The International Mathematical Olympiad (IMO) is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads.The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. About 100 countries send teams of up to six students, plus one team leader, one deputy leader, and observers. Ever since its inception in 1959, the olympiad has developed a rich legacy and has established itself as the pinnacle of mathematical competition between high school students.
The content ranges from extremely difficult precalculus problems to problems on branches of mathematics not conventionally covered at school and often not at university level either, such as projective and complex geometry, functional equations and well-grounded number theory, of which extensive knowledge of theorems is required. Calculus, though allowed in solutions, is never required, as there is a principle at play that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge. Supporters of this principle claim that this allows more universality and creates an incentive to find elegant, deceptively simple-looking problems which nevertheless require a certain level of ingenuity.
The selection process differs by country, but it often consists of a series of tests which admit fewer students at each progressing test. Awards are given to a top percentage of the individual contestants. Teams are not officially recognized—all scores are given only to individual contestants, but team scoring is unofficially compared more so than individual scores.Contestants must be under the age of 20 and must not be registered at any tertiary institution. Subject to these conditions, an individual may participate any number of times in the IMO.
The first IMO was held in Romania in 1959. Since then it has been held every year except 1980. That year, it was cancelled due to internal strife in Mongolia. It was initially founded for eastern European countries participating in the Warsaw Pact, under the Soviet bloc of influence, but eventually other countries participated as well. Because of this eastern origin, the earlier IMOs were hosted only in eastern European countries, and gradually spread to other nations.
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.
Several students, such as Christian Reiher, have performed exceptionally well on the IMO, scoring multiple gold medals. Others, such as Grigory Margulis and Grigori Perelman, have gone on to become notable mathematicians. Several former participants have won awards such as the Fields medal.
In January 2011, Google gifted $1 million to International Mathematical Olympiad organization. The donation will help the organization cover the costs of the next five global events (2011–2015).
 Scoring and format
The paper consists of six problems, with each problem being worth seven points, the total score thus being 42 points. No calculators are allowed. The examination is held over two consecutive days; the contestants have four-and-a-half hours to solve three problems per 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 such as calculus and analysis, and solutions are often short and elementary. However, they are usually disguised so as to make the process of finding the solutions difficult. Prominently featured are algebraic inequalities, complex numbers, and construction-oriented geometrical problems, though in previous years the latter has not been as popular as before. 
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. The Jury aims to select the problems so that the order in increasing difficulty is Q1, Q4, Q2, Q5, Q3 and Q6. As the leaders know the problems in advance of the contestants, they are kept strictly separated and observed.
Each country’s marks are agreed between that country’s leader and the deputy leader and coordinators 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 a jury if any disputes cannot be resolved.
 Selection process
The selection process for the IMO varies greatly by country. In some countries, especially those in east Asia, the selection process involves several difficult tests of a difficulty comparable to the IMO itself. The Chinese contestants go through a camp, which lasts from March 16 to April 2. In others, such as the USA, possible participants go through a series of easier standalone competitions that gradually increase in difficulty. In the case of the USA, the tests include the American Mathematics Competitions, the American Invitational Mathematics Examination, and the United States of America Mathematical Olympiad, each of which is a competition in its own right. For high scorers on the final competition for the team selection, there also is a summer camp, like that of China.
The former Soviet Union and other eastern European countries’ selection process consists of choosing a team several years beforehand, and giving them special training specifically for the event. However, such methods have been discontinued in some countries. In Ukraine, for instance, selection tests consist of four olympiads comparable to the IMO by difficulty and schedule. While identifying the winners, only the results of the current selection olympiads are considered.
The participants are ranked based on their individual scores. Medals are awarded to the highest ranked participants, such that slightly less than half of them receive a medal. Subsequently the cutoffs (minimum score required to receive a gold, silver or bronze medal) are chosen such that the ratio of medals awarded approximates 1:2:3. Participants who do not 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 special prize in 2005 was awarded to a student from Moldova who came up with a brilliant solution to question 3, an inequality involving three variables. He was one of only three students to achieve a perfect score for that paper.
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 ratio of gold to silver to bronze medals is generally 1:2:3, respectively.
 Current and future IMOs
- The 51st IMO was held in Astana, Kazakhstan, July 2–15, 2010.
- The 52nd IMO will be held in Amsterdam, Netherlands in July 13-24, 2011.
- The 53rd IMO will be held in Argentina in 2012.
 Notable achievements
China and Russia are the only nations that have achieved an all-members-gold IMO multiple times:
- China: 10 times in total, in 1992, 1993, 1997, 2000, 2001, 2002, 2004, 2006, 2009, 2010;
- Russia: 2 times in 2002 and 2008.
Bulgaria is the nation with the smallest population to have won IMO and it is one of four countries (with USA, China, Russia) to have won IMO by having all of its team members finish with gold medals (in 2003). The only countries to have their entire teams score perfectly on the IMO were the United States, which won IMO 1994 when it accomplished this, coached by Paul Zeitz, and Luxembourg, whose 1-member team got a perfect score in IMO 1981. This accomplishment has only been achieved twice, and the USA’s success earned a mention in TIME Magazine. Hungary won IMO 1975 in an unorthodox way when none of the eight team members received a gold medal (five silver, three bronze). Second place team East Germany also did not have a single gold medal winner (four silver, four bronze).
Several individuals have consistently scored highly and/or earned medals on the IMO: 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). In addition, he is the only person to have won both the IMO and the International Olympiad in Informatics (IOI). 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 (East Germany), Martin Härterich (West Germany) and Iurie Boreico (Moldova) 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, doing 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 highest-scoring 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. 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.
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. Tao also holds the distinction of being the youngest medalist with his 1986 bronze medal, alongside 2009 bronze medalist Raúl Chávez Sarmiento (Peru), at the age of 10 and 11 respectively. Representing the United States, Noam Elkies won a gold medal with a perfect paper at the age of 14 in 1981. Note that both Elkies and Tao could have participated in the IMO multiple times following their success, but entered university and therefore became ineligible.