The Windmill Problem: A Combinatorial-Geometric Invariant at the 2011 IMO — E8 Intelligence Research

FINDING: The 2011 IMO "windmill" problem (Q2) is the most mathematically significant item — it involves a rotating line through a finite point set, forcing a combinatorial-geometric invariant. The 2026 IMO Problem 1 is a new, as-yet-unsolved-in-this-source problem; the arxiv physics olympiad is a separate, lower-signal item. MATH: - Windmill problem (2011 IMO Q2): Given a finite set \(S\) of \(n\) points in general position, a "windmill" is a line \(l\) rotating about a pivot point \(p ∈ S\). When \(l\) passes through another point \(q ∈ S\), the pivot switches to \(q\). The invariant: the line always has exactly \( n/2 \) points on one side and \( n/2 \) on the other, after the first switch. The proof uses a parity/ordering argument on the circular sequence of points around the pivot. - No explicit constants or ratios appear in the windmill solution — it is purely combinatorial. - The 2026 IMO Problem 1 (from the video) is not yet extracted; no equ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com