How to Approach anOlympiad Problem(by Ho Jun Wei)Posted on March 23, 2012 by khorshijie

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The following article was written by Ho Jun Wei who was an IMO medalist in 2006. He

has trained several members in the Singapore IMO team in the past and was my MO

trainer when I was Sec 4. He is currently studying Mathematics in Cambridge University.

The following post presents his insights in solving an MO problem.

In this post I shall provide several tips on how to tackle a long Olympiad problem,

particularly those in SMO (senior/open) Round 2. While I draw my examples mainly from

problems in number theory, the strategies I mention are very general and apply to most

Olympiad problems. The opinions expressed in this post are entirely mine, and I accept that

everyone has a different way of approaching problems. Here I’ll share some good problem

solving habits I’ve picked up over the years, and feel free to share yours too. In trying to

keep this article general and applicable to most types of problems, I’ve refrained from going

into very specific methods for particular topics like combi or NT. I may talk about more

specific techniques in a separate post later.

First, let’s get this clear: there is no substitute for practice. There is no shortcut tobecome really good at solving math Olympiad problems. I once coached a student who later

achieved a remarkable 2 place (internationally) and Gold medal at a recent IMO. You

may know who I’m referring to. What’s his secret? He spends an average of several hours a

day (often up to 4 or 5 hours) solving Olympiad problems. Over time, these hours put into

solving problems will help you amass a huge vocabulary of Olympiad tricks, techniques,

theorems and lemmas, and hone your problem solving intuition. Doing math Olympiad is

very much like playing soccer. While a soccer match only lasts for 90 minutes, it is what

happens before the match that truly decides the game – the hours put into training daily to

gain incredible agility, stamina, and an impeccable technique. Of course, the strategies

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adopted during the 90­minute match are also important, but never as crucial as the

training and preparation that comes before it. Even if you have very good match strategies,

you won’t be able to execute them properly if you don’t have the required level of fitness or

skill. Fitness and skill comes with practice, and practice makes perfect.

A good Olympiad problem is one that uses very elementary techniques in very clever ways. It

should not require the use of any very advanced math theorems, beyond the usual handful

(e.g. Fermat’s Little Theorem, Ceva’s and Menalaus, AM­GM, etc.). There was an earlier

post by Shi Jie that touches on these (refer to the links above), which I feel is a fairly

comprehensive list (and you should make sure you already know most of these by heart).

These theorems may prove very handy in some problems, but in fact many problems in

SMO don’t even require the use of any of these theorems; they simply require keenobservation or the clever applications of simple techniques like factorization, pigeonhole

principle, or similar triangles. These simple techniques go a very long way. So myadvice is to invest your time in mastering the basics and how to spot clever ways of using

them, rather than learn fanciful stuff like quadratic reciprocity, group theory, Ramsey

theory, solving geometry problems by inversion/complex numbers or proving inequalities

using Lagrange multipliers, just to name a few. Practically all SMO problems can be solved

using elementary methods and, occasionally, one or two basic theorems. The only non­

standard method that I think may be useful is coordinate geometry, or vector geometry

(don’t worry if you don’t know these) – but even these are not necessary because any

problem that can be solved in this way can be solved using elementary methods as well.

In the course of coaching Math Olympiad, I find that many students commonly get stuck

on a problem either because they don’t know how to get started, or they’ve run out of things

to try after some time. The well­prepared problem solver would not face such situations: If

you know your usual theorems, and know your basic tools and techniques, you shouldnever really run out of things to try. You would first try to change or simplify theproblem if possible, try small cases, then pick an approach, try to prove it and maybe fail,

pick another approach or maybe try to prove a stronger result that you conjectured, find

out that your conjecture was false, move on to another approach, etc. If you have tried

enough Olympiad problems, you should be familiar with common tricks and techniques for

particular topics, and you will be able to use these as a starting point for approaching

problems.

Pre­match preparation: Theorems, Lemmas, Techniques

1. You should already know all the usual theorems by heart, and if possible,

understand the ideas behind their proofs as well. These theorems form part of the

backbone of your thinking process. They are a crucial part of your toolbox.

2. Build up your arsenal of lemmas. These are simple results that are not formaltheorems, but can be very helpful in tackling more complex problems. For instance,

did you know that the centroid , circumcentre and orthocenter of a triangle

are collinear, with ? Or that has an integer solution

if and only if the prime is of the form ? A sizeable collection of lemmas

can only be built up by doing more problems. The more lemmas you know, the

more tools you have at your disposal, the easier “standard” problems will become.

