Opportunities and beyond for mathematics students
REU programs are one of the best ways for undergraduate math students to prepare for work in mathematics and for exploration of the possible fields. These programs are held in the summer at various universities around the country. They are funded by the National Science Foundation. You are housed and fed, and paid while you work on a research problem and potentially attend seminars.
For a complete REU list visit the NSF website. Applications typically require 2 letters, transcript and maybe an essay and are usually due in February or March.
An annual exam for undergraduates held the first Saturday of every December. Considered the most prestigious university mathematics competition in the world, Hopkins competes annually.