The advice post to end all (of my) advice posts (Part 1 of 2)
by greenturtle3141, Jul 11, 2023, 7:47 PM
I keep writing way too much advice across posts and DMs, so I thought it would nice to compile everything into one post.
Just so that you have a better idea of who is writing this advice, here is my "math resume":
With that out of the way, here are my t a k e s.
Section 0: The most important piece of advice
This is what I wished I knew for the past years of my life: Take all advice given by everyone with a grain of salt. This includes the advice you're about to read, and this advice itself. In particular, I would generally advise against trusting any one person's advice too much. There are often a variety of opinions varying around, and they can be contradictory. A person's advice (including the one I am writing right now) is typically a reflection of their past experiences, which may not necessarily be well-suited for you.
All the advice I write in this post is written in good faith, as is the vast majority of advice you will ever receive. But that does not mean that you should follow my advice to a tee, unless you wholeheartedly agree and it makes a lot of sense to you. In any case, it is (in my opinion) wise to seek other sources of advice if you can before you make any conclusions for yourself.
With all that said, I bestow upon you a boatload of advice coming from a former olympian who was arguably an overachiever.
Section 1: Computational Contests (AMC/AIME)
I got a score of X on the Y, and I want to do Z by next year. What books should I read? Which contests should I look at? What classes do I take?
Do problems.
But that doesn't answer th-
Don't care. Do problems. There is no secret to getting good at math.
Whenever I show people my ability to play Tetris, they tend to stare in awe, and inevitably ask how I can play so fast. The answer is dead simple: I played Tetris every single day for 5 years. I had no training regimen, no improvement strategy, no intelligent thoughts whatsoever. I just played. And played. And played. With time, Tetris simply became second nature, and I no longer had to think to play.
Whenever people find out about my success in math competitions, they tend to be quite impressed, and often people wonder how I got so good. The answer is dead simple: I did math for 3-5 hours every single day for a year. I had no training regimen, no improvement strategy, no intelligent thoughts whatsoever. I just practiced. And practiced. And practiced. With time, my instinct for solving mathematical problems became second nature, and a multitude of problems became automatic.
(Sure, I attended a competition math class or camp here and there, but they were definitely not my main source of improvement whatsoever.)
pls give me a book to read or something
No. Just practice by doing problems. I don't care where it's from. You'll figure it out.
Here's some extra guidance on doing problems: Typically you want to try and do problems that are slightly above your level. For instance, if you can solve AIME 4's, but struggle a bit past that, then you ought to focus on AIME 5-8, and you certainly shouldn't practice on AIME 1's unless you're doing a full timed mock because that's not an efficient use of your time.
wait are books useless?
Not necessarily. But do you learn how to swim by reading a book?
A book can give some useful guidance on posture and stuff, but ultimately if you want to learn to swim, you need to be pushed into the water and struggle.
To sum up my stance: Books can be useful, but they will not be the main contribution to your problem-solving development, which is practice. I keep telling people to practice because I think it's still a bit undervalued at the time of writing.
Practice aside, here are some good things about books:
I must read books or else the world ends. What books would you recommend?
If you're asking me specifically, well, there's not a ton of books that I can recommend because I haven't read many (obviously). In fact, here's the full list.
Which contests should I look at?
Don't care. Doesn't matter. Do problems.
I must receive specific recommendations for contests or else the world ends. What would you recommend in order to save the world?
First do all the AMCs and AIMEs. If you're somehow done with that, then my recommendations are, in roughly decreasing favoritism:
How do I know if I'm ready to solve problems from contest X or read book Y?
You, uh, try it. And see if it works out.
What classes should I take?
Do problems.
I must take a class or else the world ends. What class should I take?
Let the world end.
...didn't you take a class?
I took like an AMC problem thingy maybe, maybe an AIME thingy, honestly I don't remember whatsoever. I don't think I retained much from these classes because I was not yet in my cracked phase.
The one memorable class I took was WOOT. That was kinda fun. Is it necessary? Definitely not. Would I recommend it highly? Eh. Would I lightly recommend it if you're bored and need some slightly guided mathematical stimulation? Sure go for it. They have lots of stuff going on so you'll be engaged. I will say though, WOOT is directed more towards those who are at or approaching the Olympiad level. If you're hard stuck in AMC/AIME land then it might not be so useful. But I mean, if you're really curious and you really want to, I won't stop you! Pretty sure there's a 1-week refund policy anyway, so what's the worst that can happen?
Here is everything else I've done that may count as a "class":
You might be thinking "wow this kid just dissed the idea of taking classes and yet he made USAMO partly because of classes". And sure you might have a point. But I'd say that (1) it is still true that 95% of your prep comes from practice, and (2) ultimately, there is a lot of unpredictability and random chance involved when the days of reckoning come. There is no guarantee that doing this probem, reading that book, or taking that class will contribute to your success.
I will die on the hill of "do problems" because people that ask how to prepare for competitions are typically massively undervaluing practice when it comes to maximizing the probability that you'll succeed. If you want me to be more nuanced, I will say that, like all maximization problems, you might need a mix of different things in different ratios. Taking a class can be great for exposing yourself to the mathematical views and ideas from instructors and peers. Reading a book can give you valuable perspectives on certain concepts. But in the end, the element with the largest weight is practice. Practice is eternal and has no replacement.
You keep talking about practice, practice, practice. How do I find the energy and motivation for practice?