The IMO team that I had the privilege of coaching had their personal self­

compiled “handbook of geometry lemmas” and “100 standard inequalities” which

they knew by heart, and that proved very useful because some hard problems

became almost trivial once you knew one or two of these lemmas beforehand.

3. Know your techniques!!! These are general methods or approaches to solving aparticular type of problem. For instance, when faced with a polynomial problem,

we can consider a polynomial in terms of its roots, in terms of its coefficients, or in

terms of a quotient and remainder using the remainder­factor theorem. These

multiple perspectives give us many things to try! For inequalities some common

techniques are smoothing, expand­then­AM­GM or substitution. In geometry, a

common technique is to draw one or more helping lines (cleverly chosen). Another

usual approach is to show that certain points are concyclic / collinear, and there

are other less standard methods like coordinate geometry. In combinatorics, things

like pigeonhole principle, invariance principle, extremal principle, double

counting, finding recursions or bijections are very useful. In number theory,

techniques like modular arithmetic, bounding, factorization are widely used. For

problems of the form “show that (something) is true for all integers ”, a viable

technique may be induction. This is the most important point, because many

students run out of things to try too quickly. With a large bag of techniques

available, you will always have many approaches to pry a problem from different

angles. This can only be attained by solving more Olympiad problems and learning

new techniques. I will talk about general techniques for combinatorial problems in

another post.

To illustrate what I mean by “the well­prepared problem solver never runs out of things to

try”, let us look at an SMO (senior) 2004 problem: “Find all integer pairs that satisfy

“. It is a good habit to first find easy solutions like . The fact

that a solution exists already eliminates some possible approaches (like using modular

arithmetic to prove the no solutions exist). It is easy to see that we only need to consider the

case when are non­negative, since is a solution iff is a solution and clearly

RHS > 0 so . It is also natural to first factorise the LHS as . At the

start it is intuitive to try approaches like taking mod 16, or bringing the 1 to the LHS and

factorising it as a difference of two squares to see if you can deduce anything (usually it

gives some conditions on parity). Alternatively, you may try to consider cases when or is

even or odd. Some students may run out of things to try at this point. However, if we had

done enough of such problems, we’ll have many other things to try. For instance we can tryparticular cases by putting or , to see if these particular cases give any

result or insight. Indeed we will find that gives solutions. Another fairly standard

approach is to find some bounds on the solutions (e.g. show that or or

cannot be too large). In this case, just by looking at the equation we can tell that if

becomes too large, then LHS > RHS so no solutions can exist. For instance, if

then , so by comparing with RHS we find that ,

which reduces it to just a few cases to check. In fact it turns out that we need ,

thereby reducing the problem to a few cases that are easy to solve by hand (Exercise: Verify

that the only solutions are ).

If trying to bound the solutions didn’t work, some other possible ways we could have

messed around with this problem are: take mod , we get so maybe we can

try substituting and rewrite the expression as a quartic polynomial in y and try

to factorise. Or we could conjecture that there are infinitely many solutions arising from a

clever construction (e.g. like for all integers ) which we

would try to guess and derive. These approaches may not lead to anything useful, and our

conjectures might be wrong, but at least we never run out of things to try. Given more

experience, it is easier to tell what approaches may work, which ones will lead to a dead

end, and what conjectures are more likely to be true, thereby making our problem solving

process more efficient. The point is that we don’t run out of approaches to try.

During the match: How to be effective at your game

1. Whenever possible, always try small cases (or extreme cases) first to make sureyou understand the problem and see how it works, possibly obtaining trivial

solutions (Note: In a geometry problem, “trying small cases” means drawing the

diagram in different ways or considering degenerate cases, in inequalities it means

trying to spot the equality case or testing extreme values). If you’re lucky, by trying

small cases you may be able to spot a pattern or gain some insight as to why the

problem works in a certain way, which could lead you to the solution. Or perhaps

you could make a smart conjecture by observing small cases (as in the above NT

problem, one might try out particular values e.g. gives solutions, and

trying may help you realize that cannot be too large).

