How to tackle Math Olympiad questions - 2
In the last article we talked about some general practices that will stand you in good stead while taking part in the Maths Olympiad (or any exam/test in general). In this edition, we'll talk about various additional techniques that could help you perform better in Math Olympiads.
Most of the time the questions in such competition are of the objective type question. This can make things simpler for you (on the other hand, depending on the ingenuity of the question setter, it can also make it harder for you). The way it makes it easier for you is in two ways
- One is that you can easily check if the answer you compute is part of the options provided to you
- The second is that sometimes, just sometimes, you can do some amount of guessing to eitherget the right answer, or narrow down your possibilities
For example, take the following question taken from http://www.edugain.com
Question: The number 11449 can represented by a 107 x 107 square grid. Out of the following numbers,
which number can not be represented on the square grid?
a. 14641 b. 90601
c. 9216 d. 16122
Here, you could work out the square root of all the numbers and try to figure out the answer, but a quicker way may be to notice that a square number cannot end in 2. In addition, the question reads "which number", which implies there is only one choice that is true.
So the answer here is obviously d) - 16122
Similarly, in a lot of these cases, you can eliminate at least one of the choices - there can be many variants that you could identify by just looking at the question. Ask yourself some mental questions like "Can the answer be an odd number?" etc. Of course, each mental question you ask yourself will be specific to that question.
Sometimes, a question can seem rather hard at a quick glance, but if you read it once more slowly, you can probably see past the apparent complexity into a simple solution.
Consider the question (also taken from http://www.edugain.com)
Question: Archana's roll number is a two digit number. Her friend Balvinder's roll number has the same digits as Archana's roll number, but with digits interchanged. If they add their roll number, and divide the sum by 11. Find the remainder of this division.
a. 1 b. 2
c. 0 d. Can not be determined without knowing the roll number
At first glance, it may seem really hard - how can you know without really trying? Here you can use one of two methods. One is not really what I would call mathematical, but if you are pressed for time, you could easily use it for such question. And what's that technique?
It's simple - all you do is try it out on a couple of example numbers. Imagine that Archana's roll number is 61. Then Balvinder's roll number would be 16. Add 61 and 16, and you get 77. The remainder when 77 is divided by 11 is 0.
However, you also need to know how to solve this directly (after all, you are here because you want to learn mathematics).
The way to do that is to think of it as follows
Archana's number has two digits - say they are "ab" where a and b could be any of 0,1,2,3,4,5,6,7,8 or 9.
You can represent it mathematically as
10a + b
Then Balvinder's number would be "ba" - mathematically you would say
10b + a
Add them up
10a + b + 10b + a = 11a + 11b = 11(a+b)
This is of course divisible by 11 leaving a remainder of 0.
Simple, and more satisfying to do it this way, isn't it?
So you should always keep in mind that no matter how hard the question looks, it is very likely that you have already been taught the tricks to solving it. There may be a few questions that may use concepts you haven't been taught yet (to identify the really advanced students), but there's not much you can do about that. For a majority of the questions though, you already know all the techniques you need to solve them. It's just a question of identifying them and applying them.
You should also know your strengths and use them appropriately. For instance, you may be very good with spatial questions and at visual data - then go for the geometry or figure related questions first. If numbers dance in your head, go for the arithmetic questions. Get them out of the way, and then move on to the other sections.
One final note, that I mentioned in the previous article. Once you've found a solution, put it back in the question and double check that your solution is indeed right.