# International Mathematical Olympiad

The **International Mathematical Olympiad** (**IMO** for short) is an annual mathematics contest for high school students from across the world. Each country can send a maximum of six participants. About 90 countries participate and the number of contestants is approximately 500. The contest stretches over two days. On each day, the students are given 4.5 hours to solve three challenging problems *individually*. Based on relative performance, students are awarded gold medals, silver medals, bronze medals and certificates of honorable mention.

Selection procedures for the Olympiads differ from country to country. The countries that participate regularly send six students every year, selected through tests and trained specifically for the Olympiad.

The first International Mathematical Olympiad was held in Romania in 1959. Since then, the Olympiad has been held every year except 1980. The 2007 International Mathematical Olympiad was held in Vietnam, and the 2008 International Mathematical Olympiad will be held in Spain. The IMO is usually held in the month of July.

## Contents

## History

The first International Mathematical Olympiad was held in Romania in 1959. Only seven countries participated: Romania, Hungary, German Democratic Republic, the erstwhile Soviet Union, Bulgaria, Poland and Czechoslovakia. For the first 5-6 years, participation was limited to countries in East Europe (and the Soviet Union). By 1970, many West European countries, including France, Italy and the United Kingdom, had started participating. The United States of America first participated in 1974 and China first participated in 1985. In 1989, the number of participating countries reached 50.

## Scoring and format

### Contest format

The current format of the International Mathematical Olympiad has been used since 1981. The contest spans over two days. On each day, the contestants are given a set of three problems to solve in a span of 4.5 hours. The problems are in the following mathematical disciplines: algebra, number theory, combinatorics and geometry. On each day, the problems are in increasing order of difficulty, and the problems on a particular day are all in different parts of mathematics.

### Scoring

After the contestants submit the problems, two people from the contestants' country (called the Leader and Deputy Leader) evaluate the contestants' problems in consultation with the contestants. Independently, each problem is evaluated by members of a Problem Committee, according to criteria evolved by the Committee. The Leader and Deputy leader of the country then discuss with the Problem Committee and come to a consensus on how many points each contestant gets on the problem. The maximum score per problem is 7.

### Medals

The top 1/12 of contestants (based on total score on the problems) receive a gold medal. Cutoffs for the gold medal vary from year to year, depending on the difficulty of the problems. The next 1/6 of contestants receive a silver medal, and the next 1/3 of contestants receive a bronze medal. A total of at most half the contestants receive medals. Of the remaining contestants, those who score full points on at least one problem, receive a certificate of honourable mention.

## Problem selection

### Proposal stage

Any of the participating countries can propose problems for the IMO. The problems are sent to the country that is hosting the IMO. From the *longlist* of all proposed problems, the host country selects a *shortlist* with 6-10 problems each from algebra, number theory, geometry and combinatorics.

Countries proposing problems are supposed to keep these problems confidential from the contestants.

### Selection stage

Three to four days before the actual contest, the Leaders from the countries arrive at the host country. The Leaders go through the shortlisted problems, and based on assessment of the level of difficulty of these problems, decide what problems should appear on the two contest days, and in what order. The leaders do not have contact with the contestants or Deputy Leader during this process.

### Shortlisted problems

Shortlisted problems are usually not disclosed to the world immediately after the Olympiad. These problems are used by a number of countries in their selection tests for students for the Olympiad next year. The shortlist is published officially just before the *next* year's Olympiad.

## Syllabus

There is no definite syllabus for the International Mathematical Olympiad. The four disciplines from which problems are asked are: algebra, geometry, number theory and combinatorics. Calculus is specifically avoided in the International Mathematical Olympiad. Every Olympiad problem should have an *elementary* solution that does not require any knowledge of calculus or any knowledge of higher mathematics. However, students are allowed to use methods from higher mathematics.

## Selection process

Selection process for the International Mathematical Olympiad differs from country to country. Countries that participate regularly in the Olympiad often have a two-tier (or even three-tier) selection process. The first round is open to the general public, and those selected in this round get to attend a training-cum-selection camp where the top six students are selected through a series of tests. The selected students may then be trained further.

The criterion for participating in the International Mathematical Olympiad is that the student should be under the age of 21, and should not have formally begun college education. Participating countries may impose further rules on eligibilty of contestants to appear in the camps.

## Performance of countries

The performance of countries varies year by year. Over the last decade, countries that have regularly performed well include China, the United States, Russia and Vietnam. Bulgaria topped in 2003.

The best-ever performance has been that of the United States in 1994 at Hong Kong, where all contestants got perfect scores.

## Later achievements of Olympiad contestants

Many Olympiad contestants have gone on to become mathematicians.

- 2006 Fields medalist Terence Tao had participated in the International Mathematical Olympiad as an Australian at the ages of 11, 12, 13. He then migrated to the United States of America, and stopped contesting in the Olympiads. Tao has won the Fields Medal for his work on proving a conjecture on arithmetic progressions of primes.

- Grigory Margulis, known for his work on Lie groups and also a Fields medalist like Tao, also participated in the Olympiads in his early years.

- Grigori "Grisha" Perelman, the reclusive Russian mathematician who gained prominence for his proof of the Thurston's Geometrization Conjecture (that includes the notoriously elusive Poincaré Conjecture as a special case) for which he was awarded the Millennium prize on 18 March, 2010 [1], also 2006 Fields Medal (but ultimately declined to accept this prize), and was a gold medalist at the 1982 IMO (as a member of the USSR team) where he scored a perfect paper.