It goes without saying that most of the energy and motivation will typically come from your love of math and blah blah blah. But even with this, it is natural to lose motivation.
So, where else does motivation come from, if it must come from elsewhere?
This is not a question that I can give a confident answer to. In lieu of a good answer, allow me to speak about two examples.
The first example is that of my "math senpai", Wanlin Li. She like, made MOP and also won the European Girls Math Olympiad, among other things. In her reflection, she writes about a big factor of her motivation for doing it all: When she moved to my city from Pittsburgh, she missed her friends greatly. She knew MOP would be held in Pittsburgh, and hence she reasoned that if she made MOP, she would be able to hang out with her friends again. And so she did.
It sounds silly, does it not? And yet the end truly justify the means.
The second example is me. I did not consider myself "active" in math competitions up to 10th grade. Sure, I participated in AMC, and went to HMMT November once, but competitions were not much of a priority. I was a lazy student, got quite a few B's, and supposedly was destined to not be very successful. I had very little motivation to prepare all day for math competitions. I played videogames all day instead.
But then it happened.
I became obsessed.
It all happened in a blur.
I took every class I could. I pirated and bought all the books I could find and read them (not that they necessarily helped much!). I did every problem everywhere. Hours a day. During lunch, during dinner, during class, in the shower, in my bed. I dropped videogames. I threw my life out the window for math. Nothing else mattered to me. I had one goal and I was willing to do whatever it takes to achieve it: Qualify for the USAMO. And I had one year to do it before college apps.
What was the motivation? I was honestly so blind to reality that my memory from this period is quite foggy. But I can remember bits and pieces of what happened, and I'm confident that I can pinpoint my exact rationale. I will say this: My motivation was just as silly, if not more inane, than Wanlin's. It's a deeply personal and honestly embarrassing motivation that I've told only my closest friends, and it's certainly not something I am willing to say here. But, well, if you've been paying any attention to what I've been writing, you should be able to deduce what it was.
I'm burnt out. What do I do?
If you're unfamiliar with the term, you're burnt out if you've lost a ton of motivation for math, and you're kinda starting to hate doing it. In this case, you should stop doing math. Take a good break from math for like, at least a week. Maybe more. You're not going to remedy this issue by doing math, that's for sure.
If I do X, will Y happen?
I dunno.
Keep in mind that to matter how much you prepare, there is always some variability. Maybe you get sick. Maybe your headache causes a silly. Maybe the test just isn't up your alley. To wit, there is an unfortunate difference between "being at the level to do Y" and "actually achieving Y". So I hope you can recognize when you're good enough to achieve Y and be proud of that in itself.
I am currently a X. Am I starting math too late to achieve Y?
Probably not, unless your goal is unrealistic like making MOP.
Typically, the 1-year jumps of (nothing --> AIME) and (AIME --> USA(J)MO) are within the realm of possibility. Anything narrower is certainly very possible. Any wider jump is not very realistic.
I am currently a X. How do I make MOP in Y years?
You don't.
There are more than 15,000,000 high school students in the United States of America. To make MOP, you have to beat out 14,999,970 of them in mathematical skill. Being top 30 is astronomically difficult.
Sure, maybe I'm exaggerating something. According to AMC score statistics, only around 50,000 - 60,000 students actually bother to take the AMC 10/12s each year. But as a percentage is still only --- a microscopic proportion.
Feel free to make MOP a goal, but expect to fail.
What about making the UAS(J)MO?
Qualifying for the USAMO is within the realm of reason: It more or less entails being top , and is 1%, which makes sense. Top 1% among good math students is quite prestigious and truly remarkable. In my view, the competition at this level is sufficiently cutthroat for USAMO qualification to almost never be "secure". The smallest factor can tip the scales.
Making USAMO requires quite a bit of hard work. You need to be comfortable with getting a really good score (120+) on the AMC 12 and be comfortable with getting a really good score (10+) on the AIME. In my year, I got a 136.5 + 80 = 216.5, making it over the cutoff by half a point. I got to this point after a year of constant practice for hours every day. And even then, I only made it by half a point. Even after throwing away everything to get good enough to make it, the slightest difference could have toppled everything I worked for.
That's how brutal the competition is.
So, keep in mind that there's a difference between making USAMO and getting good enough to make USAMO. Unfortunately, not everyone that deserves to make USAMO will actually make it. If you fail to make it, I hope you will keep that in mind.
What about making AIME?
Qualifying for the AIME is quite realistic if you work hard for it: The percentage is around 5 or 6% among good math students, which is still prestigious and pretty darn nice. Not something to look down on, like half of AoPS seems to do.
In most cases, I think a student that works towards making AIME (and actually wants to, rather than being forced to work toward it by a tiger mom or something) will be able to make it. Plenty of practice should be enough to attain sufficient knowledge and mathematical "vision" to reasonably solve the first 14 problems, which tends to be the goal. Once you have this knowledge, it comes down to speed and test-taking skills, which also comes from practice. Unfortunately, this is can be the main obstacle to qualifying.
If you've worked seriously toward AIME and didn't make it, I wouldn't feel too bad. It's speed math. The fast pace is unrealistic.
Also consider qualifying via the USAMTS. It's a wonderful, free opportunity.
Is X impressive?
Completely subjective and unanswerable.
I got a score of X and I feel really bad. Now what?
Do not view X as either a "good score" or a "bad score". This is a useless dichotomy. Instead, view X as simply a score. It is a single data point that you should look forward to improving upon.
