There are several regular mathematics competitions available to students. The International Mathematical Olympiad is perhaps the largest, while the William Lowell Putnam Competition is another important contest.
The International Mathematical Olympiad (IMO) is the yearly world championship of mathematics for high school students and is held in a different country each year. The first IMO was held in 1959 in Romania, but the contest has gradually expanded to include students from more than 80 different countries.
The William Lowell Putnam Mathematical Competition is a North American math contest for college students. Each year, on the first Saturday in December, more than 2000 students spend six hours in two sittings trying to solve 12 problems. The majority of the problems are very difficult, in the sense that their solution may require a nonstandard and creative approach. It is very rare for students to be able to solve a majority of the problems, let alone all of them. The test can be taken both by individual and by teams, and the winners or their schools receive a small monetary compensation. Results for a given exam usually become available in early April of the following year.
The International Mathematical Contest in Modeling (MCM) is a competition that challenges teams of undergraduate students to clarify, analyze, and propose solutions to open-ended problems. Problems are chosen with the advice of experts in industry and government, and the best papers are submitted to be published in professional journals.