The Windmill Invariant: Solving the 2011 IMO Geometry Problem — E8 Intelligence Research

Abstract

FINDING: The 2011 IMO windmill problem (Q2) is a combinatorial geometry problem whose solution hinges on a dynamic "windmill" rotation process, revealing a deep invariant about point configurations in general position. MATH: - Problem setup: Given a finite set of \( n \) points in the plane, no three collinear. A "windmill" is a line through one point \( p \) that rotates about \( p \), and when it hits another point \( q \), the pivot switches to \( q \). - Key result: For \( n \geq 3 \), there exists a point \( p \) and a starting line such that the windmill visits every point infinitely often. - The proof uses a parity/invariant argument: The number of points on each side of the rotating line changes by exactly ±1 at each pivot, and the total "signed area" or winding number of the pivot sequence is conserved modulo something. - No explicit constants or ratios appear; the essence is combinatorial invariance under a continuous rotation. CONNECTION: - The windmill process i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com