AMC tips

These are just some things that I've seemed to notice about the AMC in general:

The first 12 problems - These are really just to separate who knows basic math and who doesn't. I would say that the first 5 or so are like SAT math problems. The rest are a bit harder, but really just simple extensions of basic math. If you have any trouble with these, then either you don't know your basic math, or you're not looking at it the right way. If you're confident in your math skills and you're having trouble, then look at it again, you're probably making it harder than it is.

Stupid mistakes - Everybody makes them, and your goal should be to cut down on them as much as possible. A way to reduce a lot of them is to keep your work organized. This makes it easier to keep track of your work. If you get lost in it, then you'll probably pick out a wrong number somewhere along the way (remember, it's non-calculator) and screw up the answer. Also read the question slowly and carefully. A good way to make sure you completely understand it is to underline / box key words. Example:

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?



Ok, you've read the problem quickly, and start going through the math. If 1 is subtracted from each number with equal chance, then there's 1/2 chance of being 4 odds and 2 evens, and 1/2 chance of being 2 evens and 4 odds. so the probability is:



Ok, that's one of the choices, so mark C and go on...or maybe you should reread the problem quickly, to make sure you read it correctly. Oh, not every number has the same probability of being chosen, just each dot. That changes things a lot. Reread the question after you fill in your answer to make sure you gave them what they wanted.

Obviously it's time consuming to label and explain to yourself every single piece of work, considering you're being timed, so it's reasonable that you don't. However, this means you're more likely to make mistakes. I would think that the best solution is to remain balanced in your work organization and your speediness; that is, don't label every single thing, but perhaps draw arrows to show where work is coming from, and box important numbers in your work.

"Trick" questions - I don't think the AMC really has trick questions persay, but just questions that look difficult, but aren't. For example:

How many ordered triples of integers , with , , and , satisfy both and ?



Ugh, have you ever seen anything like that before? I haven't. Skip it...
...or look at it again. Obviously the answer is one of the multiple choices, so if we do some casework, then worst case, we have to find 4 solutions (this is also an example in which the multiple choice answers give you a hint. Obviously they aren't very useful for probability problems, but for this problem they were incredibly useful). We're given restrictions on each, so it's not too bad:

c = 0, then b = 1 (properties of logs here), and to satisfy the 2nd condition, a = 2004. Thats one solution.

c=1, then a=b, and to satisfy the 2nd condition, a=b=1002. That's two solutions.

c=2, Ugh, is large and ugly. A is at least 2, and b = , making b a ridiculously large number. Obviously no more solutions can satisfy the 2nd condition, so there are only two solutions.

This is a clear example where a little casework goes a long way.

WLOG - If you aren't familiar with that, it stands for "without a loss of generality". It means that by solving a specific case of a problem, you solve all cases of that problem. I don't really have a good example, but a problem where it would be useful is where they don't actually give you values for a problem, but want something like a ratio. In this case, you'd plug in something simple for one of the values, and then solve it. Obviously solving it this way is far from rigorous, but they don't care how complete your solution is, just that you marked the right letter on your answer sheet.

KISS - Another acronym that everybody has seen - "keep it simple, stupid". How does that apply to the AMC? Except for maybe the last couple of problems, the problems are designed to test not only your math skills, but your ability to recognize which ones to apply to a given problem. I say except for the last couple because those ones are a lot harder than the rest. Again, if you're having trouble with one of the later problems (13 - 22), and you're confident in your math skills to solve problems of that difficulty, then look at it again, and make sure you're not overlooking something. The problems can be solved without guess and check, too, but don't mix that up with casework.

Time-management - 25 questions, 75 minutes...so 3 minutes per question? I think it's absurd for even some of the smarter kids to solve the last couple of questions in 3 minutes each. The best way to manage your time is to breeze through the easier ones to get more time for the harder ones. If each of the first 12 takes you 1 minute each, then that leaves around 63 minutes left, or around 5 minutes for each of the other one. Of course, these are rough estimates. But if any of the first 12 are taking you more than a minute, box it and come back to it after looking at some of the other problems. Whatever you do, though, put priority on it. Those first 12's should be gimmes towards your score.

#24, #25 - These are the really hard ones on the test. Honestly, unless you KNOW you're capable of a perfect score, I wouldn't even bother with these, and that's that. Focus on the other ones. Remember, that question asking you to maximize the value of tangent of half of an angle on a triangle based on one of the ambiguous side lengths is worth just as much as that much more appealing one asking you how many bananas Jeff can buy if he has x amount of quarters. If you're sure that you passed the 100 point mark, then you might as well guess on them. Otherwise, I'd leave them blank, you may need that extra 3 points to push you over.

Preparation - I put this last because this applies to me last, I pretty much know the basics that I need to do well on the AMC. But if you're curious, the common topics for the problems seem to be:

Arithmetic, Quadratics, Absolute Values
Algebraic Techniques
Number Theory
Polynomials, Logarithms, and other Functions
Binomial Expansion, Inequalities, Optimization, Systems of Equations
Counting and Probability
Statistics, Sequences, and Series
Complex Numbers, Trigonometry, Law of Cosines
Right Triangles, Quadrilaterals, Pick's Theorem
Circles and Spheres
Triangles, Polygons, Polyhedra
Coordinates, Graphs, and Geometric Inequalities

I actually copy pasta'd that from an amc prep class syllabus. But those are the general things you should know. They threw in some specifics in there because they're common (I dunno about Pick's theorem though, I don't think I've EVER seen that on any amc problems, maybe one AIME problem). Geometric inequalities are things like the maximum side length of a triangle given this this and this, and other similar problems. Polyhedra are those ugly 3-dimensional problems they always ask. Number theory is properties of integers. I'm sure you know what the rest are. But if you're interested in preparing for the amc, those are the things you should be familiar with.

If I think of anything else, then I'll add it. Otherwise, that's all that comes to mind.