Is X impressive to colleges?
As long as it's not a zero on the AMC (which... honestly is impressive lol), yes. Hurdler put it better than I can:
When should I start taking the AMC 12?
Usually just start in 11th grade tbh.
But if I'm not in 11th grade yet, could it be actually easier to shoot for the USAMO if I can?
Man, if it were a few years prior then it may have been the case the the AMC 12 route was easier because the cutoffs are lower. But wow, something weird happened this year. Since I probably do not have the best judgment here, I will not give you an opinion.
Qualifying for AIME is supposed to be really easy, right?
No, no it's not!!! Stop listening to the ultra over-achievers on this site that belittle remarkable accomplishments.
When am I first supposed to qualify for AIME?
Never. The average student does not qualify for the AIME, much less take the AMC.
Don't listen to the crazy inflated statistics on AoPS. You shouldn't feel bad if you're in like 11th grade and haven't yet made AIME. It's normal. Getting to AIME is like, pretty hard! It is very much NOT normal to qualify for the AIME in middle shool. That's actually insane.
If it helps, I first qualified for AIME in 10th grade.
I am in middle school and I just failed to do X, and I am now depressed. What sh-
For christ's sake! You're in MIDDLE SCHOOL. You have your whole high school career ahead of you to do whatever you want!!! What you're supposed to do in middle school is live your life and HAVE FUN. Make friends! Play Genshin Impact!
I mean if you're certain that you want to do math all day then go ahead, but you have SO MANY ATTEMPTS at AIME and USAJMO that it's nonsensical to get worked up over not qualifying at this stage. If you're truly upset about this "setback", this is a sign that you're being competitive to an unhealthy degree. Don't let math competitions take over your life. They really, really are not that important. In fact, they're just not important in the grand scheme of things.
How much time should I spend on math competition prep every day?
That's completely up to you. Personally I think there is no use in counting hours and/or making a point of doing hours per day for some . Do as much preparation as you want to.
How do I avoid sillies?
I don't have good advice for this because I can't relate that much. I've made my fair share of sillies, but fortunately they weren't frequent enough to kill my career. Of note, I made zero sillies on the AMC the year I made USAMO, which is honestly a miracle.
If you're making at least 3 sillies on the AMC or something, that's probably not good (unless you're a genius and solving like all 25 problems). I'd say that reducing sillies can be something to incorporate into practice. Particularly, you can experiment with how fast you solve problems. If you want specific tips:
What is the strategy for taking (contest)?
What is the strategy for taking the AMC?
What is the strategy for taking the AIME?
I've finished the AMC/AIME and I'm a nervous wreck and I think I sillied half the exam. What should I do now?
Section 2: Olympiads
This section will be very, very short.
How d-
Do problems.
And maybe read EGMO.
...to be transparent, I don't think I can give very good specific advice here because I "only" made USAMO once and I "only" scored a 15.
You cut me off too soon! I was going to ask, how in the world do I write a proof??? I've never done it/I'm bad at it.
Ah. No, that's a good question.
Unfortunately this was never much of an issue for me either, I kinda just picked it up naturally. But be rest assured that struggling with proof-writing is common. Here are some tips:
Section 3: Research
Am I ready for research?
Do you like math? Then sure!
What does high-school research even consist of?
There are essentially two routes to take:
Expository research is arguably easier, in that you don't have to do "actual math". It's more about learning a topic deeply and then organizing what you have learned into a paper. One of my former math teachers calls it a "book report", perhaps with the suggestions that such a project has little value. This is not necesarily true. If you are just starting out in research, this can be a nice path to take.
"Original" research tends to be harder. This is because there is comparatively very little math that can be done using only the math you learn from high school (and computational contest topics). If you want a really good original research project, there are several ways to go about this:
This sounds daunting, but I do think that any math student is capable of doing some amount of "original" research. You don't have to prove a new theorem in order for your work to be original. You could also apply the work of others. To that end, here is the advice I wish I had known when I was first starting out with "math research" in middle school and high school.
How do I choose a topic?
As I suggested above, the possible strategies you can employ to find a topic are:
Here were the topics I've done in middle school and high school:
(Part 2)
Just so that you have a better idea of who is writing this advice, here is my "math resume":
- Track Record:
- 8th: 23 AMC 8
- 9th: 105 AMC 10
- 10th: 124.5 AMC 10, 4 AIME
- 11th: 136.5 AMC 12, 8 AIME, 15 USAMO
- 12th: whoops lol
- I think I won the Nassau Math Tournament once
- Former Nassau All-Stars Team Captain
- NYSML Tiebreakers
- ARML Tiebreakers
- Former Syosset Mathletes President
- Former President of a NPO that teaches math to middle-schoolers and below
- Ran several classes for AMC/AIME
- Among my few private students, I got one to AIME and another to USAJMO
- Regeneron STS Finalist for math research
- USAMTS Grader
- CMIMC Problem Writer
- I interned for Po-Shen Loh for a year or so
- Vice President External for the CMU Math Club (2022-23)
- Putnam Honorable Mention
- I got into a very positive number of grad schools and will be pursuing a PhD at the NYU Courant Institute in the fall
With that out of the way, here are my t a k e s.
Section 0: The most important piece of advice
This is what I wished I knew for the past years of my life: Take all advice given by everyone with a grain of salt. This includes the advice you're about to read, and this advice itself. In particular, I would generally advise against trusting any one person's advice too much. There are often a variety of opinions varying around, and they can be contradictory. A person's advice (including the one I am writing right now) is typically a reflection of their past experiences, which may not necessarily be well-suited for you.
