International Mathematical Olympiad

The International Mathematical Olympiad (IMO ) is an international student competition in mathematics, held annually since 1959 ( with one exception). Each country may appoint six participants who write two exams, each with three tasks from different areas of mathematics such as geometry, number theory, combinatorics, and others. In addition, an extensive program will be held, in which participants learn about the host country and the participants from the other countries.

On the 54th IMO in Colombia in 2013 was attended by a total of 527 students from 97 countries.

  • 2.1 Host Cities

Qualification

In order to be eligible for the IMO, should you have not yet started studying and have not yet 20 years old. The selection process for the team is different in different countries, are often made ​​of the successful participants in national Olympiads through exams, some students selected which are then promoted in training seminars, the team is determined by further examinations.

Germany

The winners of the nationwide student competitions (one price in the second round of the national competition mathematics, in the Federal round of the German Mathematical Olympiad or a country's victory in Youth Research in the field of mathematics ) are invited to write in December of the previous year's two pre-selection exams, they will be at their schools held. The best 16 of these exams take part in the preparation for the International Mathematical Olympiad: In five training seminars - the final seminar will be held at the Mathematical Research Institute Oberwolfach - participants are encouraged. Meanwhile, seven exams are written to identify the team, tied score decides a random exam. Since 2005 will take place shortly before the IMO for the team another training seminar instead.

The participants in the IMO are automatically included in the Study Foundation of the German people and invited to events such as the day of the talent.

Austria

The preparation and preliminary round takes place at schools in courses. The best of each course ( roughly the first third ) qualify for one of the three area competitions. Of these increase the successful ones ( one third of participants, representing about 15 per region competition ) and may participate in the national competition, which traditionally takes place in Raach on high mountains.

The preparation takes about three weeks, and this consists of two parts, wherein at the end of the first part, an intermediate competition is held. This is followed by the more successful half of the second section of the course and the final competition. In this, the six participants will be determined for the international competition. The next best six candidates will participate in the Central European Mathematical Olympiad (MEMO).

Switzerland

The organization holds the qualification from Imosuisse in cooperation with the Federal Institute of Technology Zurich. To this end several days of training, a training camp and several tests are held. At the same time, the Swiss Mathematical Olympiad is held. The winners qualify for the International Olympiad.

Luxembourg

The best finisher of the Belgian Mathematical Olympiad is set safely. The other places are usually given to young hopefuls. The team often starts with only a few participants; previous exceptions are the 2008 team ( 5 participants), as well as 2009 and 2011 (each 6 participants).

History

The first IMO was held in 1959 in Brasov in Romania, it is the oldest science Olympiad. At the first Olympiad 52 students participated in the seven countries of Bulgaria, Czechoslovakia, East Germany, Hungary, Poland, Romania and the USSR. Originally the competition was intended as a one-off event for young mathematicians of the socialist countries in which mathematical talents were encouraged. After Romania but also in the following year IMO hosted and then with Hungary is another country took over the organization, was an annual event.

As the first non- socialist country took part in 1965 Finland. It was followed by 1967 Great Britain, Italy, Sweden and France, in 1969 the Netherlands and Belgium, 1970 Austria, 1974, the United States and Greece in 1975. The Federal Republic of Germany has been increasing since 1977 with a team consisting of students, Switzerland since 1991. The first Olympiad, which was held in a non- socialist country, was in 1976 in Austria, in 1979 was followed by the United Kingdom as host.

In 1980, the Olympics should have taken place in Mongolia, but was shortly canceled by the organizer, so this year no IMO took place. Instead, some spare Olympiads were organized, including in Mersch ( Luxembourg ) and in Mariehamn ( Finland), where, however, attended by only a few countries. In order not to jeopardize the continued existence of the IMO, Hungary and France took over at very short notice to organize for the years 1982 and 1983. To cope with this, the team size of the original eight participants had to be reduced. 1982 was therefore a team of only four participants, since 1983, from six. This team size has been maintained until today.

In the IMO 1995 in Canada, the current logo was introduced, it is modeled after the Olympic rings and showing the sign for infinity.

The number of participants and the countries rose sharply over time. At the Olympic Games in Germany in 2009 for the first time took part in over 100 countries, there were 565 participants from 104 countries to send observers other countries to send a team the following year. The proportion of girls among participants was approximately 10%.

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

While Austria has only been one time host, IMO four times took place in Germany already taken: the GDR was in Berlin in 1965 and 1974 in Erfurt hosts, the Federal Republic of Germany notified the IMO 30, 1989 in Braunschweig and the 50th IMO 2009 Bremen from.

Venues

The venues for the past five Olympic Games and the already allocated for the future include:

Already in use are:

  • 2014 - Cape Town ( South Africa)
  • 2015 - Thailand
  • 2016 - Hong Kong
  • 2017 - Brazil
  • 2018 - Romania

Expiration

A few days before the official start of the IMO, the head of delegation of the participating countries come together for the first jury session. The IMO jury selects from the different proposals that have been introduced by the countries, the six exam questions; the other activity suggestions are used by the individual countries often for the selection and preparation of the team to the next IMO. Based on the tasks in the official IMO Languages ​​English, German, French, Russian and Spanish translations finished the heads of delegation in the native languages ​​of the participants.

After the opening ceremony, the two exams are written on two consecutive days. Each takes 4 ½ hours. As an aid only ruler and compass are allowed except writing materials, so no particular set square and no calculator. During the first half hour, students have the opportunity to ask questions for clarification in the task.

