The Windmill Problem: Rotational Symmetry and Invariant Structure in IMO Geometry — E8 Intelligence Research

FINDING: The "windmill" problem (2011 IMO Q2) is the standout mathematical artifact — a deceptively simple combinatorial geometry problem whose solution reveals deep rotational symmetry and invariant structure. The 2026 IMO Problem 1 is too new for verified content; the arxiv link is a physics olympiad proceedings, not IMO. MATH: The windmill problem: Given a finite set of points in the plane, no three collinear, show there exists a line (the "windmill pivot") that, when rotated continuously, always passes through exactly one point of the set, and visits every point infinitely often. Key invariant: the number of points on each side of the rotating line changes by ±1 at each pivot step; the total "winding number" of the line's orientation over a full cycle is 2π. The solution uses a parity/ordering argument — the line's pivot point sequence forms a Hamiltonian cycle in the point set's "allowable sequence" (the order of projections onto a rotating axis). No explicit constants, but the s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com