How to solve P2 and P5’s at the IMO
The structure of most problems in the Maths Olympiad is straightforward; they give you some sort of structure with a set of constraints, perhaps a sequence with some condition between terms, a set of points constructed from triangle \(ABC\), a function with a functional equation attached to it, or even just the fact that \(a\), \(b\), and \(c\) are integers satisfying some equation.
They then ask you to prove some particular results from the information you are given. But sometimes it seems like there is nothing you can do. How do you even start? How do you make progress?
Remember that all the information you need to solve the problem is given to you. What we are trying to do should be possible. Perhaps after drawing the diagram a couple of times, we confirm that some result is indeed true; maybe we tried our small cases, and our answers do indeed follow a pattern. If this is the case, then something about the information you are given must force the desired result to be true.
All the information you need to solve a problem is in the problem; sometimes it’s just in a form that is not particularly useful to you. You should ask yourself: Have I used all the information given to me? And if the answer is no, then you should ask: How could I use this piece of information?
Oftentimes, the information you get is “negative, meaning it tells you that something isn’t the case, and it can help to convert this into “positive” information, which tells you something is the case. For example, the fact that \(p\) doesn’t divide \(n\) is negative, but it can be turned into something positive by realizing this implies that there exists \(x\) and \(y\) such that \(xp + by = 1\).
Sometimes, to prove a certain claim, you don’t need all the information given to you; it helps to figure out which information is actually relevant to a claim and ignore everything that isn’t. (Of course with proof by contradictions even the desired result can also be used as information).
A technique I learned is that if you want to be sure if a particular piece of information is necessary, just remove it. Then you can draw your diagram again, try your cases again, and if things stop working, you know that condition was actually important; otherwise, it’s not strictly necessary.
Sometimes, when we use a piece of information, the new information we get somehow contains “less” information than before. Most of the time, you can intuitively feel when there is no new information to gain from the information you are given, but be careful not to toss something away too early when you might need to use it later.
Even if the information you have left over is technically sufficient to solve a problem, sometimes it is just not as nice to work with; the obvious example of this is converting a geometry problem to a purely projective statement. It might technically use fewer points and thus appear simpler, but angles are just so nice to work with. You need to carefully balance using up your information while still having control over the information you have.
The key to solving every problem is to use your information in ways that give you a lot of control.
The key to constructing points is to pick points that give you the most useful information. Constructions are not created equal, and the key to figuring out the best one is finding the one that gives you a lot of information you can actually work with, in the context of geometry it often means having angles that are easy to chase.
In a combinatorics problem, the key to deciding whether to view the problem as a graph, whether to try a global bounding argument or whether to analyze moves individually comes down to which method gives you the most useful information about the problem.
Sometimes you need to lose information to gain control, it is a trade-off but it’s not permanent one and it’s up to you to decide.
You can’t tell people anything, So this is advice I’ve written for myself: