International Mathematics Olympiad

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The International Mathematics Olympiad (IMO) is an international student competition in the field of mathematics that has been held annually since 1959 (with one exception). Each country is allowed to send six participants who write two exams, each with three tasks from different areas of mathematics such as algebra , combinatorics , geometry and number theory . In addition, there is an extensive supporting program in which the participants get to know the host country and the participants from the other countries.

A total of 621 students from 112 countries took part in the 60th IMO in Great Britain 2019.

qualification

In order to be eligible to participate in the IMO, one must not have started a degree and be less than 20 years old. The selection process for the team differs in the individual countries, often some students are selected from the successful participants in national Olympiads through exams, who are then promoted in training seminars, the team is determined through further exams.

Germany

German team 2016 during the final seminar in Oberwolfach, with the team mascot, the "MathemaTigerin"

The winners of the nationwide school competitions (a prize in the second round of the federal mathematics competition , in the federal round of the German Mathematical Olympiad or a state victory at Jugend forscht in the field of mathematics) are invited to write two pre-selection exams in December of the previous year Schools held. The best 16 of these exams take part in the preparation for the International Mathematical Olympiad: The participants are supported in five training seminars - the final seminar takes place at the Mathematisches Forschungsinstitut Oberwolfach . In the meantime, seven exams are written to determine the team; in the event of a tie, a tie-off exam decides. Since 2005 another training seminar has been held for the team shortly before the IMO.

The participants in the IMO are automatically accepted into the German National Academic Foundation and invited to events such as the Day of Talents .

Austria

The preparation and qualification take place in courses at the schools. The best of each course (roughly the first third) qualify for one of the three regional competitions. Of these, the most successful (a third of the participants, that is approx. 15 per regional competition) climb up and are allowed to take part in the national competition, which traditionally takes place in Raach am Hochgebirge .

The preparation time for this takes about three weeks and consists of two parts, with an intermediate competition at the end of the first part. This is followed by the second part of the course and the final competition for the more successful half. This will determine the six participants for the international competition. Six other applicants are taking part in the Central European Mathematical Olympiad (MEMO) .

Switzerland

The Imosuisse organization holds the qualification in cooperation with the Swiss Federal Institute of Technology in Zurich . For this purpose, several days of training, a training camp and several exams are held. The Swiss Mathematical Olympiad will be held at the same time . The winners qualify for the International Olympiad.

Luxembourg

The best placed at the Belgian Mathematical Olympiad is sure to be placed. The other places are mostly given to young hopefuls. The team often starts with only a few participants; The exceptions to date are the 2008 teams (5 participants) and 2009, 2011, 2017 and 2019 (6 participants each).

history

The first IMO took place in Brașov in Romania in 1959 , making it the oldest Science Olympiad . 52 students from the seven countries Bulgaria, Czechoslovakia, GDR, Hungary, Poland, Romania and the USSR took part in the first Olympiad. Originally, the competition was intended as a one-off event for young mathematicians in the socialist countries, where mathematical talents were intensively promoted. But after Romania organized an IMO the following year and another country, Hungary, took over the organization, an annual event took place.

Finland was the first non-socialist country to take part in 1965. Great Britain, Italy, Sweden and France followed in 1967, the Netherlands and Belgium in 1969, Austria in 1970, the USA in 1974 and Greece in 1975. The Federal Republic of Germany has been participating with a school team since 1977, Switzerland since 1991. The first Olympics to be held in a non-socialist country was in Austria in 1976, and Great Britain followed in 1979 as the host.

In 1980 the Olympics should have taken place in Mongolia, but the organizers canceled it at short notice, so that no IMO took place this year. Instead, some replacement Olympiads were organized, including in Mersch (Luxembourg) and Mariehamn (Finland), in which only a few countries took part. In order not to endanger the continued existence of the IMO, Hungary and France took over the organization for 1982 and 1983 at very short notice. In order to be able to cope with this, the team size had to be reduced from the original eight participants. In 1982 a team consisted of only four participants, since 1983 it has consisted of six. This team size has been retained until today.

The present logo was introduced at the IMO 1995 in Canada, it is based on the Olympic rings and shows the symbol for infinity .

The number of participants and countries increased sharply over time. At the 2009 Olympics in Germany, over 100 countries took part for the first time, 565 participants from 104 countries came, and other countries sent observers to send a team in the following year. The proportion of girls among the participants was around 10%.

So far a team has been disqualified twice, namely North Korea in 1991 and 2010.

