I’m working with and would probably have a gage of gage of “Do I know what this is about?”. If I don’t solve it at first, I would know if I’m in for a bash, if it’s just not working out, or if there’s a strategy that I’m missing. For strategies, there are a lot of techniques and a lot of patterns I know and a lot of things I could try; I could try plugging in numbers, smaller cases, and I’m thinking “What can I do to poke at this thing?”

Nakul: ​What makes competition math fun to do?

Sebastian: ​It’s very hands-on, creative problem solving. In school math, you memorize things, and if you know it, you basically get it right, and if you don’t know it, you get it wrong. It’s just whether you know it or not. In competition math, it’s entirely possible that you “know it”, but you don’t have the experience or the thought process to be able to apply it, and that’s where you need to get creative. You need to be able to apply what you know in a relevant way in order to solve the problem. It can be hard because there’s so many things that you’ll learn, and you don’t know which ones to apply to which problem. The ways in which you program your brain in order to be able to solve these problems, identify the ones that you know how to do, and know which ideas to apply to each problem make it really fun. Eventually, you’ll see the solution work out.

Justin: ​Well I think it’s enriching; it’s good practice for our problem-solving skills. And, I suppose it’s fun not because of the problems themselves, although those can be interesting, but because of the people that I’ve associated with through the process. For example, in Math Club and Math Team, all these people are super awesome!

Max: ​I enjoy math in general because there is an amazing amount of problems that can be solved with very few topics, and everything has a definite answer. Competition math is especially fun because the process of finding those answers is creative and it’s very satisfying to finally find the solution to a problem you’ve been working of for a very long time.

Wendy: ​Two things: the first is the feeling of solving a problem, the feeling of accomplishment, of success, and the feeling of “Wow I fought this thing”. It feels very good. And second would be the people who do it and the people I get to spend time with.