The Hidden Projective Symmetry Behind IMO's Windmill Problem — E8 Intelligence Research

FINDING: The corpus reveals no single "hardest problem" but a cluster of structurally distinct challenges — Putnam's elegant-combinatorics, IMO's windmill (dynamic geometry), Collatz (iterative arithmetic), and the MATH dataset (benchmarking) — with the deepest mathematical content in the windmill problem's hidden projective symmetry. MATH: - **Windmill (IMO 2011 P2):** For odd \(n\), a finite set \(S\) of points in general position. A "windmill" is a line through a pivot point \(p ∈ S\), rotating continuously; when it hits another point \(q ∈ S\), the pivot switches to \(q\) (line passes through \(q\) and continues rotating). **Key result:** There exists a starting line such that every point of \(S\) is a pivot infinitely often. The proof uses a parity/involution argument: the number of points on each side of the rotating line changes by \(± 1\) at each pivot switch, and for odd \(n\), the line must pass through exactly one point at all times (since \(n-1\) is even, sides ba Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com