The Curriculum

Theorem, example, method.

The Academy's curriculum traces the four classical pillars of olympiad mathematics through algebra, geometry, number theory, and combinatorics. Pedagogy organises the curriculum. We move from theorem to worked example before naming the method beneath the calculation.

Algebra

Olympiad algebra lives at the intersection of inequalities, identities and structure. Teachers study how estimates, rewrites, and purposeful substitutions reveal the structure of a problem. Symmetric polynomials, factorisations, and functional equations provide the working language.

Classical inequalities, arithmetic and geometric mean, Cauchy Schwarz, rearrangement, and power mean
Symmetric and homogeneous polynomials, Newton's identities
Functional equations and the discipline of substitution
Polynomial roots, Vieta's relations and Lagrange interpolation
Algebraic structure in number theoretic problems

ii.

Geometry

Geometry is the most visible olympiad subject and the easiest to teach badly. Our approach is patient and synthetic. A clear diagram and a focused set of techniques support steady progress toward transformations and projective methods.

Classical synthetic geometry, circles, angles, similar triangles
Power of a point and the radical axis
Transformations, rotation, reflection, spiral similarity
Projective and inversive geometry as olympiad tools
The pedagogy of the diagram

iii.

Number Theory

Number theory is where the Academy spends much of its time, because it is where the gap between a willing student and a trained one is widest. Year Two has worked through divisors and gcd identities in Cycle 1 with Melih Üçer, then exponential congruences and multiplicative orders later in the year with Dr. Dušan Djukić.

Modular arithmetic and the residue permutation method
Fermat's Little Theorem, statement, proof and the meaning of cycles
Euler's Theorem and the totient function
Multiplicative order, ord_n(a), and the divisibility ord_n(a) | φ(n)
Exponential Diophantine equations and the smallest prime divisor method
Lifting the exponent, preparation and applications

iv.

Combinatorics

Combinatorics rewards the imaginative student more than the computational one, but it is also the most teachable olympiad subject, because its methods are short and its proofs visible. We emphasise the named methods, the canonical examples, and the move from counting to bounding.

Counting arguments, bijections, double counting, generating functions
Pigeonhole, in its elementary and parametric forms
Extremal arguments and the smallest counterexample method
Graphs, colourings, and Ramsey problems
Probabilistic arguments at the olympiad level

A regional inheritance

The four subjects have a long memory in this part of the world.

Each of the four subjects has a documented lineage in the medieval mathematical tradition of the Arab world. That tradition produced vocabulary, notation, and techniques that teachers still share with students. This connection gives trainees historical grounding and confidence in the classroom.

Algebra

The word itself comes from Al-Khwārizmī's ninth century treatise on al-jabr and al-muqābala, which classified quadratic equations into six types and proved each geometrically. Two centuries later Al-Karajī extended the calculus of polynomials, worked out the closed form for the sum of cubes, and used what modern historians read as an early inductive argument. Al-Samaw'al al-Maghribī wrote Al-Bāhir fi'l-jabr in the twelfth century, credited the induction idea back to Al-Karajī, and pushed algebraic manipulation further than anyone before him in the region.

ii.

Geometry

Thābit ibn Qurra proved a generalisation of the Pythagorean theorem that still appears in problem sets, and translated the Greek geometric canon into Arabic with corrections that shaped how it reached Europe. Ibn al-Haytham's Kitāb al-manāẓir reworked the geometry of vision using rigorous constructions, the same instinct for building a figure carefully before reasoning about it that we ask trainees to model in their classrooms.

iii.

Number Theory

Thābit gave a rule for producing amicable number pairs that carried his name for centuries. Ibn al-Haytham stated and worked with a congruence system whose reasoning matches what we now call Wilson's theorem, arriving at the result more than seven centuries before the eighteenth century statement Wilson's name is attached to. Al-Karajī's Al-Fakhrī extended and corrected Diophantus's arithmetical problems , a direct ancestor of the number theoretic problem style olympiads still use.

iv.

Combinatorics

Al-Samaw'al set out a problem in ten unknowns with 210 equations, an early and deliberate piece of combinatorial reasoning about systems. He also produced a closed form for the sum of squares of the first n integers, the kind of summation identity that still opens olympiad counting problems. Al-Bīrūnī's geodetic work, meanwhile, is a masterclass in the extremal habit of mind: reduce a hard measurement to a clever configuration, then bound the error.

We name these figures where the historical record supports the specific claim. Each scholar has a fully sourced standalone entry in the Historical Profiles, and the shorter Heritage narrative lives on the About page.

The pedagogical method

A consistent rhythm across every cycle.

Independent of subject, every session of the Academy follows the same structure of three connected stages. Once teachers internalise the rhythm, they can carry it back to their classrooms.

First

Theorem

We state the theorem, motivate its hypotheses, and prove it in a form that reveals the underlying mechanism.

Then

Example

We work several olympiad problems where the theorem appears in concrete form. The calculation comes first; the meaning comes after.

Finally

Method

We name the hidden idea behind the examples. Once the method has a name, problems become cleaner, more systematic, and easier to teach.

Always

Written record

Every session is paired with a full LaTeX document, the kind of notes a teacher can take into a classroom and teach from directly.

Curriculum becomes practice.

The Academy's curriculum lives in the working notebooks of its teachers. Apply to join the next cycle and become part of that working community.

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