2010 Math Prize for Girls, Problems 1-5

Since the 2011 Math Prize for Girls competition is coming up next month, I thought I’d go over last year’s contest.  The problems can be found here.  Because solutions are provide there too, here, I will indicate what students can do if they had trouble with a certain problem and briefly comment on the mathematics involved. I’ll spread these comments over a few posts since there are 20 questions total.  You might want to open up a separate window with the problems because I won’t repeat the problems here.  (By the way, in Math Contest Prep, we will rarely, if ever, go through a contest like this. Contest problem examples will be more thematically selected.)

Problem #1 This problem checks to see if you can simplify algebraic expressions. There isn’t a lot of mathematical substance to this kind of question.  The thing you have to guard against are careless errors.  To avoid making them, write neatly, use ample paper, and don’t give in to the urge to do too much in your head.

If you have trouble and it’s not because you don’t write neatly, you’ll want to improve your skill at algebraic manipulation and the best way to do that is through exercises.  Also, become familiar with some oft-used algebraic identities, such as the binomial theorem or the identities a^2 - b^2 = (a+b)(a-b) and a^3-b^3 = (a-b)(a^2 + ab + b^2), which can be verified by multiplying out.

Cammie Smith Barnes’ Errorbusters! column in the Girls’ Angle Bulletin offers lots of useful advice on just this topic.  See for instance her column in Volume 4, Number 4.

Problem #2 A discrete probability problem.  If you’re having trouble, start with simpler problems (ones where the range of possible outcomes is easier to determine) and gradually increase the complexity. Discrete probability is intimately connected with the art of counting and there are many concepts that help one count more efficiently and in a more organized manner, such as the principle of inclusion and exclusion.  Even multiplication can be viewed in this way, and understanding when to multiply is important.  (In probability, this is related to independent and dependent events.)

Katherine Sanden’s article Just for Kicks describes a tool called an outcome tree that can help with problems like these. A nice introductory book on probability is Warren Weaver’s Lady Luck: The Theory of Probability. And for a nifty collection of probability problems, check out Frederick Mosteller’s Fifty Challenging Problems in Probability.

Problem #3 A key observation is that because perfect squares are nonnegative, none of the squares on the left-hand-side of the equation can exceed 34. This fact limits the number of possibilities considerably.

If you have trouble, perhaps it didn’t occur to you to work this problem out case by case. That determination is aided by experience, but as a rough guideline, if you can see that there are a few solutions and you can systematically work through them, then it’s worth considering. On contests, you sometimes have little choice but to bite the bullet and do some laborious computation. This problem, however, is really not so bad to work out.

Problem #4 This problem tests whether you are comfortable with arithmetic means and medians. Because x is unknown, you do not know where it sits in the order of the six numbers, so you have to split into cases depending on whether or not x is involved in the value of the median. Technically, breaking this problem into such cases is the key idea. When you break a problem like this into cases, you get separate equations for each case and you have to remember that after you get your answers, you must check these answers to make sure that they satisfy the conditions that define the case they’re supposed to belong to.

If you’re having trouble beyond the understanding of the definitions of mean and median, it might be because you made an assumption about the median without realizing it and this caused you to assume a specific value for the median and neglect other possibilities. Often when solving math problems, an impasse is met when it isn’t clear how to proceed because there isn’t a unique choice. The way out is to split into a complete set of cases where, in each case, you make a definite choice. You have to abandon the notion that you can always solve problems by solving a single nice equation. This is actually a very common theme in mathematics. Another place where this kind of casing is common is in problems that involve absolute value.

Problem #5 All those divisibility rules come in handy for this problem, although it wouldn’t be outrageous to solve this problem by just starting at 10 and working your way up until you find the first two digit number that does divide into the given number. There are lots of divisibility rules and techniques for checking for divisiblity quickly that are fun to learn about, and many of these are applications of modular arithmetic and a few general facts about divisibility, such as that if both x and y are divisible by a, then both x + y and xy are too.

If you’re having trouble, learn properties of divisibility and the divisibility rules (such as “casting out nines”). They can be a lot of fun and after you’ve learned them, you’ll be able to do things like say right away that the number 550,451,352,253,154 is divisible by 11! You can read about these rules in almost any introduction to Number Theory.

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