All the advice I write in this post is written in good faith, as is the vast majority of advice you will ever receive. But that does not mean that you should follow my advice to a tee, unless you wholeheartedly agree and it makes a lot of sense to you. In any case, it is (in my opinion) wise to seek other sources of advice if you can before you make any conclusions for yourself.
With all that said, I bestow upon you a boatload of advice coming from a former olympian who was arguably an overachiever.
Section 1: Computational Contests (AMC/AIME)
I got a score of X on the Y, and I want to do Z by next year. What books should I read? Which contests should I look at? What classes do I take?
Do problems.
But that doesn't answer th-
Don't care. Do problems. There is no secret to getting good at math.
Whenever I show people my ability to play Tetris, they tend to stare in awe, and inevitably ask how I can play so fast. The answer is dead simple: I played Tetris every single day for 5 years. I had no training regimen, no improvement strategy, no intelligent thoughts whatsoever. I just played. And played. And played. With time, Tetris simply became second nature, and I no longer had to think to play.
Whenever people find out about my success in math competitions, they tend to be quite impressed, and often people wonder how I got so good. The answer is dead simple: I did math for 3-5 hours every single day for a year. I had no training regimen, no improvement strategy, no intelligent thoughts whatsoever. I just practiced. And practiced. And practiced. With time, my instinct for solving mathematical problems became second nature, and a multitude of problems became automatic.
(Sure, I attended a competition math class or camp here and there, but they were definitely not my main source of improvement whatsoever.)
pls give me a book to read or something
No. Just practice by doing problems. I don't care where it's from. You'll figure it out.
Here's some extra guidance on doing problems: Typically you want to try and do problems that are slightly above your level. For instance, if you can solve AIME 4's, but struggle a bit past that, then you ought to focus on AIME 5-8, and you certainly shouldn't practice on AIME 1's unless you're doing a full timed mock because that's not an efficient use of your time.
wait are books useless?
Not necessarily. But do you learn how to swim by reading a book?
A book can give some useful guidance on posture and stuff, but ultimately if you want to learn to swim, you need to be pushed into the water and struggle.
To sum up my stance: Books can be useful, but they will not be the main contribution to your problem-solving development, which is practice. I keep telling people to practice because I think it's still a bit undervalued at the time of writing.
Practice aside, here are some good things about books:
- They can be fun! If there is a book you want to read, I encourage you to read it. Satisfying your mathematical curiosity is an important part of enjoying mathematics.
- They can increase your breadth of knowledge.
I must read books or else the world ends. What books would you recommend?
If you're asking me specifically, well, there's not a ton of books that I can recommend because I haven't read many (obviously). In fact, here's the full list.
- Art of Problem Solving vol. 2: I credit this book for giving me like one extra solve in my entire career. To be transparent, I learned some 3D coordinate geometry which let me execute like one (unnecessary) bash. Fun book I guess.
- Euclidean Geometry in Mathematical Olympiads: This is a book I would actually recommend. It's not strictly necessary at the AMC/AIME level, though it certainly becomes a more useful resource as you progress towards Olympiads. Speaking of which, if you are approaching the Olympiad level, my advice would shift sligtly towards "ok maybe there's a book or two that you might want to look at", which would include EGMO.
- Problems in Plane and Solid Geometry by Prasalov: Did not help me at all for my competition math ventures, but it was very fun to read and I have no regrets. Give it a spin if you're bored and have a desire for projective geometry (which is full of black magic).
- (insert various "books" which are actually just collections of problems)
Which contests should I look at?
Don't care. Doesn't matter. Do problems.
I must receive specific recommendations for contests or else the world ends. What would you recommend in order to save the world?
First do all the AMCs and AIMEs. If you're somehow done with that, then my recommendations are, in roughly decreasing favoritism:
- Math Prize for Girls: If you haven't heard of it, don't think for a second that this is an "easier contest for girls". This competition is brutal. Ranges from mid-AIME to late-AIME.
- Online Math Open: Wide range of difficulty, from trivial AMC to... late IMO? The site for this and NIMO is dead, but fortunately the complete archive is available on evan's site: https://web.evanchen.cc/problems.html
- National Internet Math Olympiad: It's fun!
- CMIMC: I'm from CMU, of course I'll put this first among the "big three". I am shamelessly biased. (And I wrote some of the problems )
- HMMT: November and February combined will cast a wide net of difficulties.
- PUMaC: I'm putting this 3rd among the "big three" only because one of the past problems was plagiarized
and they never sent me my trophy. Fun contest though. - Korean Math Olympiad Round 1: It's a fun AIME-like contest. See here for the problems: https://artofproblemsolving.com/community/c155599h1148649_problemsanswer_key_pdf
- ARML: Give yourself the 10-minute time pressure for fun! And try doing the relays solo for extra fun!
- Purple Comet: A very solid contest.
- San Diego Math League: Also quality.
- MMATHS: (I no longer can guarantee quality past this point, mainly due to my lack of knowledge, so the contests from now on are not very well-sorted.) Anyways, MMATHS is run by Columbia students.
- Lehigh Math Tournament: This was pretty fun when I looked at it.
- PRMO: An Indian contest that keeps popping up on AoPS.
- Stanford Math Tournament: I know of at least one good problem from SMT.
