In which film does the star find himself dangling from Mount Rushmore and dodging crop dusters?

The film where the star finds himself dangling from Mount Rushmore and dodging crop dusters is titled "North by Northwest". It is a 1959 American thriller which was directed and produced by Alfred Hitchcock. The film' screenplay was written by Ernest Lehman. The three key stars are Cary Grant, Eva Marie Saint, and James Mason.

This classic suspense film finds New York City ad executive Roger O. Thornhill (Cary Grant) pursued by ruthless spy Phillip Vandamm (James Mason) after Thornhill is mistaken for a government agent. Hunted relentlessly by Vandamm's associates, the harried Thornhill ends up on a cross-country journey, meeting the beautiful and mysterious Eve Kendall (Eva Marie Saint) along the way. Soon Vandamm's henchmen close in on Thornhill, resulting in a number of iconic action sequences. One of these sequences will find the film's protagonist at Mount Rushmore.

Now the film ranks at number 98 in Empire magazine's list of the 500 Greatest Films of All Time. The Writers Guild of America ranked the screenplay No. 21 on its list of 101 Greatest Screenplays ever written. It is also ranked the 40th greatest American film by the American Film Institute.