The Alice T. Schafer Prize goes to an outstanding woman undergraduate each year. The National Security Agency offers a popular internship for those interested in cryptography, signals, and number theory. Go to scenic Budapest for the Budapest Semesters in Mathematics for a semester or a year. Take classes from famous Hungarian mathematicians. Focus is on problem solving.
Classes are in English.
The Putnam exam each December is a nation-wide collegiate competition in mathematics. Each school gives out the exam simultaneously. Each summer the Institute for Advanced Study holds a joint workshop called the Park City Mathematics Institute in Park City, Utah, where there are separate programs for undergraduates, graduate students, researchers, and high school teachers. Generally a lot of opportunities for undergraduates to learn math and meet lots of people.