What Is a Math Olympiad and How Do You Get Started?

A math olympiad is a timed, proof-based competition in which high school students solve a small number of unusually hard problems drawn from geometry, number theory, combinatorics, and algebra. You get started by working through past problems and handouts rather than by taking a course: the standard entry points are the AMC/AIME sequence in the United States, and the free beginner materials collected on coach Evan Chen's site (web.evanchen.cc) are a reasonable first stop. This applies if you are a high school student with solid algebra and geometry who wants to test whether competition math is for you; it does not require having finished calculus or any university-level material.

What makes olympiad problems different

School math asks you to apply a known method to many similar exercises. Olympiad problems ask you to find the method. A typical problem gives a short statement and expects a written proof, not a numerical answer, and the intended solution often uses an idea you have not seen before.

The four recurring subject areas are:

• Geometry — circles, triangles, collinearity, concyclicity; synthetic proofs are the norm.
• Number theory — divisibility, modular arithmetic, primes, Diophantine equations.
• Combinatorics — counting, pigeonhole, extremal arguments, invariants, graph problems.
• Algebra — inequalities, functional equations, polynomials, sequences.

A useful illustration of the flavor: Evan Chen's own homepage displays a geometry fact — "The nine-point circle of $\triangle BIC$ is orthogonal to the circle with diameter $ST$" — attributed to Taiwan TST Quiz 2015. That is a competition problem statement: precise, self-contained, and requiring a construction and a proof rather than a computation.

The main competitions, roughly in order of difficulty

The names above are the ones listed in the navigation of Evan Chen's site, which also links to a Mock AIME, an archive of past problems and solutions, and pages for MOP, ELMO, and USEMO. If you are outside the US, the same structure exists under different names — most countries have a national olympiad feeding into the IMO team — but the preparation material is largely interchangeable.

How to actually start

1. Check your baseline. Try a past AMC 10 or 12 under time. If you score well but find the last few problems hard, you are ready for proof-based work. If the early problems are already difficult, spend a few months on fundamentals first.
2. Pick one subject and go deep. Beginners usually do best starting with geometry or number theory, because the toolkit is finite and the feedback is fast.
3. Work problems before reading solutions. Give a problem 30–60 minutes. A solution you read without struggling teaches much less than one you nearly found.
4. Keep a written record. Write full proofs, not sketches. Olympiad grading rewards complete arguments, and the habit is built only by practice.
5. Use structured handouts. Evan Chen's site hosts a collection of handouts and a beginner's page aimed exactly at this stage, plus an archive of past problems with solutions.
6. Find feedback. Solutions you cannot verify yourself need a second reader — a coach, a club, or an online community. His site also lists a Discord link.

Expected result: after a few months of consistent work you should be able to solve mid-level AIME problems reliably and produce a coherent proof attempt on a USAMO-level geometry or number theory problem, even if it does not fully close.

Common sticking points

• Reading solutions too early. The most frequent failure mode. The struggle is the training.
• Skipping the basics to chase hard problems. Inequality tricks and combinatorial invariants are built on fluency with algebra and counting.
• Neglecting writing. Many strong solvers lose points for gaps in logic, not for missing ideas.
• Treating it as a solo grind. Feedback on your proofs is hard to generate alone.

Where to find materials

Evan Chen's homepage (web.evanchen.cc) collects, in its own navigation: a beginner's page, a page for coaches, handouts, an archive of past problems and solutions, a Mock AIME, and pages for MOP, ELMO, and USEMO. It also links his textbook Euclidean Geometry in Math Olympiads, published in 2016, and course notes. His background — IMO gold medalist, 2014 USA Math Olympiad winner, and now a coach who has trained the US team — is the reason these materials are widely used as a starting point.

For a first week, a workable plan is: one past AMC under time, then one geometry handout worked problem by problem with full written proofs.