2. Restate the problem in different ways. A geometry problem that asks you to

show that is perpendicular to may be a disguise for asking you to show

that (check that the two conditions are indeed

equivalent, this is a lemma!). This restates a problem of angles to one of length,

which may be useful if the conditions in the problem give you information about

length. An inequality problem may sometimes be transformed by means of a

suitable substitution. In other words, try to write equivalent formulations of the

same problem, thereby changing the original problem to something that’s possibly

easier to prove. (take care to make sure that the new problem is indeed equivalent

to, and not stronger than, the original!)

3. “Wishful thinking”. This is a very useful strategy to keep in mind. By carefulobservation or by trying small cases you may conjecture that certain things may be

true, upon which the problem would be a lot easier. For instance, one may wish “if

only the circumcentre lies on this line…” or “if only this number is always a perfect

square…” or one may conjecture, after trying small cases, that a particular function

is surjective/monotonic. Another example: If you’re given a combi problem that

asks “for what values of n does (a certain property) hold”? You might try small

cases then guess that it works only when n is a triangular number or power of 2,

for instance. Now try to prove that wish/conjecture, because it might just be true

and make your life much simpler.

4. Prove something stronger than the original problem. This is rare and difficult to

spot, but sometimes can be the only way to solve a problem. For instance, Shi­Jie

has shown in a previous post how some inequalities can be proven by proving a

stronger inequality. There are some inductive proofs that work only if you prove a

stronger statement (because it also strengthens your induction hypothesis).

5. Don’t be too fixated on any particular approach too early in the game.When first exploring a problem, keep an open mind, think of different approaches

to the problem, and see what information each of them will give you. Do not spend

too much time on any single approach or conjecture, especially if you’re not sure

it’s the right one. For instance, if you make a conjecture and try to prove it but keep

failing, it may not even be true; try to disprove it instead. Having said that, don’t be

a schizophrenic and keep jumping from one approach to another – you risk giving

up the right idea too quickly. How long you should spend on one particular

approach is a very subjective matter, and in fact it is a judgment skill that comes

with experience i.e. more practice!

6. Always try easy/obvious approaches first before moving on to more adventurous

things.

7. Look at the conditions or constraints given. If you haven’t used a particularcondition yet, ask yourself why it may be helpful. If an inequality has constraints

like , it is useful for homogenizing or substitution. If it has a constraint like

, this may be a clue that you should use some trigonometric stubstitution. If

a combi problem has conditions like “every two students did 5 questions in

common”, then clearly you have to consider some suitable quantity where this

constraint would give you some kind of equality/inequality. If you’re given a

number theory problem like find all integer solutions to , then

taking mod 4 or mod 3 should be useful.

Finally, let’s look at a recent SMO (Open) 2010 problem: Let be a two

sequences of integers defined by and for , we have

. Prove that is the difference of two cubes.

How do we start to prove something like that? Clearly we don’t need stuff like Fermat’s

Little Theorem or the Chinese Remainder Theorem here. The experienced problem solver

will immediately see two viable strategies for this problem: induction on (we still need to

figure out what the hypothesis should be), or by clever manipulation / guesswork to guess a

closed form for the recursive sequence. In any case, the most obvious thing to do first is to

look at small cases: It is easy to work out that and

. We also find easily that and . At this

point, based on the cases the sharp problem solver might already have a suspicion

and make a conjecture, which one can confirm using the case. If not, we can try to

work out the cubes for . It is less obvious what consecutive cubes this will give,

because the numbers get big and unmanageable very quickly for large . But by letting

and some simple calculations we obtain

which we easily solve to get . Look at the value for again… smell something

fishy? It becomes very natural to “wish” that , which is true and very

easy to show by induction. Moral of the story? Try small cases to see how the problem

works! This is what I mean when I say that many SMO problems don’t require any

theorems at all, but simply require astute observation skills and familiarity with elementary

techniques and approaches for such problems.

If you’ve read the entire post till here, thanks for taking the time. I hope I managed to at

least shed some light on how to approach an Olympiad problem. I’d like to reemphasize

that the only way to improve is to do more problems (it’s good to look at solutions and learn

from them when you’re completely stuck, or even when you managed to solve the problem).

For starters, if you don’t know the usual handful of theorems, go learn them first. Then

keep building up your bank of tricks, techniques and lemmas (for instance by reading this

blog often and trying more problems), and you’ll find that in time to come, you won’t run

out of ideas and approaches so easily when faced with these pesky long problems.