While Austria has hosted only once, the IMO has already taken place four times in Germany: The GDR hosted Berlin in 1965 and Erfurt in 1974, the Federal Republic of Germany hosted the 30th IMO in Braunschweig in 1989 and the 50th IMO in Bremen in 2009 .

Venues

See list of International Mathematical Olympiads .

procedure

A few days before the official start of the IMO, the heads of delegation from the participating countries meet for the first jury meeting. The IMO jury selects the six test tasks from the task proposals submitted by the federal states; the other proposed tasks are often used by the individual countries to select and prepare the teams for the next IMO. Based on the tasks in the official IMO languages ​​of English, German, French, Russian and Spanish, the heads of delegation produce translations into the participants' mother tongues.

After the opening ceremony, the two exams are written on two consecutive days. Each takes 4½ hours. Apart from writing utensils, only compasses and rulers are permitted as aids, in particular no set square or pocket calculator . During the first half hour, the students have the opportunity to ask questions if the task is unclear.

The participants' solutions are then corrected by the respective heads of delegation and their deputies. There are seven points for a completely solved task, so that a total of 42 points can be achieved. In order to ensure a uniform assessment, the points are awarded in consultation with coordinators; in the event of disputes, the jury decides in the last instance by majority decision. During the correction, participants have the opportunity to get to know the host country and other participants.

In its final meeting, the jury decides on the point limits for the prizes. In addition, at the suggestion of the elected IMO Advisory Board , it decides on the award of the IMO to future hosts and invitations to new countries to send a team of students. The prizes will then be ceremoniously handed over in a closing ceremony. The gold medals are usually presented by special figures in public life, for example Andrew Wiles (2001 in the USA) or Crown Prince Felipe (2008 in Spain).

The costs for the IMO are borne by the host country, only the arrival and departure must be paid by the participating countries themselves, observers have to bear part of the costs themselves. At the 50th IMO 2009 in Germany, the costs were around 1.5 million euros. A fund was set up in 1995 for emergencies.

tasks

On each of the two days of the exam, three tasks are set. Usually there is a geometry problem in both exams ; other areas are number theory , inequalities , combinatorics and functional equations . The tasks often have short, elegant solutions that require a lot of creativity from the participants. On the other hand, tasks that require terms from higher mathematics , such as differential calculus or algebra , are excluded .

Since the beginning of the 1980s, the maximum number of points for all tasks, regardless of their difficulty, has been 7 points. Before that, the number of points was determined by the jury depending on the assessed difficulty. The tasks are usually classified according to difficulty, so that the first and fourth tasks are comparatively easy, while the sixth task is traditionally the hardest.

Over the years the difficulty of the tasks increased, so today the first task in the first IMO would be considered too easy. The task was:

Show that the fraction for all natural numbers is completely reduced .

With the help of the Euclidean algorithm , the greatest common divisor of numerator and denominator can be determined very easily as 1, so that the fraction is always shortened.

The third task of the IMO in 1986 was one of the most difficult tasks:

At each corner of a pentagon there is a whole number, the sum of all numbers is positive. If the numbers are at three consecutive corners , where is negative, they can be replaced by . Will this process break off at some point?

The task was set by Elias Wegert ; only eleven students were able to complete the task.

A similarly difficult task was set two years later:

If and are natural numbers, so that is also a natural number, it is even a square number.

No member of the task committee was able to solve this problem, so they presented the task to some university mathematicians familiar with number theory who, with a limited working time of 6 hours, also found no proof. Nevertheless, the task was set and solved by eleven students.

The task, in which the fewest points were awarded until 2019, dates from 2017 and deals with a geometric game between a hunter and a hare. Only two students managed to solve this problem, one student received 5 points, one 4 and 3 others one. The other 608 students did not receive any point in this task, so that only 0.6% of the theoretically possible points were awarded.

Prices

Teodor von Burg with his first two gold medals

The most successful participants are honored with gold, silver and bronze medals, these are awarded in a ratio of 1: 2: 3, whereby no more than half of the students should receive a medal. A gold medal is awarded to the best twelfth of the participants, the next sixth receives a silver medal and a further quarter bronze. Anyone who does not receive a medal, but has completed at least one of the six tasks in full, receives an honorable mention (recognition, awarded since 1988). Special prizes can be awarded for particularly elegant solutions; this occurred a total of 53 times up to 2019.

