International Mathematical Olympiad
The International Mathematical Olympiad (IMO) is an annual sixproblem mathematical olympiad for precollege 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. About 100 countries send teams of up to six students,^{[2]} plus one team leader, one deputy leader, and observers.^{[3]}
The content ranges from extremely difficult algebra and 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 wellgrounded 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 simplelooking 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 topscoring 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]}
Contents
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 Soviet 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.^{[clarification needed]} 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 Zhuoqun 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, JeanChristophe 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 examination consists of six problems. Each problem is worth seven points, so the maximum total score is 42 points. No calculators are allowed. The examination is held over two consecutive days; each day the contestants have fourandahalf 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 constructionoriented 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
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 USA, possible participants go through a series of easier standalone competitions that gradually increase in difficulty. In 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 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^{[clarification needed]}. While identifying the winners, only the results of the current selection olympiads are considered.^{[clarification needed]}
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%)^{[clarification needed]} of the 517 contestants (excluding the 6 from North Korea — see below) a medal),^{[20]} 2012 (when the choice was to give either 226 (46.35%) 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.
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. There is some debate as to whether North Korea was actually cheating, especially in the 51st IMO. ^{[22]}
Summary
#^{[23]}  Venue  Year  Date^{[23]}  Topranked country^{[24]}  Refs 

1  Brașov and Bucharest  1959  July 23 – July 31  Romania  ^{[25]} 
2  Sinaia  1960  July 18 – July 25  Czechoslovakia  ^{[25]} 
3  Veszprém  1961  July 6 – July 16  Hungary  ^{[25]} 
4  České Budějovice  1962  July 7 – July 15  Hungary  ^{[25]} 
5  Warsaw and Wrocław  1963  July 5 – July 13  Soviet Union  ^{[25]} 
6  Moscow  1964  June 30 – July 10  Soviet Union  ^{[25]} 
7  East Berlin  1965  July 13 – July 13  Soviet Union  ^{[25]} 
8  Sofia  1966  July 3 – July 13  Soviet Union  ^{[25]} 
9  Cetinje  1967  July 7 – July 13  Soviet Union  ^{[25]} 
10  Moscow  1968  July 5 – July 18  East Germany  ^{[25]} 
11  Bucharest  1969  July 5 – July 20  Hungary  ^{[25]} 
12  Keszthely  1970  July 8 – July 22  Hungary  ^{[25]} 
13  Žilina  1971  July 10 – July 21  Hungary  ^{[25]} 
14  Toruń  1972  July 5 – July 17  Soviet Union  ^{[25]} 
15  Moscow  1973  July 5 – July 16  Soviet Union  ^{[25]} 
16  Erfurt and East Berlin  1974  July 4 – July 17  Soviet Union  ^{[25]} 
17  Burgas and Sofia  1975  July 3 – July 16  Hungary  ^{[25]} 
18  Lienz  1976  July 2 – July 21  Soviet Union  ^{[25]} 
