International Mathematical Olympiad


The International Mathematical Olympiad is a mathematical olympiad for pre-college 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. More than 100 countries, representing over 90% of the world's population, send teams of up to six students, plus one team leader, one deputy leader, and observers.
The content ranges from extremely difficult algebra and pre-calculus 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, combinatorics, 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 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 approximately the top-scoring 50% of the individual contestants. Teams are not officially recognized—all scores are given only to individual contestants, but team scoring is unofficially compared more 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 International Mathematical Olympiad is one of the most prestigious mathematical competitions in the world. In January 2011, Google sponsored €1 million to the International Mathematical Olympiad organization.

History

The first IMO was held in Romania in 1959. Since then it has been held every year except in 1980. That year, it was cancelled due to internal strife in Mongolia. It was initially founded for eastern European member countries of the Warsaw Pact, under the USSR bloc of influence, but later other countries participated as well. Because of this eastern origin, the IMOs were first 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 Zhuo Qun Song, Teodor von Burg, Lisa Sauermann, and Christian Reiher, have performed exceptionally well in the IMO, winning multiple gold medals. Others, such as Grigory Margulis, Jean-Christophe Yoccoz, Laurent Lafforgue, Stanislav Smirnov, Terence Tao, Sucharit Sarkar, Grigori Perelman, Ngô Bảo Châu and Maryam Mirzakhani have gone on to become notable mathematicians. Several former participants have won awards such as the Fields Medal.

Scoring and format

The competition consists of six problems. Each problem is worth seven points for a maximum total score of 42 points. No calculators are allowed. The competition is held over two consecutive days; each day the contestants have four-and-a-half hours to solve three problems. 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 solutions difficult. Prominently featured are algebraic inequalities, complex numbers, and construction-oriented geometrical problems, though in recent 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 order 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 deputy leader and coordinators provided by 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 tests of a difficulty comparable to the IMO itself. The Chinese contestants go through a camp. In others, such as the United States, possible participants go through a series of easier standalone competitions that gradually increase in difficulty. In the United States, 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 in the final competition for the team selection, there also is a summer camp, like that of China.
In countries of the former Soviet Union and other eastern European countries, a team has in the past been chosen several years beforehand, and they are given 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.

Awards

The participants are ranked based on their individual scores. Medals are awarded to the highest ranked participants; slightly fewer than half of them receive a medal. The cutoffs are then chosen so that the numbers of gold, silver and bronze medals awarded are approximately in the ratios 1:2:3. Participants who do not win a medal but who score seven points on at least one problem receive an honorable mention.
Special prizes may be awarded for solutions of outstanding elegance or involving good generalisations of a problem. This last happened in 1995 and 2005, but was more frequent up to the early 1980s. The special prize in 2005 was awarded to Iurie Boreico, a student from Moldova, who came up with a brilliant solution to question 3, which was an inequality involving three variables.
The rule that at most half the contestants win a medal is sometimes broken if it would cause the total number of medals to deviate too much from half the number of contestants. This last happened in 2010 or 266 of the 517 contestants, 2012 or 277, and 2013, when the choice was to give either 249 or 278 of the 528 contestants a medal. In these cases, slightly more than half the contestants were awarded a medal.

Penalties

was disqualified for cheating at the 32nd IMO in 1991 and again at the 51st IMO in 2010. It is the only country to have been accused of cheating.

Summary

Notable achievements

The following nations have achieved the highest team score in the respective competition:
The following nations have achieved an all-members-gold IMO with a full team:
Also noteworthy is that the United States was a single point away from achieving all gold medals in 2012, 2014, and 2015 and was just two points away in 2018, in each of these years obtaining 5 gold medals and 1 silver medal.
The only countries to have their entire team score perfectly in the IMO were the United States in 1994 ; and Luxembourg, whose 1-member team had a perfect score in 1981. The US'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. Second place team East Germany also did not have a single gold medal winner.
Several individuals have consistently scored highly and/or earned medals on the IMO: Zhuo Qun Song is the most successful participant with five gold medals and one bronze medal. Reid Barton was the first participant to win a gold medal four times. Barton is also one of only eight four-time Putnam Fellows. Christian Reiher, Lisa Sauermann, Teodor von Burg, and Nipun Pitimanaaree are the only other participants to have won four gold medals ; Reiher also received a bronze medal, Sauermann a silver medal, von Burg a silver medal and a bronze medal, and Pitimanaaree a silver medal. Wolfgang Burmeister, Martin Härterich, Iurie Boreico, and Lim Jeck are the only other participants besides Reiher, Sauermann, von Burg, and Pitimanaaree to win five medals with at least three of them gold. Ciprian Manolescu managed to write a perfect paper for gold medal more times than anybody else in the history of the competition, doing it all three times he participated in the IMO. Manolescu is also a three-time Putnam Fellow. Eugenia Malinnikova is the highest-scoring female contestant in IMO history. She has 3 gold medals in IMO 1989, IMO 1990 and IMO 1991, missing only 1 point in 1989 to precede Manolescu's achievement.
Terence Tao participated in IMO 1986, 1987 and 1988, winning bronze, silver and gold medals respectively. He won a gold medal when he just turned 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, followed by 2009 bronze medalist Raúl Chávez Sarmiento, 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.
The current ten countries with the best all-time results are as follows:
RankCountryAppearanceGoldSilverBronzeHonorable Mentions
1341573560
245130112291
3289957120
459821679910
5327970277
660771441056
7297767450
84362108731
9605411910911
1042511007915

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