The International Mathematical Olympiad (IMO) is an annual mathematical olympiad for high school students. It is the oldest of the international science olympiads.

The first IMO was held in Romania in 1959. Since then it has been held every year except 1980. About 90 countries send teams of (at most) six students each (plus one team leader, one deputy leader and observers). Teams are not officially recognized - all scores are given only to individual contestants. Contestants must be under the age of 20 and must not have any post-secondary school education. Subject to these conditions, an individual may participate any number of times in the IMO.

The paper consists of six problems, with each problem being worth seven points. The total score is thus 42 points. The examination is held over two consecutive days; the contestants have four-and-a-half hours to solve three problems on each day. The problems chosen are from various areas of secondary school mathematics, broadly classifiable as geometry, number theory, algebra, and combinatorics. They require no knowledge of higher mathematics, and solutions are often short and elegant. Finding them, however, requires exceptional ingenuity and mathematical ability.

Each participating country, other than the host country, may submit suggested problems to a Problem Selection Committee provided by the host country, which reduces the submitted problems to a shortlist. The team leaders arrive at the IMO a few days in advance of the contestants and form the IMO Jury which is responsible for all the formal decisions relating to the contest, starting with selecting the six problems from the shortlist. As the leaders know the problems in advance of the contestants, they are kept strictly separated from the contestants until the second examination has finished; the contestants are accompanied to the IMO by their deputy leaders (and maybe observers as well).

Each country's marks are agreed between that country's leader and deputy leader and Co-ordinators provided by the host country (the leader of the team whose country submitted the problem in the case of the marks of the host country), subject to the decisions of the Chief Coordinator and ultimately the Jury if any disputes cannot be resolved.

The participants are ranked based on their individual scores.

• The total number of awarded medals is as close as possible to but not more than half the total number of contestants.
• Subsequently the number of gold, silver and bronze medals is chosen such that their ratio approximates 1:2:3.
• Participants who don't win a medal but who score seven points on at least one problem get an honorable mention.

Special prizes may be awarded for solutions of outstanding elegance or involving good generalisations of a problem. This last happened in 2005, 1995 and 1988, but was more frequent up to the early 1980s.

The rule that at most half the contestants win a medal is sometimes broken if adhering to it causes the number of medals to deviate too much from half the number of contestants. This last happened in 2006 when the choice was to give either 188 or 253 of the 498 contestants a medal.

• The 48th IMO is being held in Hanoi, Vietnam from 19-31 July, 2007.
• The 49th IMO will be held in Madrid, Spain in 2008.
• The 50th IMO will be held in Bremen, Germany in 2009.
• The 52th IMO will be held in the Netherlands in 2011.

Sources differ about the cities hosting some of the early IMOs. This may be partly because leaders are generally housed well away from the students, and partly because after the competition the students did not always stay based in one city for the rest of the IMO. The exact dates cited may also differ, because of leaders arriving before the students, and at more recent IMOs the IMO Advisory Board arriving before the leaders.

• Grigory Margulis (USSR) received IMO 1962 Silver medal. He became only the seventh mathematician ever to receive both a Fields Medal, in 1978, and a Wolf Prize in Mathematics, in 2005.
• Grigori Perelman (USSR) wrote a perfect paper in IMO 1982 and received a Gold medal. He was awarded but declined to accept a Fields medal in 2006 for his work on proving the Poincaré conjecture, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics. For that, he is eligible for a share of the $1,000,000 Millennium Prize offered by Clay Mathematics Institute.
• Vladimir Drinfel'd (USSR) at the age of 15 wrote a perfect paper in IMO 1969 and received a Gold medal. He was awarded a Fields Medal in 1990.

Complete list of Fields Medal winners who also received IMO medals with corresponding years and medals received noted (G for Gold, S for Silver, and B for Bronze medal):

• Reid Barton (USA) was the first participant to win a Gold medal four times (1998-1999-2000-2001). Barton is also one of only seven four-time Putnam Fellow (2001-2002-2003-2004).
• Christian Reiher (Germany) is the only other participant to have won four Gold medals (2000-2001-2002-2003); Reiher also received a Bronze medal (1999).
• Wolfgang Burmeister (GDR) and Martin Harterich (FRG) are the only other participants besides Reiher to win five Medals with at least three of them Gold.
• Ciprian Manolescu (Romania) managed to write a perfect paper (42 points) for Gold medal more times than anybody else in history of competition. He did it all three times he participated in IMO (1995-1996-1997). Manolescu is also a three-time Putnam Fellow (1997-1998-2000).
• Eugenia Malinnikova (USSR) is the best female contestant in IMO history. She has 3 gold medals in IMO 1989 (41 points), IMO 1990 (42) and IMO 1991 (42), missing only 1 point in 1989 to precede Manolescu's achievement.
• Terence Tao (Australia) participated in IMO 1986, 1987 and 1988, winning Bronze, Silver and Gold medals respectively. He won a Gold medal at the age of thirteen in IMO 1988, becoming the youngest person to receive a Gold medal. He received a Fields medal in 2006.
• Oleg Golberg (Russia/USA) is the only participant in IMO history to win Gold medals for different countries: he won two for Russia in 2002 and 2003, then one for USA in 2004.
• Team USA won IMO 1994 in unique style when all six members of the team wrote a perfect paper and thus received six Gold medals. This accomplishment has never been repeated and earned a mention in TIME Magazine.
• Other teams that won IMO and had all members receive Gold medals are China 8 times (1992-1993-1997-2000-2001-2002-2004-2006) and Bulgaria once (2003).
• Team Hungary won IMO 1975 in completely opposite and totally unorthodox way when none of the eight team members received a Gold medal (5 Silver, 3 Bronze). Second place team GDR also did not have a single Gold medal winner (4 Silver, 4 Bronze).

The following table lists all IMO Winners who have won at least three Gold Medals, with corresponding years and non-Gold Medals received noted (S for Silver medal, B for Bronze medal).

• Steve Olson. Count Down. Houghton Mifflin, 2004. ISBN 0-618-25141-3. Describes the IMO (based on IMO 2000) from the viewpoint of the contestants, with general background information on various related issues (such as competitiveness).
• Tom Verhoeff. The 43rd International Mathematical Olympiad: A Reflective Report on IMO 2002. Computing Science Report 02-11, Faculty of Mathematics and Computing Science, Eindhoven University of Technology. August 2002. PDF Describes the IMO (based on IMO 2002) from the viewpoint of the leaders, with a comparison to the International Olympiad in Informatics.

• International Science Olympiad
• Provincial Mathematical Olympiad

• Official IMO web site
• Old central IMO web site
• Introduction to the IMO Business, including information on IMO ShortLists/LongLists by Orlando Döhring
• IMO Resources - IMO problems and solutions, IMO Shortlists, IMO Longlists and one of the largest collection of Olympiad problems in the world.
• IMO problems and solutions
• Partial IMO results
• IMO scores (since 1980's) with Statistical analysis
• IMO winners Hall of Fame - IMO winners who have won Fields Medal or Nevanlinna Prize
• IMO contestants who have won three or more gold medals
• MathLinks IMO Forums - discuss the latest developments in the IMO with former, current or future contestants, leaders and other people involved in the IMO
• The IMO Compendium - substantial collection of shortlisted and long listed problems for the IMO's; huge collection of problems from other contests.