19  Belgrade  1977  July 1 – July 13  United States  ^{[25]} 
20  Bucharest  1978  July 3 – July 10  Romania  ^{[25]} 
21  London  1979  June 30 – July 9  Soviet Union  ^{[25]} 
The 1980 IMO was due to be held in Mongolia. It was cancelled, and split into two unofficial events in Europe.^{[26]}  
22  Washington, D.C.  1981  July 8 – July 20  United States  ^{[25]} 
23  Budapest  1982  July 5 – July 14  West Germany  ^{[25]} 
24  Paris  1983  July 3 – July 12  West Germany  ^{[25]} 
25  Prague  1984  June 29 – July 10  Soviet Union  ^{[25]} 
26  Joutsa  1985  June 29 – July 11  Romania  ^{[25]} 
27  Warsaw  1986  July 4 – July 15 
Soviet Union United States 
^{[25]} 
28  Havana  1987  July 5 – July 16  Romania  ^{[25]} 
29  Sydney and Canberra  1988  July 9 – July 21  Soviet Union  ^{[25]} 
30  Braunschweig  1989  July 13 – July 24  China  ^{[25]} 
31  Beijing  1990  July 8 – July 19  China  ^{[25]} 
32  Sigtuna  1991  July 12 – July 23  Soviet Union  ^{[25]}^{[n 1]} 
33  Moscow  1992  July 10 – July 21  China  ^{[25]} 
34  Istanbul  1993  July 13 – July 24  China  ^{[25]} 
35  Hong Kong^{[n 2]}  1994  July 8 – July 20  United States  ^{[25]} 
36  Toronto  1995  July 13 – July 25  China  ^{[27]} 
37  Mumbai  1996  July 5 – July 17  Romania  ^{[28]} 
38  Mar del Plata  1997  July 18 – July 31  China  ^{[29]} 
39  Taipei  1998  July 10 – July 21  Iran  ^{[30]} 
40  Bucharest  1999  July 10 – July 22 
China Russia 
^{[31]} 
41  Daejeon  2000  July 13 – July 25  China  ^{[32]} 
42  Washington, D.C.  2001  July 1 – July 14  China  ^{[33]} 
43  Glasgow  2002  July 19 – July 30  China  ^{[34]} 
44  Tokyo  2003  July 7 – July 19  Bulgaria  ^{[35]} 
45  Athens  2004  July 6 – July 18  China  ^{[36]} 
46  Mérida  2005  July 8 – July 19  China  ^{[37]} 
47  Ljubljana  2006  July 6 – July 18  China  ^{[38]} 
48  Hanoi  2007  July 19 – July 31  Russia  ^{[39]} 
49  Madrid  2008  July 10 – July 22  China  ^{[40]} 
50  Bremen  2009  July 10 – July 22  China  ^{[41]} 
51  Astana  2010  July 2 – July 14  China  ^{[42]} 
52  Amsterdam  2011  July 13 – July 24  China  ^{[43]} 
53  Mar del Plata  2012  July 4 – July 16  South Korea  ^{[44]} 
54  Santa Marta  2013  July 18 – July 28  China  ^{[45]} 
55  Cape Town  2014  July 3 – July 13  China  ^{[46]} 
56  Chiang Mai  2015  July 4 – July 16  United States  ^{[47]} 
57  Hong Kong  2016  July 6 – July 16  United States  ^{[48]} 
58  Rio de Janeiro  2017  July 12 – July 23  South Korea  ^{[49]} 
59  ClujNapoca  2018  July 3 – July 14  ^{[50]}  
60  Bath  2019  July 11 – July 22  ^{[51]}  
61  2020  ^{[52]}  
62  2021  ^{[53]}  
63  2022  ^{[54]} 
Notable achievements
The following nations have achieved the highest team score in the respective competition:
 China, 19 times (from the first participation in 1985 until 2014): in every year from 1989 to 2014 except 1991, 1994, 1996, 1998, 2003, 2007, 2012;
 Soviet Union, 14 times: in 1963, 1964, 1965, 1966, 1967, 1972, 1973, 1974, 1976, 1979, 1984, 1986, 1988, 1991;
 Hungary, 6 times: in 1961, 1962, 1969, 1970, 1971, 1975;
 United States, 6 times: in 1977, 1981, 1986, 1994, 2015, 2016;
 Romania, 5 times: in 1959, 1978, 1985, 1987, 1996;
 West Germany, twice: in 1982 and 1983;
 Russia, twice: in 1999 and 2007;
 South Korea, twice: in 2012 and 2017;
 Bulgaria, once: in 2003;^{[55]}
 Iran, once: in 1998;
 East Germany, once: in 1968.
