9th CanMO 1977 Problem 5 A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K?
The key is to cut the cone along KP and unroll it to give the sector of a circle. If the sector is bounded by KP and KP' and the arc PP', then evidently the particle travels along the straight line PP'. It is closest to K when it is at the midpoint M. The arc PP' has length 2π, so the ∠PKP' is 2π/3. Hence ∠KPP' = π/6, so KM = 3/2. 9th CanMO 1977 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002