For contests like, say, Putnam, would Rudin's Principles or Courant's Calculus be a better fit? Courant's book has a lot more problems, and the mathematics involved much more closely resembles what's there is contest calculus. On the other hand, the problems seem mostly trivial or bruteforce computation-based. I am entirely sure that is not true of Rudin.
Please help.
If you want Putnam Style problems a putnam book may be better suited. Have you looked at Problem Solving through Problems (Larson) or Putnam and Beyond?