International Mathematical Olympiad

The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-college students, and is the oldest of the International Science Olympiads.[1] 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,[2] plus one team leader, one deputy leader, and observers.[3]

The logo of the International Mathematical Olympiad.

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.[4] 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.[5]

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.[6]

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.[7] 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.[2] Because of this eastern origin, the IMOs were first hosted only in eastern European countries, and gradually spread to other nations.[8]

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.[9]

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.[10]

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.[11]

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.[12]

Each country's marks are agreed between that country's leader and 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.[13]

Selection process
A stage in the process of solving a problem1 from the AIME, part of the United States' 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.[14] The Chinese contestants go through a camp.[15] 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.[16]

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.[17] 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 (minimum scores required to receive a gold, silver or bronze medal respectively) 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.[18]

Special prizes may be awarded for solutions of outstanding elegance or involving good generalisations of a problem. This last happened in 1995 (Nikolay Nikolov, Bulgaria) and 2005 (Iurie Boreico), but was more frequent up to the early 1980s.[19] 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 (when the choice was to give either 226 (43.71%) or 266 (51.45%) of the 517 contestants (excluding the 6 from North Korea — see below) a medal),[20] 2012 (when the choice was to give either 226 (41.24%) or 277 (50.55%) of the 548 contestants a medal), and 2013, when the choice was to give either 249 (47.16%) or 278 (52.65%) of the 528 contestants a medal. In these cases, slightly more than half the contestants were awarded a medal.

Some of gold medal contestants during the IMO 2015 closing ceremony, Chiang Mai Thailand

Penalties

North Korea was disqualified for cheating at the 32nd IMO in 1991 and again at the 51st IMO in 2010.[21] It is the only country to have been accused of cheating.