The most successful participant with five gold and one bronze medal is the Canadian Zhuo Qun (Alex) Song. Together with him there are six participants who have won at least four gold medals:

Surname country Period Medals
Zhuo Qun (Alex) Song Canada 2010-2015 5 × gold, 1 × bronze
Teodor von Burg Serbia 2007–2012 4 × gold, 1 × silver, 1 × bronze
Lisa Sauermann Germany 2007-2011 4 × gold, 1 × silver
Nipun Pitimanaaree Thailand 2009-2013 4 × gold, 1 × silver
Christian Heron Germany 1999-2003 4 × gold, 1 × bronze
Reid Barton United States 1998-2001 4 × gold

The first German who managed to win three gold medals was Wolfgang Burmeister from the GDR in 1971, who was the most successful participant until 2000. In total, he won three gold medals, two silver medals and two special prizes in five participations. In addition to these seven participants, 38 more had managed to win at least three gold medals by 2019.

Several successful mathematicians are among the winners. By 2018, 16 Fields Medal winners had taken part in the IMO during their school days:

Terence Tao won his gold medal at the age of twelve, making him the youngest gold medalist to date.

Although the IMO is an individual competition, there are also unofficial ranking lists of the countries. China, Russia, the USA and South Korea usually take first places here. Germany occupied a place between 10 and 20 in most years, Austria is usually around 50th place, as is Switzerland. German teams have so far won the competition three times: the GDR in 1968, the Federal Republic of Germany in 1982 and 1983.

In 2019, China and the USA finished first, tied on points, and South Korea third. Germany reached 32nd place, Austria 61st place, Switzerland 58th place.

literature

  • Samuel L. Greitzer: International Mathematical Olympiads 1959–1977. Mathematical Association of America, Washington 1978, ISBN 0-88385-627-1 .
  • Murray S. Klamkin: International Mathematical Olympiads 1978–1985. Mathematical Association of America, Washington 1986, ISBN 0-88385-631-X .
  • Marcin E. Kuczma: International Mathematical Olympiads 1986–1999. Mathematical Association of America, Washington 2003, ISBN 0-88385-811-8 .
In addition to the tasks and the solutions, the books also contain general information about the IMO.
The first part of the book describes the process of the 50th IMO in Germany, the second part is devoted to the history of the IMO with detailed statistics.

Individual evidence

  1. ↑ Eric Müller: Report on the 50th International Mathematical Olympiad. P. 4. (PDF), accessed on July 13, 2018.
  2. ↑ German selection competition. Retrieved July 13, 2018.
  3. ↑ Austrian Mathematical Olympiad with selection for IMO. Retrieved July 14, 2018.
  4. ↑ Swiss Mathematical Olympiad with selection for the IMO. Retrieved July 13, 2018.
  5. ^ A b Hans-Dietrich Gronau, Hanns-Heinrich Langmann, Dierk Schleicher: 50th IMO - 50 Years of International Mathematical Olympiads. Springer-Verlag, 2011, ISBN 978-3-642-14564-3 . P. 229.
  6. ↑ The data are taken from the complete overview of the participations in the individual countries. Retrieved July 13, 2018.
  7. ↑ These locations are mentioned in the Art of Problem Solving exercise book . Retrieved July 13, 2018.
  8. ↑ General information on the official IMO website. Retrieved July 13, 2018.
  9. ^ Hans-Dietrich Gronau: Report on the 51st International Mathematical Olympiad. P. 4. (PDF), accessed on July 13, 2018.
  10. ↑ Eric Müller, Hans-Dietrich Gronau: Report on the 49th International Mathematical Olympiad. P. 3. (PDF), accessed on July 13, 2018.
  11. ↑ General Regulations. (PDF; 110 kB). Annual Regulations for IMO 2017. (PDF) Retrieved July 13, 2018.
  12. ↑ Press release ( memento of November 29, 2014 in the Internet Archive ) of July 6, 2009. Accessed on September 7, 2017.
  13. ^ Regulations. Retrieved July 13, 2018.
  14. ↑ mathematik-olympiaden.de . Retrieved July 13, 2018.
  15. ↑ The Pentagon Problem by Elias Wegert (abstract). (PDF; 25 kB), accessed on July 13, 2018.
  16. ↑ The Pentagon Problem by Elias Wegert (lecture). (PDF; 5 MB), accessed on July 13, 2018.
  17. ↑ Arthur Engel: Problem Solving Strategies. Springer 1998, ISBN 0-387-98219-1 , p. 127.
  18. ^ Jürgen Prestin: Report on the 58th International Mathematical Olympiad. (PDF), accessed on July 13, 2018.
  19. ↑ Total calculated according to the official list . Retrieved July 13, 2018.
  20. ^ List on the IMO page at the German Mathematical Olympiad. Retrieved July 13, 2018.
  21. ↑ On the page about the IMO at the German Mathematical Olympiad (archive version from 2017), the Nevanlinna Prize winners are listed in addition to the field medal winners.

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