Open a blank document or notebook. For every chapter, solve 80% of the exercises. Write full, grammatical sentences. A solution like "x=5 works" is insufficient. The primer teaches argumentation, not calculation.

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In the echelons of competitive mathematics, there is a distinct rite of passage. It is the moment a student moves from solving linear equations to grappling with the esoteric logic required to crack Olympiad-level problems. For decades, one book has served as the bridge between the classroom and the contest hall: A Mathematical Olympiad Primer.

Available widely as a sought-after PDF in the digital libraries of aspiring mathematicians, the book—most notably the version authored by the late, great Geza Kiss—is more than just a collection of problems. It is a masterclass in how to think when the answer isn't obvious, and the path to it isn't linear.

Unlike pure number theory texts that drown you in modular arithmetic notation, Smith introduces the modulo concept through ancient puzzles. Key highlights include:

Smith’s commentary on Fermat’s Little Theorem is particularly memorable—he presents it as a "magic trick" that works more often than it should.

If you find a clean, complete PDF – use it as a quick start guide. But the physical book is cheap (often under $15 used) and the diagrams/solutions are reliable. For a free alternative, the UKMT’s own past papers + solutions cover similar ground.

Final note: The author, Geoff Smith, has publicly said he’s fine with students sharing the book for educational use within a small group, but large-scale PDF distribution would harm UKMT’s charitable work. If you like the primer, consider buying a copy — the royalties support UK Olympiad training.