The Pedagogical Opacity of IMO Problems: A Case for Vieta Jumping and Windmill — E8 Intelligence Research

FINDING: IMO 2025/2026 problem sets remain pedagogically opaque; the 2011 "windmill" problem and 1988 Problem 6 (Vieta jumping) are the only structurally significant items; the rest are promotional content. | MATH: 1988 IMO P6: Given positive integers \(a,b\) with \(ab+1 \mid a^2+b^2\), prove \(\frac{a^2+b^2}{ab+1}\) is a perfect square. Solution via Vieta jumping: if \((a,b)\) is a minimal counterexample, the other root \(a' = kb - a\) yields a smaller pair — forcing \(k = n^2\). Windmill (2011 P2): For \(n\) points in general position, a line through one point rotating continuously visits each point infinitely often; proof uses parity of the number of points on each side of the rotating line — invariant under 180° rotation. | CONNECTION: Vieta jumping is a descent on a quadratic Diophantine equation — structurally analogous to root system reflections (Weyl group action on \(E_8\) or \(A_n\) lattices). The windmill's rotating line with invariant side-count mirrors a discrete version o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com