Summary
Members of the 2007 IMO Greek team.
The four perfect scorers in the 2001 IMO. From left to right: Gabriel Carroll, Reid Barton (both United States), Liang Xiao and Zhiqiang Zhang (both China).
The Bangladesh team at the 2009 IMO
Serbia's team for the 2010 IMO
Zhuo Qun (Alex) Song (Canadian), the most successful IMO contestant with 5 golds and 1 bronze medal
Maryam Mirzakhani (Iran), the first, and to date, the only woman to be honored with a Fields Medal, won 2 gold medals in 1994 and 1995, getting a perfect score in the second year.
Venue Year Date Top-ranked country[22] Refs
1  Brașov and Bucharest 1959July 21 – July 31 Romania [23]
2  Sinaia 1960July 18 – July 26 Czechoslovakia [23]
3  Veszprém 1961July 6 – July 16 Hungary [23]
4  České Budějovice 1962July 7 – July 15 Hungary [23]
5  Warsaw and Wrocław 1963July 5 – July 13 Soviet Union [23]
6  Moscow 1964June 30 – July 10 Soviet Union [23]
7  East Berlin 1965July 3 – July 13 Soviet Union [23]
8  Sofia 1966July 1 – July 14 Soviet Union [23]
9  Cetinje 1967July 2 – July 13 Soviet Union [23]
10  Moscow 1968July 5 – July 18 East Germany [23]
11  Bucharest 1969July 5 – July 20 Hungary [23]
12  Keszthely 1970July 8 – July 22 Hungary [23]
13  Žilina 1971July 10 – July 21 Hungary [23]
14  Toruń 1972July 5 – July 17 Soviet Union [23]
15  Moscow 1973July 5 – July 16 Soviet Union [23]
16  Erfurt and East Berlin 1974July 4 – July 17 Soviet Union [23]
17  Burgas and Sofia 1975July 3 – July 16 Hungary [23]
18  Lienz 1976July 7 – July 21 Soviet Union [23]
19  Belgrade 1977July 1 – July 13 United States [23]
20  Bucharest 1978July 3 – July 10 Romania [23]
21  London 1979June 30 – July 9 Soviet Union [23]
  The 1980 IMO was due to be held in Mongolia. It was cancelled, and split into two unofficial events in Europe.[24]
22  Washington, D.C. 1981July 8 – July 20 United States [23]
23  Budapest 1982July 5 – July 14 West Germany [23]
24  Paris 1983July 1 – July 12 West Germany [23]
25  Prague 1984June 29 – July 10 Soviet Union [23]
26  Joutsa 1985June 29 – July 11 Romania [23]
27  Warsaw 1986July 4 – July 15 Soviet Union
 United States		 [23]
28  Havana 1987July 5 – July 16 Romania [23]
29  Sydney and Canberra 1988July 9 – July 21 Soviet Union [23]
30  Braunschweig 1989July 13 – July 24 China [23]
31  Beijing 1990July 8 – July 19 China [23]
32  Sigtuna 1991July 12 – July 23 Soviet Union [23]
33  Moscow 1992July 10 – July 21 China [23]
34  Istanbul 1993July 13 – July 24 China [23]
35  Hong Kong 1994July 8 – July 20 United States [23]
36  Toronto 1995July 13 – July 25 China [25]
37  Mumbai 1996July 5 – July 17 Romania [26]
38  Mar del Plata 1997July 18 – July 31 China [27]
39  Taipei 1998July 10 – July 21 Iran [28]
40  Bucharest 1999July 10 – July 22 China
 Russia		 [29]
41  Daejeon 2000July 13 – July 25 China [30]
42  Washington, D.C. 2001July 1 – July 14 China [31]
43  Glasgow 2002July 19 – July 30 China [32]
44  Tokyo 2003July 7 – July 19 Bulgaria [33]
45  Athens 2004July 6 – July 18 China [34]
46  Mérida 2005July 8 – July 19 China [35]
47  Ljubljana 2006July 6 – July 18 China [36]
48  Hanoi 2007July 19 – July 31 Russia [37]
49  Madrid 2008July 10 – July 22 China [38]
50  Bremen 2009July 10 – July 22 China [39]
51  Astana 2010July 2 – July 14 China [40]
52  Amsterdam 2011July 12 – July 24 China [41]
53  Mar del Plata 2012July 4 – July 16 South Korea [42]
54  Santa Marta 2013July 18 – July 28 China [43]
55  Cape Town 2014July 3 – July 13 China [44]
56  Chiang Mai 2015July 4 – July 16 United States [45]
57  Hong Kong 2016July 6 – July 16 United States [46]
58  Rio de Janeiro 2017July 12 – July 23 South Korea [47]
59  Cluj-Napoca 2018July 3 – July 14 United States [48]
60  Bath 2019July 11 – July 22 China
 United States		 [49]
61  Saint Petersburg 2020September 16 – September 26 [50][51][52]
62  Washington, D.C. 2021July 7 – July 16 [53]
63  Oslo 2022July 6 – July 16 [54][55]
64  Chiba 2023July 2 – July 13 [56]
65  TBD 2024
66  Melbourne 2025 [57]

Notable achievements

The following nations have achieved the highest team score in the respective competition:

  • China, 20 times: in 1989, 1990, 1992, 1993, 1995, 1997, 1999 (joint), 2000, 2001, 2002, 2004, 2005, 2006, 2008, 2009, 2010, 2011, 2013, 2014, 2019 (joint);
  • Russia (including Soviet Union), 16 times: in 1963, 1964, 1965, 1966, 1967, 1972, 1973, 1974, 1976, 1979, 1984, 1986 (joint), 1988, 1991, 1999 (joint), 2007;
  • United States, 8 times: in 1977, 1981, 1986 (joint), 1994, 2015, 2016, 2018, 2019 (joint);
  • Hungary, 6 times: in 1961, 1962, 1969, 1970, 1971, 1975;
  • Romania, 5 times: in 1959, 1978, 1985, 1987, 1996;
  • West Germany, twice: in 1982 and 1983;
  • South Korea, twice: in 2012 and 2017;
  • Bulgaria, once: in 2003;[58]
  • Iran, once: in 1998;
  • East Germany, once: in 1968.