Then the solutions of the participants of the respective delegations and their representatives to be corrected. For a completely solved problem, there are seven points, so that a total of 42 points can be achieved. In order to ensure a uniform assessment, the points will be awarded in consultation with coordinators in disputes decides in the last instance, the jury, by majority decision. Participants will have the opportunity to learn about the host country and other participants during the correction.

In its final meeting, the panel will decide on the points limits for the prices. They also decide on a proposal of the selected IMO Advisory Board on the award of the IMO to future hosts and invitations to new countries to send a team of students. The prices are then ceremoniously handed over in a graduation ceremony. The gold medals are usually awarded by special persons in public life, for example, by Andrew Wiles (2001 in the U.S.) or Crown Prince Felipe (2008 in Spain).

The cost of the IMO shall be borne by the host country, only the arrival and departure is required to pay the participating countries, observers must bear part of the costs yourself. In the 50th IMO 2009 in Germany the costs were about 1.5 million euros. For emergencies, a fund was set up in 1995.

Tasks

At each of the two exam days three duties. Usually there are two exams in a geometry task; other areas are number theory, inequalities, combinatorics and functional equations. The tasks often have short, elegant solutions that require a lot of creativity by the participants. Excluded, however, are tasks that terms of higher mathematics, or about differential calculus or algebra, require.

Since the early 80s, the maximum number of points is equal for all tasks regardless of their difficulty 7 points, before the scores were determined by the jury in response to the assessed difficulty. The tasks are usually ordered by difficulty, so that the first and fourth task is comparatively easy, while the sixth task is the hardest traditional.

Over the years, increased the difficulty of the tasks, the first task would now be considered in the first IMO as too easy. The task is:

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

Using the Euclidean algorithm, the greatest common divisor of the numerator and denominator determined very easily as 1, so the fraction is always truncated.

One of the most difficult tasks the sixth object of the IMO in 1986,:

At the vertices of a pentagon is each an integer, the sum of all numbers is positive. Standing on three consecutive corners of the figures, which is negative, we may replace it with. Cancels this procedure from sometime?

Was set the task of Elias Wegert, only eleven students were able to solve the problem completely.

A similarly difficult task was made two years later:

And are natural numbers, so also is a natural number, is an even square number.

No member of the task committee was able to solve this task, so that they agree with number theory familiar university mathematicians can furnish a task that, with a limited processing time of 6 hours, also found no evidence. Nevertheless, the task was set and solved by eleven students.

The task with the currently (as of 2013), the fewest points were awarded, dates from the year 2007:

It is a positive integer. Given a set of points of the three -dimensional space. Determine the minimum number of levels, the association comprises the amount, but not to the point ( 0, 0, 0).

Only five students to solve this task, two students received 2 points, 40 more one succeeded. The other 473 students received no point in this task, so that only 2.2% of the theoretically possible points were awarded.

Prices

The successful participants will be awarded with gold, silver, and bronze medals will be awarded this 1:2:3 ratio, with no more than half the student will receive a medal. A gold medal is awarded to the best so -twelfth of the participants, the next sixth receives a silver medal and another quarter bronze. Those who gain no medal, but has completely dissolved at least one of the six tasks, receives an honorable mention ( recognition, awarded since 1988). For the more elegant solutions Special rates may be awarded, this came so far (as of 2013) a total of 53 times before.

The most successful participant is currently the Serb Teodor von Burg, the 2007-2012 won four gold, one silver and one bronze medals over the years. Besides him, there are four participants, who won four gold medals: The German Lisa Sauermann, the year before got to the top of the medal table after they had won in the years 2007-2011 four gold and one silver medal, the Thais Nipun Pitimanaaree, who won from 2009 to 2013 also four gold medals and one silver, the German Christian Heron 1999-2003 won four gold medals and one bronze in the years and the Americans Reid Barton, who is up in 1998 2001 as the first participant managed to win four gold medals. The first German who managed to win three gold medals, in 1971 Wolfgang Burmeister from the GDR, which was so until 2000, the most successful participants. Overall, he obtained three gold medals, two silver medals and two special prizes at five participants. In addition to these six participants, it still have 33 more managed to win at least three gold medals.

Among the winners are a number of successful mathematicians. Twelve Fields Medals winners received in their school days at the IMO in part, including eight that won at least one gold medal IMO:

  • Vladimir Drinfeld (1969: Gold with full score )
  • Jean -Christophe Yoccoz (1973: Silver 1974: Gold )
  • Richard Borcherds (1977: Silver 1978: Gold )
  • William Timothy Gowers (1981: Gold )
  • Grigori Perelman (1982: Gold )
  • Terence Tao (1986: Bronze, 1987 Silver, 1988 Gold)
  • Stanislav Smirnov (1986, 1987, each gold with full score )
  • Ngô Bao Châu (1988: Gold with full score, 1989: Gold)

Terence Tao won while his gold medal at the age of twelve years, making it the youngest Goldmedaillist.

Although the IMO is an individual competition, there are also unofficial rankings of countries. Here usually occupy China, Russia, the U.S. and South Korea the first places. Germany occupied a place in most years 10-20, Austria is usually classified around the 50th place, just as Switzerland. German teams succeeded on three occasions, winning the competition: In 1968, the German Democratic Republic, 1982 and 1983, the Federal Republic of Germany.

2013, China ranked the first place, South Korea the second and the USA third. Germany reached number 27, number 48 Austria, Switzerland 40th place

International Science Olympiad Felipe, Prince of Asturias Abstract algebra Jean-Christophe Yoccoz
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