APMO · 9. Combinatorial geometry and graphs
Extremal choices for point sets (sorting along a direction, convex hull), parity colourings of lattice points, double counting in graphs and greedy colouring of conflict graphs, through five complete APMO proofs.
Use a generic projection and the convex hull to make strict extremal choices in point sets.
Use two-colourings and odd cycles to force a parity property.
Model relations as graphs and double count with degrees and Cauchy–Schwarz.
Prove both halves of an extremal colouring problem: a clique lower bound and a greedy upper bound.
Write complete olympiad proofs with constructions that are checked condition by condition.