Sof International Mathematics Olympiad Class 2

[Author Metcalfe]

The content ranges from extremely difficult algebra and pre-calculus problems to problems in branches of mathematics not conventionally covered in secondary or high 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, often times a great deal of ingenuity to net all points for a given IMO problem.

The first IMO was held in Romania in 1959. Since then it has been held every year (except in 1980, when it was cancelled due to internal strife in Mongolia)[6] 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.[7]

Sources differ about the cities hosting some of the early IMOs. This may be partly because leaders and students are generally housed at different locations, and partly because after the competition the students were sometimes based in multiple cities 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.[8]

Several students, such as Lisa Sauermann, Reid W. Barton, Nicușor Dan and Ciprian Manolescu have performed exceptionally well in the IMO, winning multiple gold medals. Others, such as Terence Tao, Artur Avila, Grigori Perelman, Ng Bảo Chu and Maryam Mirzakhani have gone on to become notable mathematicians. Several former participants have won awards such as the Fields Medal.[9]

The competition consists of 6 problems. The competition is held over two consecutive days with 3 problems each; each day the contestants have four-and-a-half hours to solve three problems. Each problem is worth 7 points for a maximum total score of 42 points. Calculators are banned. Protractors were banned relatively recently.[10] Unlike other science olympiads, the IMO has no official syllabus and does not cover any university-level topics. 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 elementary. However, they are usually disguised so as to make the solutions difficult. The problems given in the IMO are largely designed to require creativity and the ability to solve problems quickly. Thus, the prominently featured problems are algebraic inequalities, complex numbers, and construction-oriented geometrical problems, though in recent years, the latter has not been as popular as before because of the algorithmic use of theorems like Muirhead's inequality, and Complex/Analytic Bash to solve problems.[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, where the First day problems Q1, Q2, and Q3 are in increasing difficulty, and the Second day problems Q4, Q5, Q6 are in increasing difficulty. The team leaders of all countries are given the problems in advance of the contestants, and thus, 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]

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 Junior Mathematical Olympiad/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]

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 7 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, for his solution to Problem 3, a three variable inequality.

North Korea was disqualified twice for cheating, once at the 32nd IMO in 1991[23] and again at the 51st IMO in 2010.[24] However, the incident in 2010 was controversial.[25][26] There have been other cases of cheating where contestants received penalties, although these cases were not officially disclosed. (For instance, at the 34th IMO in 1993, a contestant was disqualified for bringing a pocket book of formulas, and two contestants were awarded zero points on second day's paper for bringing calculators.[27])

Russia has been banned from participating in the Olympiad since 2022 as a response to its invasion of Ukraine.[28] Nonetheless, a limited number of students (specifically, 6) were allowed to take part in the competition, but only remotely and with their results being excluded from the medal tally.[28] No other country has been banned on similar grounds.

The only countries to have their entire team score perfectly in the IMO were the United States in 1994, China in 2022, and Luxembourg, whose 1-member team had a perfect score in 1981. The US's success earned a mention in TIME Magazine.[84] Hungary won IMO 1975 in an unorthodox way when none of the eight team members received a gold medal (five silver, three bronze).[76] Second place team East Germany also did not have a single gold medal winner (four silver, four bronze).[82]

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[92] 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 Ral Chvez Sarmiento (Peru), at the age of 10 and 11 respectively.[93] Representing the United States, Noam Elkies won a gold medal with a perfect paper at the age of 14 in 1981. Both Elkies and Tao could have participated in the IMO multiple times following their success, but entered university and therefore became ineligible.

The International Mathematical Olympiad is the pinnacle of all high school mathematics competitions and the oldest of all international scientific competitions. Each year, countries from around the world send a team of 6 students to compete in a grueling competition.

The competition takes place over 2 consecutive days. Each day 3 problems are given to the students to work on for 4.5 hours. Following the general format of high school competitions, it does not require calculus or related topics, though proofs using higher mathematics are accepted.

Scoring on each problem is done on a 0-7 scale (inclusive and integers only). Full credit is only given for complete, correct solutions. Each solution is intended to be in the form of a mathematical proof. Since there are 6 problems, a perfect score is 42 points.

The IMO started in 1959 as a competition among Eastern European countries. Since then, it has been held every year (except 1980) and has evolved into the premier international competition in mathematics.

Each year nearly every country proposes several problems in consideration for the International Mathematical Olympiad. All submissions are compiled into a Longlist, the length of which can easily exceed 100 problems. Then the IMO deputy leaders convene on site and discuss which problems should be used on the International Mathematical Olympiad test that year. Eventually most of the problems on the Longlist are eliminated from consideration, and what is left is a shortlist, with a length between 26 problems and 32 problems, spread out across the topics of Algebra, Combinatorics, Geometry, and Number Theory. The six problems are then chosen out of these.

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