The following nations have achieved an allmembersgold IMO with a full team:
 China, 11 times: in 1992, 1993, 1997, 2000, 2001, 2002, 2004, 2006, 2009, 2010 and 2011.^{[56]}
 United States, 3 times: in 1994, 2011, and 2016.^{[57]}
 Russia, 2 times: in 2002 and 2008.^{[58]}
 South Korea, twice: in 2012 and 2017.^{[59]}
 Bulgaria, once: in 2003.^{[60]}
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 1member team had a perfect score in 1981. The US's success earned a mention in TIME Magazine.^{[61]} 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^{[62]} with five gold medals (including one perfect score in 2015) and one bronze medal.^{[63]} Reid Barton (United States) was the first participant to win a gold medal four times (19982001).^{[64]} Barton is also one of only eight fourtime Putnam Fellow (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).^{[65]} 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).^{[66]} Manolescu is also a threetime Putnam Fellow (1997, 1998, 2000).^{[67]} Eugenia Malinnikova (Soviet Union) is the highestscoring 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.^{[68]}
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^{[69]} to receive a gold medal (a feat matched in 2011 by Zhuo Qun Song of Canada). 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.^{[70]} 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.
All Time Medal Table
The top 20 countries with the best alltime results are as follows:^{[71]}
 As 2017
Rank  Country  Appearance  Gold  Silver  Bronze  Total  Honorable Mentions 

1  China  32  147  33  6  186  0 
2  United States  43  119  111  29  259  1 
3  Russia  26  92  52  12  156  0 
4  Hungary  57  81  160  95  336  10 
5  Soviet Union  29  77  67  45  189  0 
6  Romania  58  75  141  100  316  4 
7  South Korea  30  70  67  27  164  7 
8  Vietnam  41  59  109  70  238  1 
9  Bulgaria  58  53  111  107  271  10 
10  Germany  40  49  98  75  222  11 
11  United Kingdom  50  46  103  122  271  16 
12  Iran  32  43  92  39  174  3 
13  Japan  28  39  77  41  157  5 
14  Taiwan  26  37  88  22  147  8 
15  Ukraine  25  34  57  44  135  9 
16  Canada  37  32  51  87  170  19 
17  Poland  57  28  73  134  235  27 
18  East Germany  29  26  62  60  148  0 
19  France  48  23  58  109  190  26 
20  Thailand  29  21  50  47  118  23 
http://www.imoofficial.org/results_country.aspx?column=awards&order=asc
Team Ranking
Year  1  2  3 

2000  CHN  RUS  USA 
2001  CHN  RUS  USA 
2002  CHN  RUS *  USA * 
2003  BUL  CHN  USA 
2004  CHN  USA  RUS 
2005  CHN  USA  RUS 
2006  CHN  RUS  KOR 
2007  RUS  CHN  KOR 
2008  CHN  RUS  USA 
2009  CHN  JPN  RUS 
2010  CHN  RUS  USA 
2011  CHN  USA  SIN 
2012  KOR  CHN  USA 
2013  CHN  KOR  USA 
2014  CHN  USA  TPE 
2015  USA  CHN  KOR 
2016  USA  KOR  CHN 
2017  KOR  CHN  VIE 
http://www.imoofficial.org/results.aspx
 2002 shared between RUS & USA
Media coverage
 A documentary, "Hard Problems: The Road To The World's Toughest Math Contest" was made about the United States 2006 IMO team.^{[72]}
 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
 PanAfrican Mathematics Olympiads
Notes
Citations
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References
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 Verhoeff, Tom (August 2002). "The 43rd International Mathematical Olympiad: A Reflective Report on IMO 2002" (PDF). Computing Science Report, Faculty of Mathematics and Computing Science, Eindhoven University of Technology, Vol. 2, No. 11.
 Djukić, Dušan (2006). The IMO Compendium: A Collection of Problems Suggested for the International Olympiads, 19592004. Springer. ISBN 9780387242996.
 Lord, Mary (July 23, 2001). "Michael Jordans of math  U.S. Student whizzes stun the cipher world". U.S. News & World Report. 131 (3): 26.
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 Vakil, Ravi (1997). A Mathematical Mosaic: Patterns & Problem Solving. Brendan Kelly Publishing. p. 288. ISBN 9781895997286.
 Liu, Andy (1998). Chinese Mathematics Competitions and Olympiads. AMT Publishing. ISBN 1876420006.
External links
Wikimedia Commons has media related to International Mathematical Olympiad. 
Official
 Official IMO web site
 Old central IMO web site