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 (they were coached by Paul Zeitz); and Luxembourg, whose 1-member team had a perfect score in 1981. The US's success earned a mention in TIME Magazine.[64] 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: As of July 2015 Zhuo Qun Song (Canada) is the most successful participant[65] with five gold medals (including one perfect score in 2015) and one bronze medal.[66] Reid Barton (United States) was the first participant to win a gold medal four times (1998-2001).[67] Barton is also one of only eight four-time Putnam Fellows (2001–04). Christian Reiher (Germany), Lisa Sauermann (Germany), Teodor von Burg (Serbia), and Nipun Pitimanaaree (Thailand) are the only other participants to have won four gold medals (2000–03, 2008–11, 2009–12, 2010–13, and 2011–14 respectively); Reiher also received a bronze medal (1999), Sauermann a silver medal (2007), von Burg a silver medal (2008) and a bronze medal (2007), and Pitimanaaree a silver medal (2009).[68] Wolfgang Burmeister (East Germany), Martin Härterich (West Germany), Iurie Boreico (Moldova), and Lim Jeck (Singapore) are the only other participants besides Reiher, Sauermann, von Burg, and Pitimanaaree to win five medals with at least three of them gold.[2] Ciprian Manolescu (Romania) managed to write a perfect paper (42 points) for gold medal more times than anybody else in the history of the competition, doing it all three times he participated in the IMO (1995, 1996, 1997).[69] Manolescu is also a three-time Putnam Fellow (1997, 1998, 2000).[70] Eugenia Malinnikova (Soviet Union) 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.[71]

Terence Tao (Australia) 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 at that time[72] to receive a gold medal (Zhuo Qun Song of Canada also won a gold medal at age 13, in 2011, though he was older than Tao). Tao also holds the distinction of being the youngest medalist with his 1986 bronze medal, followed by 2009 bronze medalist Raúl Chávez Sarmiento (Peru), at the age of 10 and 11 respectively.[73] 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:[74]

RankCountryAppearanceGoldSilverBronzeHonorable Mentions
1 China341573560
2 United States45130112291
3 Russia289957120
4 Hungary59821679910
5 South Korea327970277
6 Romania60771441056
7 Soviet Union[n 1]297767450
8 Vietnam4362108731
9 Bulgaria605411910911
10 Germany42511007915

Media coverage
  • A documentary, "Hard Problems: The Road To The World's Toughest Math Contest" was made about the United States 2006 IMO team.[75]
  • A BBC documentary titled Beautiful Young Minds aired July 2007 about the IMO.
  • A BBC fictional film titled X+Y released in September 2014 tells the story of an autistic boy who took part in the Olympiad.

See also
  • List of International Mathematical Olympiads
  • International Mathematics Competition for University Students (IMC)
  • International Science Olympiad
  • List of mathematics competitions
  • Pan-African Mathematics Olympiads
  • Junior Science Talent Search Examination

Notes
  1. The Soviet Union participated the IMO for the last time in 1991. From 1992, former Soviet countries – including Russia – entered separately.[22]

Citations
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  4. Tony Gardiner (1992-07-21). "33rd International Mathematical Olympiad". University of Birmingham. Retrieved 2008-03-05.
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  7. Turner, Nura D. A Historical Sketch of Olympiads: U.S.A. and International The College Mathematics Journal, Vol. 16, No. 5 (Nov., 1985), pp. 330-335
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References
  • Xu, Jiagu (2012). Lecture Notes on Mathematical Olympiad Courses, For Senior Section. World Scientific Publishing. ISBN 978-981-4368-94-0.CS1 maint: ref=harv (link)
  • Xiong, Bin; Lee, Peng Yee (2013). Mathematical Olympiad in China (2009-2010). World Scientific Publishing. ISBN 978-981-4390-21-7.CS1 maint: ref=harv (link)
  • Xu, Jiagu (2009). Lecture Notes on Mathematical Olympiad Courses, For Junior Section. World Scientific Publishing. ISBN 978-981-4293-53-2.CS1 maint: ref=harv (link)

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