Brief explanations of advanced topics such as the Pigeonhole Principle, modular arithmetic, and Euclidean geometry.

n=p1e1p2e2…pkekn equals p sub 1 raised to the e sub 1 power p sub 2 raised to the e sub 2 power … p sub k raised to the e sub k power

Olympiad geometry emphasizes properties of triangles and circles over coordinate-based calculations.

Furthermore, the existence of these primers as free or low-cost PDFs represents a profound shift in educational equity. Not every talented student has access to a specialized math coach or a well-funded summer program. The PDF bypasses geographic and economic barriers. A student in a rural village with an internet connection can download the same primer as a student in a metropolitan math circle. This accessibility transforms the primer from a mere book into a global equalizer. It provides a structured, self-guided curriculum that teaches not just what to think, but how to think—cultivating the patience to wrestle with a single problem for hours and the humility to learn from elegant, concise solutions. The digital nature of the resource allows for hyperlinks to video explanations or community forums, creating an interactive ecosystem around a static document.

This is the heart of the book. It features a selection of problems primarily from the British Mathematical Olympiad (BMO) rounds 1 and 2.

(nk)=n!k!(n−k)!the 2 by 1 column matrix; n, k end-matrix; equals the fraction with numerator n exclamation mark and denominator k exclamation mark open paren n minus k close paren exclamation mark end-fraction 4. Synthetic Geometry

| Book | Comparison | | :--- | :--- | | | AoPS books are more comprehensive and voluminous. The Primer is more concise. If you want a 300-page deep dive, choose AoPS. If you want a sharp 150-page overview, choose Smith. | | Problem-Solving Strategies (Arthur Engel) | Engel's book is the "Bible" of Olympiad prep. It is denser, harder, and more systematic. Smith's book is a prerequisite to Engel; read Smith first, or Engel will crush you. | | Mathematical Olympiad Challenges (Andreescu/Gelca) | Andreescu’s books are excellent but often jump straight to hard problems. Smith focuses more on the writing and structure of proofs, making it better for beginners. |

a4+4b4=(a2+2b2+2ab)(a2+2b2−2ab)a to the fourth power plus 4 b to the fourth power equals open paren a squared plus 2 b squared plus 2 a b close paren open paren a squared plus 2 b squared minus 2 a b close paren