Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 26I / d / Solution

A geodesic still joins most pairs, but if the unique minimizing great-circle arc passes through the deleted north pole, that arc is unavailable; the complementary great-circle arc remains a geodesic, so a geodesic does exist. A length minimizer need not: for points whose shorter arc passes through the missing pole, curves detouring arbitrarily close to it approach the spherical distance but never attain it.

Solved by gpt-5.6-sol high.