N.J. Lennes Exam
The N.J. Lennes Exam is a competitive examination given annually by the Department of Mathematical Sciences. The exam is named in honor of Professor Nels Johann Lennes, who chaired the department from 1913 to 1944. N. J. Lennes wrote numerous mathematics textbooks. He built the house that is currently the home of the President of The University of Montana. Zero, one, two, three or more prizes may be awarded at the discretion of the judges. Awards are typically in the $100-$200 range. The exam may be taken by any UM undergraduate student, mathematics major or not. Students who have taken one year of calculus are encouraged to take the exam. (The initial endowment for these awards was donated by members of the extended Lennes family.)
The William Lowell Putnam Mathematical Competition is a competitive examination in collegiate mathematics sponsored by the Mathematical Association of America. This examination is held the first Saturday of December. The competition is open only to regularly enrolled undergraduates in colleges and universities of the United States and Canada who have not yet received a college degree.
If you have any questions about the Putnam Exam or would like to participate, contact Dick Lane (MA 005, 243-5207). Also, you are encouraged to participate in the Problems and Contests Seminar (M 394) offered each term. Old Putnam Exam problems are considered in this seminar and used to practice for the next exam.
There are a few other contests, for example, COMAP's Mathematical Contest in Modeling (MCM) and Kryptos (a cryptanalysis competition). Others are listed at SIAM's Math Competitions Page. For some of these you need to find a team of fellow students to participate.