- CHMMC: I know nothing about this one.
- Berkeley MT: idk
- Duke Math Meet: idk
- NYSML: Past problems are not public I think, unless you decide to do some pirating
- CEMC: Go Canada!!!
How do I know if I'm ready to solve problems from contest X or read book Y?
You, uh, try it. And see if it works out.
What classes should I take?
Do problems.
I must take a class or else the world ends. What class should I take?
Let the world end.
...didn't you take a class?
I took like an AMC problem thingy maybe, maybe an AIME thingy, honestly I don't remember whatsoever. I don't think I retained much from these classes because I was not yet in my cracked phase.
The one memorable class I took was WOOT. That was kinda fun. Is it necessary? Definitely not. Would I recommend it highly? Eh. Would I lightly recommend it if you're bored and need some slightly guided mathematical stimulation? Sure go for it. They have lots of stuff going on so you'll be engaged. I will say though, WOOT is directed more towards those who are at or approaching the Olympiad level. If you're hard stuck in AMC/AIME land then it might not be so useful. But I mean, if you're really curious and you really want to, I won't stop you! Pretty sure there's a 1-week refund policy anyway, so what's the worst that can happen?
Here is everything else I've done that may count as a "class":
- AMSP for two summers. The most important thing I got out of AMSP was a better foundation of number theory, because apparently I was very weak in basic NT and this filled in that knowledge gap quite nicely.
- Some weekly AMC/AIME course where we just did problems. I will say that this was one factor that was directly responsible for causing my qualification to USAMO, beacuse literally a few days prior to the AMC I learned a useful perspective for simplifying expressions of the form where possible, which got me an extra 4.5 points... which was crucial because I made the USAMO cutoff by half a point.
- Some random Winter course for AIME/USAMO. It was not useful at all.
You might be thinking "wow this kid just dissed the idea of taking classes and yet he made USAMO partly because of classes". And sure you might have a point. But I'd say that (1) it is still true that 95% of your prep comes from practice, and (2) ultimately, there is a lot of unpredictability and random chance involved when the days of reckoning come. There is no guarantee that doing this probem, reading that book, or taking that class will contribute to your success.
I will die on the hill of "do problems" because people that ask how to prepare for competitions are typically massively undervaluing practice when it comes to maximizing the probability that you'll succeed. If you want me to be more nuanced, I will say that, like all maximization problems, you might need a mix of different things in different ratios. Taking a class can be great for exposing yourself to the mathematical views and ideas from instructors and peers. Reading a book can give you valuable perspectives on certain concepts. But in the end, the element with the largest weight is practice. Practice is eternal and has no replacement.
You keep talking about practice, practice, practice. How do I find the energy and motivation for practice?
It goes without saying that most of the energy and motivation will typically come from your love of math and blah blah blah. But even with this, it is natural to lose motivation.
So, where else does motivation come from, if it must come from elsewhere?
This is not a question that I can give a confident answer to. In lieu of a good answer, allow me to speak about two examples.
The first example is that of my "math senpai", Wanlin Li. She like, made MOP and also won the European Girls Math Olympiad, among other things. In her reflection, she writes about a big factor of her motivation for doing it all: When she moved to my city from Pittsburgh, she missed her friends greatly. She knew MOP would be held in Pittsburgh, and hence she reasoned that if she made MOP, she would be able to hang out with her friends again. And so she did.
It sounds silly, does it not? And yet the end truly justify the means.
The second example is me. I did not consider myself "active" in math competitions up to 10th grade. Sure, I participated in AMC, and went to HMMT November once, but competitions were not much of a priority. I was a lazy student, got quite a few B's, and supposedly was destined to not be very successful. I had very little motivation to prepare all day for math competitions. I played videogames all day instead.
But then it happened.
I became obsessed.
It all happened in a blur.
I took every class I could. I pirated and bought all the books I could find and read them (not that they necessarily helped much!). I did every problem everywhere. Hours a day. During lunch, during dinner, during class, in the shower, in my bed. I dropped videogames. I threw my life out the window for math. Nothing else mattered to me. I had one goal and I was willing to do whatever it takes to achieve it: Qualify for the USAMO. And I had one year to do it before college apps.
What was the motivation? I was honestly so blind to reality that my memory from this period is quite foggy. But I can remember bits and pieces of what happened, and I'm confident that I can pinpoint my exact rationale. I will say this: My motivation was just as silly, if not more inane, than Wanlin's. It's a deeply personal and honestly embarrassing motivation that I've told only my closest friends, and it's certainly not something I am willing to say here. But, well, if you've been paying any attention to what I've been writing, you should be able to deduce what it was.
I'm burnt out. What do I do?
If you're unfamiliar with the term, you're burnt out if you've lost a ton of motivation for math, and you're kinda starting to hate doing it. In this case, you should stop doing math. Take a good break from math for like, at least a week. Maybe more. You're not going to remedy this issue by doing math, that's for sure.
If I do X, will Y happen?
I dunno.
Keep in mind that to matter how much you prepare, there is always some variability. Maybe you get sick. Maybe your headache causes a silly. Maybe the test just isn't up your alley. To wit, there is an unfortunate difference between "being at the level to do Y" and "actually achieving Y". So I hope you can recognize when you're good enough to achieve Y and be proud of that in itself.
I am currently a X. Am I starting math too late to achieve Y?
Probably not, unless your goal is unrealistic like making MOP.
