Here is another challenging problem from the Polish Mathematical Olympiads. Its generality will cause more thought than for a simpler, specific problem.

“A cyclist sets off from point *O* and rides with constant velocity *v* along a rectilinear highway. A messenger, who is at a distance *a* from point *O* and at a distance *b* from the highway, wants to deliver a letter to the cyclist. What is the minimum velocity with which the messenger should run in order to attain his objective?”

See the Tired Messenger Problem