Maths Olympiad Master-Class

Explore quadratic function analysis using modulus bounds, express coefficients from f(-1), f(0), f(1), and determine fixed points, stable fixed points, and parameter ranges.

Explore quadratic functions through vertex form, discriminant, and factoring, then apply symmetry and root concepts to solve Olympiad problems with practical problem-solving strategies.

Investigate functions and their graphs, focusing on modulus transformations and symmetry about lines and points. Apply these ideas to area calculations between graphs and the x-axis.

Master basic inequalities including a^2+b^2 ≥ 2ab, a^2+b^2+c^2 ≥ ab+bc+ca, the AM-GM and Cauchy inequalities, and the triangle inequality with two-dimensional forms and vector interpretations.

Explore how to apply am-gm inequalities for two, three, and four terms, and use Cauchy and triangle inequality to solve olympiad problems with clever term additions.

Explore trigonometric problem solving through olympiad style challenges, revising core techniques in trigonometry, including monotonicity, triangle identities, and cosine and sine rule applications.

This lecture introduces functional equations, teaching elimination to solve for f(x) with two unknowns, and examines all functions f: R→R satisfying f(x)+f(y)=f(f(x))f(y) using substitutions and proofs of uniqueness.

Explore fixed-point and bridge-function techniques for solving functional equations. Derive nth iterates for linear maps and polynomial like f(x)=2x^2−1, using fixed points, bridge functions, and arccos identities.

Explore a functional equations recap through fixed-point methods and bridge functions, applying composition f^n and solving natural and nonzero real-valued cases.

Explore the construction method for solving inequalities by building a function (linear or quadratic), then use its endpoints or discriminant to conclude positivity, with olympiad-style examples.

Explore the five triangle centers—centroid, orthocenter, circumcenter, incenter, and excenter—and their key properties, including medians, altitudes, angle bisectors, and Euler line, in Olympiad geometry.

explores triangle geometry problems: uses a parallelogram with centroid and perpendiculars to yield a cyclic configuration and bisector, then analyzes isosceles triangle with incenter and concyclic condition yielding cd-bd relation.

Explore an isosceles triangle with AB = AC where D lies inside so B, C, incenter I, D are concyclic. Use parallels and tangents to relate BD and CD via similarities.

Explore triangle geometry with isosceles ABC, incenter I, and circles T1, T2, T3 to prove BR is perpendicular to CR, highlighting cyclicity and circle-triangle relationships in olympiad problems.

Learn core olympiad geometry theorems, including shiva's theorem on concurrency of cevians via product of ratios, and menelaus for collinearity. Explore ptolemy, simpson, stewart, monge, and their trigonometric forms.

Study the extreme principle, a method for proving existence of maximal or minimal elements. Apply it to geometry, number theory, and combinatorics through examples like round-robin tournaments and Sylvester's problem.