Windmill Problem: The Sole Mathematical Core of IMO 2011 P2 — E8 Intelligence Research

FINDING: The 2011 IMO P2 "windmill" problem is the only mathematically substantive item; the rest are procedural proceedings (LOCO, MOASEI, HLPP) with no mathematical content. The windmill problem involves a finite point set in general position and a rotating line that "switches" pivot points, proving that for any starting line, a windmill can be configured to visit every point infinitely often. MATH: - **Core structure**: Given \(n\) points in general position (no 3 collinear), a windmill is a line \(L\) rotating about a pivot point \(p_i\). When \(L\) sweeps past another point \(p_j\), the pivot switches to \(p_j\). - **Key invariant**: The number of points on each side of \(L\) remains constant during rotation (except at the instant of switching, where the pivot lies on \(L\)). - **Solution insight**: Choose a line with exactly \(\lfloor (n-1)/2 \rfloor\) points on one side. As the line rotates \(180^\circ\), the pivot must switch at least once per half-turn; by parity/contin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com