Typically, the 1-year jumps of (nothing --> AIME) and (AIME --> USA(J)MO) are within the realm of possibility. Anything narrower is certainly very possible. Any wider jump is not very realistic.
I am currently a X. How do I make MOP in Y years?
You don't.
There are more than 15,000,000 high school students in the United States of America. To make MOP, you have to beat out 14,999,970 of them in mathematical skill. Being top 30 is astronomically difficult.
Sure, maybe I'm exaggerating something. According to AMC score statistics, only around 50,000 - 60,000 students actually bother to take the AMC 10/12s each year. But as a percentage is still only --- a microscopic proportion.
Feel free to make MOP a goal, but expect to fail.
What about making the UAS(J)MO?
Qualifying for the USAMO is within the realm of reason: It more or less entails being top , and is 1%, which makes sense. Top 1% among good math students is quite prestigious and truly remarkable. In my view, the competition at this level is sufficiently cutthroat for USAMO qualification to almost never be "secure". The smallest factor can tip the scales.
Making USAMO requires quite a bit of hard work. You need to be comfortable with getting a really good score (120+) on the AMC 12 and be comfortable with getting a really good score (10+) on the AIME. In my year, I got a 136.5 + 80 = 216.5, making it over the cutoff by half a point. I got to this point after a year of constant practice for hours every day. And even then, I only made it by half a point. Even after throwing away everything to get good enough to make it, the slightest difference could have toppled everything I worked for.
That's how brutal the competition is.
So, keep in mind that there's a difference between making USAMO and getting good enough to make USAMO. Unfortunately, not everyone that deserves to make USAMO will actually make it. If you fail to make it, I hope you will keep that in mind.
What about making AIME?
Qualifying for the AIME is quite realistic if you work hard for it: The percentage is around 5 or 6% among good math students, which is still prestigious and pretty darn nice. Not something to look down on, like half of AoPS seems to do.
In most cases, I think a student that works towards making AIME (and actually wants to, rather than being forced to work toward it by a tiger mom or something) will be able to make it. Plenty of practice should be enough to attain sufficient knowledge and mathematical "vision" to reasonably solve the first 14 problems, which tends to be the goal. Once you have this knowledge, it comes down to speed and test-taking skills, which also comes from practice. Unfortunately, this is can be the main obstacle to qualifying.
If you've worked seriously toward AIME and didn't make it, I wouldn't feel too bad. It's speed math. The fast pace is unrealistic.
Also consider qualifying via the USAMTS. It's a wonderful, free opportunity.
Is X impressive?
Completely subjective and unanswerable.
I got a score of X and I feel really bad. Now what?
Do not view X as either a "good score" or a "bad score". This is a useless dichotomy. Instead, view X as simply a score. It is a single data point that you should look forward to improving upon.
Is X impressive to colleges?
As long as it's not a zero on the AMC (which... honestly is impressive lol), yes. Hurdler put it better than I can:
hurdler wrote:
...unless you explicitly say so in your essays or something (which you can if you want), the admissions officers aren't going to know if took the AMC casually, semicasually, seriously (prepared a lot for it) or exteremely seriously. All they will know is that you probably put in a nonzero amount of effort (aka positive amount of effort) into studying for math competitions. Having more info about an applicant is always better than having less info about an applicant. They should be looking at applications on an individual basis, not comparing you to others. The cutoff is more subjective than objective usually.
When should I start taking the AMC 12?
Usually just start in 11th grade tbh.
But if I'm not in 11th grade yet, could it be actually easier to shoot for the USAMO if I can?
Man, if it were a few years prior then it may have been the case the the AMC 12 route was easier because the cutoffs are lower. But wow, something weird happened this year. Since I probably do not have the best judgment here, I will not give you an opinion.
Qualifying for AIME is supposed to be really easy, right?
No, no it's not!!! Stop listening to the ultra over-achievers on this site that belittle remarkable accomplishments.
When am I first supposed to qualify for AIME?
Never. The average student does not qualify for the AIME, much less take the AMC.
Don't listen to the crazy inflated statistics on AoPS. You shouldn't feel bad if you're in like 11th grade and haven't yet made AIME. It's normal. Getting to AIME is like, pretty hard! It is very much NOT normal to qualify for the AIME in middle shool. That's actually insane.
If it helps, I first qualified for AIME in 10th grade.
I am in middle school and I just failed to do X, and I am now depressed. What sh-
For christ's sake! You're in MIDDLE SCHOOL. You have your whole high school career ahead of you to do whatever you want!!! What you're supposed to do in middle school is live your life and HAVE FUN. Make friends! Play Genshin Impact!
I mean if you're certain that you want to do math all day then go ahead, but you have SO MANY ATTEMPTS at AIME and USAJMO that it's nonsensical to get worked up over not qualifying at this stage. If you're truly upset about this "setback", this is a sign that you're being competitive to an unhealthy degree. Don't let math competitions take over your life. They really, really are not that important. In fact, they're just not important in the grand scheme of things.
How much time should I spend on math competition prep every day?
That's completely up to you. Personally I think there is no use in counting hours and/or making a point of doing hours per day for some . Do as much preparation as you want to.
How do I avoid sillies?
I don't have good advice for this because I can't relate that much. I've made my fair share of sillies, but fortunately they weren't frequent enough to kill my career. Of note, I made zero sillies on the AMC the year I made USAMO, which is honestly a miracle.
