Try this beautiful problem from the Pre-RMO II, 2019, Question 27 based on Shortest Distance.
A conical glass is in the form of a right circular cone. The slant height is 21 and the radius of the top rim of the glass is 14. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If d the shortest distance it should crawl to reach the honey drop, what is the integer part of d?
But try the problem first...
Answer: is 36.
PRMO II, 2019, Question 27
Higher Algebra by Hall and Knight
Rotate \(\Delta\)OAP by 120\(^\circ\) in anticlockwise then A will be at B, P will be at P'
or, \(\Delta\)OAP is congruent to \(\Delta\)OBP'
or, PB+PA=P'B+PB \(\geq\) P'P
Minimum PB+PA=P'P equality when P on the angle bisector of \(\angle\)AOB