The eight stages
Olympiad maths changes as you get better at it.
At first, the challenge is spotting the right idea. Then it becomes combining ideas, persisting when the obvious approach fails, and eventually constructing proofs from scratch.
The track begins at Stage 3, around AMC 10 level. Below that, you will usually learn more from building the underlying mathematics than from doing harder Olympiad problems.
Stage 3
Learn to spot the idea.
The mathematics is usually familiar. The challenge is recognising the observation, substitution or small trick that makes the problem fall into place.
Stage 4
Learn to connect ideas.
The route is less obvious now. Problems increasingly ask you to combine two ideas, change perspective halfway through, or notice something that was deliberately left unstated.
Stage 5
Learn to keep going when the first idea fails.
Problems stop yielding to a single observation. You will try approaches that go nowhere, abandon them, and start again. Persistence becomes part of the mathematics.
Stage 6
Learn to turn an idea into a proof.
Finding the answer is no longer enough. You have to explain why it must be true — clearly, rigorously and without gaps.
This is where proof-based Olympiad mathematics begins to take over.
Stage 7
Learn to find the structure the problem is hiding.
The useful idea may be several steps away from anything obvious in the question. Standard methods still matter, but knowing them is no longer enough.
You have to discover how the problem wants to be attacked.
Stage 8
Learn to work without knowing whether you are close.
At this level, an hour of work can produce very little visible progress.
You learn to explore, test ideas and follow partial observations without knowing whether any of them will lead to the solution.
Stage 9
Learn to stay with one problem for hours.
A difficult problem may take an evening rather than a sitting.
The breakthrough can come after several failed approaches, one useful lemma, and a long period in which it feels as though nothing is happening.
Stage 10
Face problems you may not solve at all.
These sit at the far end of the collection.
You may spend hours on one and still not reach a complete solution. At this level, even partial progress — finding the right transformation, proving one key lemma, seeing why an approach fails — matters.
A note on Stage 10
Stages 3–9 are calibrated against measured difficulty. Stage 10 is different.
The source data does not reliably distinguish between problems at the very top of the difficulty range. So Stage 10 should not be read as a precise level above Stage 9.
It is simply where we have placed the hardest group of problems in the collection.
Treat the number as a signpost, not a measurement.