If you're making at least 3 sillies on the AMC or something, that's probably not good (unless you're a genius and solving like all 25 problems). I'd say that reducing sillies can be something to incorporate into practice. Particularly, you can experiment with how fast you solve problems. If you want specific tips:
- Try reading the problem multiple times. For example, a general piece of advice for the AIME is to read the question TWICE before you solve the problem, and then a THIRD TIME after you solve the problem.
- Sanity-check your thought process and computations. If you have a formula, plug in a small number to see if it holds up. If you computed , take the residues mod 9 on each side to figure out that you messed up. If you counted ways to order something, try reasoning why it would make sense for it to be that order of magnitude and divisible by 4.
What is the strategy for taking (contest)?
- The WEEK before the contest, start getting good sleep. Not just the day before! What you want to do is try and prevent sleep debt from accumulating.
- The day before the contest, don't do ANY MATH WHATSOEVER. Just STOP. NO MORE PRACTICING. Play games. Watch anime. Play sports. Don't play Genshin Impact. Hang out with your friends. Touch the local grass. Do homework. Drink boba tea. Make pasta from scratch. Literally do anything but competition math.
- Will eating X or drinking Y help? I don't know. You know your body better than I do! You should test this on yourself quite a ways prior to the contest if this is something you really think can help. In my case, I figured out that eating chocolate increases my performance in all tasks by some non-measurable amount. As an anecdote, I half-jokingly enjoy telling people the (true) story I made NYSML tiebreakers by downing an entire chocolate bar and an entire bottle of water before the Individual Round. Though, don't think that you need to consume something to get an edge. I don't think a ton of serious math people actually do that unless you're weird like me.
What is the strategy for taking the AMC?
- Do not be afraid to skip problems. I once skipped a #4, and it was actually a very good idea because in the end I genuinely had no clue how to solve it.
- In general, don't be afraid to work on problems that you find are more attractive than others.
- If you want to guess, only do so when you can eliminate at least two answer choices.
- Organize your work. Here's how I do it:
- I subdivide the first page into 6 sections, for problems 1-6.
- I subdivide the second page into 6 sections, for problems 7-12.
- I subdivide the third page into 4 sections, for problems 13-16.
- I subdivide the fourth page into 4 sections, for problems 17-20.
- I use all remaining pages arbitrarily for any remaining problems.
- The idea is to force my work to be contained in boxes. This forces a level of organization and makes it easy to go back to previous problems if I want to. You don't have to do this exact procedure, but feel free to take ideas from this philsophy to help you organize your work, if organization is a problem for you.
What is the strategy for taking the AIME?
- Generally go in order, but feel free to skip around and solve whatever you find most attractive.
- Read every problem three times. Read it twice in a row before you start the problem. Read it a third time after you have obtained an answer.
- If you're using a scantron, make sure to bubble instead of .
- Organize your work. Here's how I do it:
- I use 16 sheets of paper, plus any extra fodder that I might need.
- The 0th sheet of paper is a "hub" paper. It is a chart with 15 rows for each of the 15 problems. In each row, I record the problem's number, its answer, and my certainty for the answer. This is useful for telling me which problems to check if I choose to check my work at the end.
- Every other sheet's work is for exactly one problem. No more, no less.
- Every other sheet has a circled number on the top-left indicating which problem that sheet is reserved for.
- Every other sheet is always kept in the same cyclic order. The hub sheet is kept elsewhere.
- Usually there's a problem in the final five that's misplaced. So don't be afraid to read the final 5.
- Please don't stress yourself too much. Whatever happens, happens. Making AIME is already insane, and the effort you put in to going further will always speak for itself. I hope you know that.
I've finished the AMC/AIME and I'm a nervous wreck and I think I sillied half the exam. What should I do now?
- Please do not visit AoPS for the next week.
- Play games and have fun. Do not think about math. Whatever happened, happened. There's nothing you can do now and no amount of worrying and stress will change the result. Your score is ultimately a number that, honestly, is not that useful in the long run, even if you don't believe me at the moment.
Section 2: Olympiads
This section will be very, very short.
How d-
Do problems.
And maybe read EGMO.
...to be transparent, I don't think I can give very good specific advice here because I "only" made USAMO once and I "only" scored a 15.
You cut me off too soon! I was going to ask, how in the world do I write a proof??? I've never done it/I'm bad at it.
Ah. No, that's a good question.
Unfortunately this was never much of an issue for me either, I kinda just picked it up naturally. But be rest assured that struggling with proof-writing is common. Here are some tips:
- A proof is just a line of reasoning. It's an explanation that uses math. What I tell my teammates is to pretend that they are writing a letter to a friend which explains why the problem statement at hand is true.
- Don't be afraid to write a lot of English words. Most proofs are mostly English words! They make things easier to read.
- Do the USAMTS. It is basically the only good opportunity for high-schoolers to write proofs in a competition setting. We give feedback on your proofs for free. And you win prizes. And you can qual for AIME if you do well enough.
- Anything that has a name can be cited without proof. For everything else, use your experience to tell you whether or not the fact or lemma you wish to employ is well-known. If you suspect that it is not well-known but perhaps on the cusp of being so, sketch the proof. If you think it is niche, prove it.
- Practice writing proofs! Try writing a proof for some problem, then compare what you wrote to the official solution (or perhaps, to the proofs of other people, if there is a thread on the problem).
- Don't be afraid to ask for feedback. Users like me will gladly critique your writing for free if they have time.
Section 3: Research
Am I ready for research?
