How to prepare for math olympiad proof questions
- Chang Liu
- Jul 18
- 5 min read
Updated: Jul 19
How How to Prepare for Math Olympiad Competition
This article discusses how to prepare for math olympiad type questions. How to prepare for proof type questions? how to win math olympiad? How to become successful in math olympiad contests?
Here is an outline
There is no such thing as genius and smarts
Difference between AMC and olympiad contests
Fundamentals first
Shake the tree
How to prepare and improve?
Links to resources and fundamentals
Conclusions
There is no such thing as smarts and genius
When someone becomes really good at MathCon, Math Circles, and AMC math, that person faces a new wall to climb - it is called the Math Olympiad. This is a final challenge. The nature of the test and contest changed permanently. Here is how.
If AMC math is like attending an individual sport events and become very competent, the Olympiad is designed to be a heavy duty obstacle course, with multiple courses. If you see a question and immediately know if you can do it, then the question is not well designed. In math olympiad, every question is designed to disguise the true intent.
In AMC math, you are to compete on a given path and win a race. You have seen these questions before. Your practice can make you comfortable.
In math olympiad, you are to choose and explore multiple paths and finish a series of races by trials and error. It is more like riddles and maze. You stumble upon the answer, or not.
There is no such thing as "smarts". People who are smart do not automatically exist as a birth right. If smart works, then everyone on top of society must be truly smart and have great grades at school. Conversely, everyone great at school will also rise to the top of society. But you know that is not the case. Never over emphasize "smarts" or IQ.
Math Olympid therefore resembles real business and real life. You must stumble upon a path from uncertainty, by exploring vigorously all paths with discipline, intuition, and luck. You must read something you don't know and try to understand it quickly and methodically. You must have fantastic fundamentals and also quick reflexes.
Difference of AMC and Olympiad contest
If AMC is a test, then Olympiad math resembles a battle or game. You are competiting with the question maker and with yourself.
The Olympiad math contest is designed to test your ability to learn and work with the unknown, quickly.
For olympiad type questions, its solution typically consist of multiple stages. For example, you may need to prove a theorem or lemma in the first stage, and then apply it to solve the problem. In other words, if the question is to crack the lock to the fortress, you must make a black smith shop first before forging special tools.
Fundamentals first
Fundamentals are the most important. The more math tools (facts) you have in your mind map (tool box), the better. When you find the right tool, you can crack the code easily.
It is important to know the SOP (standard operating procedure), common theorems, lemmas and facts. The more you know, the more likely your mind will jump into the right abstract path.
Think of an abstract path as allowing you briefly elevate yourself above the question, so you see the whole picture.
Shake the tree
It is actually not easy to prepare an Olympiad math questions. There are committees of people whose job is to come up with good questions. A good question must satisfy two criteria:
It must not be understandable on first sight. If it is not hard at first sight, it is wrong.
it must be accessible, i.e., solvable by a measured effort for someone with descent preparation.
With this in mind, consider the Olympiad questions a riddle solving contests. First, you must understand the question. Shake the tree until something reveals itself.
Consider the olympic math question (or Physics, Chemistry, IT) as a test to figure out the key to unlock a castle. It is an innovative process, not a "smart process".
How to prepare and improve for math olympiad contests
Every question is tailor made by a committee and create problem makers. It is unlikely to resemble any questions that have been used before. Therefore just cramming more past questions does not automatically increase your odds.
When you work on past problems, quality beams quantity.
Don't spend forever working on a past problem. Instead, set a time limit before you consult the answer key.
First, don't read the entire answer key. Just getting some good hint of direction is enough.
Instead of reading the proof, try to write the proof all by yourself as if you are teaching.
Contribute to a community and friends.
The best way to learn something truly is to teach it. Contribute to social network sites with your answers and assistance. You will understand a lot better.
The following video has some quotes I like:
Q: "I get stuck on one problems for hours ... what should I do?" "How do I improve my problem solving skills"
A: Look at key techniques and key theorems that are used. Don't spend hours and hours stuck on a problem. Try to reflect on how you can come to solve it on your own.
In the video below, the interviewers ask a three time Olympic gold medalists about how to learn to prepare for hard question.
Q: "What is the approach to tackle tough problems"
Q: "Do you follow a certain framework"
A: "I try to understand the problem and then stumble onto an Aha moment". "Once you understand the structure of the question, the solution will be trivial".
The ten year US Olympiad team Poshen Lo saidd, there is no such thing as genius.
Poshen Lo's talking point are valuable. I list some below:
Poshen Lo: What makes some people look really smart, is that they are good at figuring out something they have never seen before. They have deep passion about the subject and they are attracted to work on things they don't understand.
People are out of their comfort zone. They are not working on things they have seen before. The normal education is to show you enough things that you feel comfortable.
I interview high school students for 30 minutes. One is to look at how they approach something they don't know how to do. The second is to look into their heart. Some people try everything and shakes the whole tree. Some people hesitate.
This interview by Math Wise is incredible revealing. A student from Thailand revealed how he improved in the camp. The team and open sharing of ideas are critical. He learned everyone follows roughly same thought process development.
He worked hard to get to the point where he can make one IMO level question independantly.
A love of math is important as a driving force. Each question can take 1-2 hours and it is incredible exercise of mental stamina.
This genius discussed that he was raised mathematically by his mother. He find a couple of days to work on a problem at the beginnging, because he does not want to look at solutions. However, sometimes you need to be time efficient.
Links to resources
The OTIS Except, a collection of 202 questions and answers
"How to write proofs" https://zimmer.csufresno.edu/%7Elarryc/proofs/proofs.html
Conclusions
Math Olympiad events test and build characters, much like entrepreneurs and science. The name of the game is to catch lighting in the bottle. This is why winners are highly values and recognized.
For interactive individualized coaching, we are ready to help your star. To contact Coach Chang Liu, please email changliu@stembay.org.
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