Where to actually find good IMO problems and solutions
I spend more time than I want to admit digging through archives, PDFs, and forum threads looking for complete problem sets with solid solutions. The internet is flooded with incomplete compilations. Here is what actually works. The official site for IMO problems goes back to 1959. The IMO official webpage has a problems archive sorted by year. You can download PDFs directly. Each year lists the six problems with full statements and usually a proof sheet or at least an outline. It is not always complete for every single year, especially the older ones from the 1960s and early 1970s, but the coverage gets reliable around 1980 onward. The next reliable source is the IMO Shortlist. These are the problems that were considered but did not make it into the final exam. They tend to be harder than actual exam problems. The Shortlist is organized by topic: algebra, combinatorics, geometry, number theory. If you are preparing seriously, work through the Shortlist after you have exhausted the actual exams. The IMO official archive also links to the Shortlist PDFs.
There are third-party sites that repost everything. IMO Online is one of the more reliable ones. It aggregates problems, solutions, and sometimes videos. Another useful resource is the HMMT and PUMaC problem archives. These are not IMO problems but they follow the same style and difficulty curve, and they fill gaps when the IMO archive is missing a year.
How to actually use these problems
Most people download a PDF and stare at it. That does not work. You need a process. Start with one problem. Give it forty-five minutes of real effort. Not thirty seconds of thinking followed by a search. Forty-five minutes of writing, diagramming, trying cases, failing, restarting. If you cannot solve it, move on. Mark it. Come back to it later if you have time. The solutions matter as much as the problems. After you have spent your time, read the official solution or an alternative one from a reputable source. Do not just glance at it. Write out the full proof yourself. You will catch gaps in your understanding immediately. A solution that looks clean on paper often hides a non-obvious substitution or a case split you would never have found on your own.
Get the Full Details
I once spent three weeks working through the 2008 IMO Shortlist geometry problems. One particular problem asked you to prove that a certain configuration of circles and tangents produced a concurrency point. The official solution used an inversion centered at one of the circle intersections. I kept trying coordinate bashing because inversions felt like a trick. I eventually learned the inversion approach, but it took me longer than it should have because I was resistant to using it. The lesson was straightforward: do not force a method that does not fit. Recognize when a problem is asking for a transformation and switch tactics within the first ten minutes instead of burning an hour.
Common mistakes people make
The biggest mistake is solving too many easy problems. The early IMO exams from the 1960s and 1970s have some problems that feel approachable but do not teach you the right techniques. The real value is in problems that force you to construct an argument from scratch. Focus on problems from 1990 to 2024. That period has the most consistent difficulty and the best coverage of modern problem types. Another mistake is only working on your strong topic. If you are good at number theory, you will avoid geometry and combinatorics. The exam rewards breadth. A single weak area can cost you the difference between a medal and no medal. Rotate your practice so you touch all four topics every week. A third mistake is reading solutions too quickly. When you look at a solution, pause before each step and ask whether you could have derived it. If the answer is no, write down why. The insight is usually in the step you missed, not in the algebra that follows.
What the exam actually tests
The IMO does not test advanced math. You do not need calculus, linear algebra, or any university-level material. The four areas are high school level but pushed to an extreme depth. Geometry requires knowledge of projective techniques, radical axes, and power of a point beyond what is taught in standard classes. Combinatorics involves constructive counting and extremal arguments. Number theory relies on modular arithmetic, Diophantine methods, and order arguments. Algebra covers inequalities, functional equations, and polynomial techniques. The counter-intuitive part is that some of the hardest problems use only elementary tools. A clean application of the pigeonhole principle or a well-chosen invariant can unlock a problem that looks like it requires heavy machinery. Beginners often reach for advanced techniques because they think that is what the exam demands. It does not. It demands precise, elegant reasoning with minimal tools.
Resources and links
The IMO official problems archive is at imo-official.org. The Shortlist is also hosted there. The resource book "The IMO Compendium" by Djurdjevic, Kuiken, and Vukmanovic compiles every problem up to 2009 with solutions. It is expensive but worth it if you are serious. For free alternatives, the Art of Problem Solving forums have extensive solution threads for every IMO problem. The volume links thread is particularly useful for seeing multiple solution approaches. If you need a starting point, begin with the 2010 through 2020 exams. Work one exam per week. Time yourself for seven hours. Then review the solutions thoroughly. Repeat. This method typically builds enough problem-solving stamina and skill to be competitive at the national team level within two years of consistent practice.