Do you like math? Then sure!
What does high-school research even consist of?
There are essentially two routes to take:
- Expository research
- "Original" research
Expository research is arguably easier, in that you don't have to do "actual math". It's more about learning a topic deeply and then organizing what you have learned into a paper. One of my former math teachers calls it a "book report", perhaps with the suggestions that such a project has little value. This is not necesarily true. If you are just starting out in research, this can be a nice path to take.
- Expository writing is certainly not mindless. There is an intellectual challenge in organizing the results you learn.
- Expository writing can be intellectually very valuable! They can help lower the barrier of entry that exists for learning some obscure or tough topics.
- "Math Competition Handouts" are an example of expository writing. They help people learn a topic.
- Expository research is a fun way to learn math! After all, one of the best ways to learn a topic is to teach it.
"Original" research tends to be harder. This is because there is comparatively very little math that can be done using only the math you learn from high school (and computational contest topics). If you want a really good original research project, there are several ways to go about this:
- Be very good, somehow, and actually prove original things by yourself.
- Participate in research programs that accept high school students. (Which may be harder than it sounds...)
- Spam emails to (possibly local) professors and beg them to accept you as a research mentee.
- Pick a topic which is so obscure that nobody has really done math on it. (That's what I did!)
This sounds daunting, but I do think that any math student is capable of doing some amount of "original" research. You don't have to prove a new theorem in order for your work to be original. You could also apply the work of others. To that end, here is the advice I wish I had known when I was first starting out with "math research" in middle school and high school.
- Pick a topic you're legitimately interested in and start by doing some literal research --- go read up on the topic, gather various sources on the topic, look at articles on the topic to see what has been done. The topic shouldn't be too broad, such as "linear algebra".
- Once you have a comfortable understanding of the topic, ask yourself:
- Is there a new "spin" I can take on the topic?
- What would happen if instead of using X, we used Y? Would it work?
- What would happen if we applied this topic to (thing)?
- Can I find a different proof for (theorem in topic)?
- Is (topic) connected to any other topics? How well-explored is this connection?
- What if we try accounting for another factor to complicate (topic)?
- How else can we generalize (topic)?
- You can also consider using a more numerical approach.
- Can I write a program to simulate (something in topic)?
- Are there any broad questions within (topic) that I can shed light on by using the help of programming?
- If any of the ideas you come up seem promising, give it a shot! You won't know if it will be a success until you try. If you fail, you would have learned something, and you could even include the contents of your failure in the "paper" you write.
- Try multiple of these ideas if you have free time and you want to!
- Speaking of the "paper", if your goal for the research is to participate in something not necessarily too formal, like a math fair, then I think it's totally fine for the paper itself to not be too professional. In that case, you can include things like what you tried and what things failed.
How do I choose a topic?
As I suggested above, the possible strategies you can employ to find a topic are:
- Just pick something you're interested in and roll with it.
- For a higher chance of being able to prove novel results, you could dig around the internet until you find a topic that is sufficiently obscure so that not much work has been done.
- For a higher chance of being able to prove novel results, you could email a professor who could assign you a topic based on your interests.
Here were the topics I've done in middle school and high school:
- 7th grade: The Rubik's Cube (pretty terrible topic because either you know group theory or you don't and you perish. But also I was in 6th grade so it's fine. It ended up being purely expository and I just listed how many possible combinations there are)
- 8th grade: RSA (This was a better topic. There were things I could learn including modular arithmetic and various other number theory things that are applied in RSA. My "paper" was expository and described the process in my own way. This is a perfect topic for a dumb 8th grader that's getting into math.)
- 9th grade: The Fourier Transform (The topic was okay, but my execution was not good. I did a decent job of motivating the Fourier Transform, at least for a 9th grader, but I tried to cover too many aspects of it and my content ended up being too widespread and flimsy. Instead of covering its applications and how it is used in ODEs and covering several theorems, I should have picked one aspect and studied it more in depth. For example, I could have tried to use it to solve various types of differential equations and see which ones it can work well on. That would've been a good topic! The work may not be groundbreaking, but the point is that there's some independent work going on instead of just a "book report".)
- 10th grade: The Lights Out puzzle (I prove several theorems. Turns out I wasn't the first to prove any of them, but I did it myself. How did I do it? Well, I started by studying how mathematicians solved the original 5x5 standard Lights Out puzzle. Since they used linear algebra, it gave be a nice opportunity to learn linalg, which made for a fun expository section. But I wanted to do my own work too. I applied their methods to solve other some variants, and also read other related literature. By combining several ideas, I realized that I could think about whether one particular strategy I saw in a paper would be viable in a variant called the Rook-Toggle variant. This lead to one theorem. Then, I decided what other deranged variations I could come up with, and that's when I came up with infinite Lights Out, leading to my second theorem. In sum, reading various papers and playing with variants on a concept can get you a cool topic that's accessible to you!)
- 11th grade: The Number Rotation Puzzle (This combinatorial puzzle in one of my phone puzzle apps annoyed me, and I wanted a solving algorithm. I solved it over square boards. I wrote it up. Then I found out, through extensive internet searching, that someone already did it on square boards. So I was forced to solve it for rectangular boards (which is extremely hard) or trash the project. Against all odds, I did it.)
(Part 2)
This post has been edited 3 times. Last edited by greenturtle3141, Jan 18, 2024, 4:53 PM
by Heavytoothpaste, Jul 11, 2023, 9:36 PM