How to Do Well on Math Contests Like the AMC

Steps

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   Decide what your goal is. Someone who wants to get a 150 on the AMC is going to have a different way of training than someone just looking to qualify for the AIME. Know your goal, and you can prepare for it well.
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   Get some old tests. All math contests need a source of questions, and oftentimes the same types of questions appear over and over again. Just this year, the AIME II had a geometry question almost identical to one on an AIME a while back. The AMC returns to the same themes and question types year after year.
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   Enter some contests. The best way to get good at something is to do it, right? So go ahead and enter some math contests. You would be surprised how many there are out there-and the experience you can gain is invaluable. Remember, different contests teach you different skills-USAMTS is going to teach you how to write proofs, while ARML individual round tests are going to test your speed.
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   Budget time to work on math problems, especially in the areas that you are struggling. Say you take a practice AMC from a few years ago, and you can't get any geometry problem in the last 10 questions or so. Well, it sounds like you need to work on your geometry skills. Beef up on theorems and do some basic problems that apply these theorems. Books like Art of Problem Solving are excellent for this-they have theorems presented, and then problems that you can try applying them in. Remember, the only way to learn how to solve problems is to solve some.
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   A few weeks before, do a few timed sessions of old tests you have never seen before. Make it as true to the actual test experience as possible. Make an answer sheet and fill in the bubbles just like they will be when you take the test. This might make you realise how long or short the test really will feel like.
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   Don't try to solve a lot of problems right before the big test. Chances are, you will stumble on a couple that you can't get or that really get you interested-and you don't need any extra distractions.
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   Try to get as much sleep as possible. A tired brain does not think as well as a well rested brain.
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   Make sure you bring everything you need on test day and have it ready well beforehand. Have pencils, and also a ruler, protractor, or compass if you feel you might need it. Plan for every contingency. Bring tissues.
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   When you get into the test, pump yourself up. If you've prepared, you have every right to feel confident. Chances are, 99% of the people you are taking the test with(especially the AMC level) haven't prepared as much as you, and that gives you the edge.
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    Once you get the test, stay calm. Fill in the information carefully and completely.
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    Bring a watch and time the test yourself. That way, you won't have to worry about calculating how many minutes you have left with an analog clock on the wall-one less thing to worry about.
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    Make sure you have enough scrap paper. If you want, divide the scrap paper into parts with a box for each question to keep your work organised. You can also make boxes for each question as you go.
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    OK, the test is in front of you and the stage is set. The proctor has finished checking to see that everyone is ready. She looks at the clock-and waits for the second hand to get to the top of the minute. A few minutes later and she says - "You may begin."
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    Open the test book, and get started. Here is where the training should kick in. You should start at the beginning, and read the first question. Don't read it so fast you miss details-this is a possibility if it is a speed test. Read slowly and deliberately, and solve the problem on the scrap paper. When you finish, read the problem again, and make sure you did what they wanted. Once you are sure, bubble it in or otherwise, and continue. Depending on the contest, the act of solving the problem could take 30 seconds to an hour. That is the part that should be savored-the actual solving process. Hopefully, you will be able to solve the problems easily at first, and you will find elements of problems that you have done or practiced before. Hopefully also you will discover solutions to problems you have never seen before, and you will wonder in awe at their elegance. After all, that is what math contests are all about.
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    If you come across a question you can't do, give it one more thought. In lower level contests, there is usually a trick or an easy way to solve the problems. Remember that 99% of the time, you will use all of the information given to you, so if you haven't used something, think about how you might. Oftentimes, one more look will make you realise that there is a solution after all, and it isn't such a hard problem.
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    There will always (well, almost always) be problems you can't solve. Sometimes, you need to calculate whether to guess or not--be wary. If you have eliminated it down to 2, oftentimes the wrong answer will look much more appealing, especially up around AMC questions #20 and up. Make yourself have a logical basis for all your answers, and you will usually get the right answers.
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    Write clearly. When time is winding down, and you scan back to previous work, you need to be able to read it. Don't let misreading a digit keep you from your goal.
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    If you feel yourself slowing down, or getting tired, or even giving up, ask yourself this - How many more chances will I get? Why not push to the finish? I've seen kids walk out of the AIME an hour early. Don't do that. If you re-energise yourself, and give yourself another boost of confidence, you can keep going. A mathlete in this sense is not unlike an athlete, with the test before you as an adversary. Don't give that piece of paper a fighting chance. You know there has to be a way to get every question, and this information alone should help you when tackling problems where you don't even know where to start.
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    Don't be afraid of success. Just because you've never qualified for the USAMO doesn't mean this can't be the year. Just because you've never won the school title for the AMC doesn't mean you shouldn't be the kid who does. Someone has to do it every year-and you have just as much chance as the kid last year. If you are intimidated by someone who you judge to be better than you, you are just putting another distraction on your shoulders you don't need. Never look over to see if they have already turned the page when you are still on the first question-you don't need to think about that because you only need to worry about what you can do.
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    When it's all over, and you've checked over one last time, it's time to let go. Turn in the sheet and get on with life. Hopefully, you saw some problems that interested you. Worry about your score when you get it-don't waste time worrying about it now. Chances are, your training and focus paid off.
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    Evaluate your performance. Did you make your goal? Was the test unusually hard or easy? Did you still miss a bunch of geometry questions? Every test is an opportunity to improve. If you can take it again next year, analyse it and see how you can improve. But don't kill yourself over the stupid mistakes. Making one or more per test is normal, so take this into account when considering your goal. Just look at it and try to find a bright side.
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