Zadanie 2b, Paper 3, maj 2026, TZ2, Matematyka AA IB HL

A robot makes three consecutive mutually perpendicular moves of positive integer lengths u,v,w. Its straight-line distance from start to finish is a positive integer t, so u² + v² + w² = t². Such a route is recorded as (u,v,w;t). One way to construct routes uses z₁ = a + bi, z₂ = c + di, where a,b,c,d are positive integers: u = ∣z₁∣² − ∣z₂∣², $v=2\operatorname{Re}(z_1\overline{z_2})$, $w=2\operatorname{Im}(z_1\overline{z_2})$, and t = ∣z₁∣² + ∣z₂∣². Only choices with u,v,w>0 are allowed.