Problems from olympiads (main page)

Problem Collection for International Mathematical Olympiads


The International Mathematical Olympiad (IMO) is the most important and prestigious mathematical competition for high-school students. It has played a significant role in generating wide interest in mathematics among high school students, as well as identifying talent.

In the beginning, the IMO was a much smaller competition than it is today. In 1959, the following seven countries gathered to compete in the first IMO: Bulgaria, Czechoslovakia, German Democratic Republic, Hungary, Poland, Romania, and the Soviet Union. Since then, the competition has been held annually. Gradually, other Eastern-block countries, countries from Western Europe, and ultimately numerous countries from around the world and every continent joined in. (The only year in which the IMO was not held was 1980, when for financial reasons no one stepped in to host it. Today this is hardly a problem, and hosts are lined up several years in advance.) In recent years, more than 100 countries participate in the event.


The total number of available problems is 314.

Year IMO
1959 IMO
1960 IMO
1961 IMO
1962 IMO
1963 IMO
1964 IMO
1965 IMO
1966 IMO
1967 IMO
1968 IMO
1969 IMO
1970 IMO
1971 IMO
1972 IMO
1973 IMO
1974 IMO
1975 IMO
1976 IMO
1977 IMO
Year IMO
1978 IMO
1979 IMO
1981 IMO
1982 IMO
1983 IMO
1984 IMO
1985 IMO
1986 IMO
1987 IMO
1988 IMO
1989 IMO
1990 IMO
1991 IMO
1992 IMO
1993 IMO
1994 IMO
1995 IMO
1996 IMO
Year IMO
1997 IMO
1998 IMO
1999 IMO
2000 IMO
2001 IMO
2002 IMO
2003 IMO
2004 IMO
2005 IMO
2006 IMO
2007 IMO
2008 IMO
2009 IMO
2010 IMO
2011 IMO
2012 IMO
2013 IMO
2014 IMO

The Competition

The format of the competition quickly became stable and unchanging. Each country may send up to six contestants and each contestant competes individually (without any help or collaboration). The country also sends a team leader, who participates in problem selection and is thus isolated from the rest of the team until the end of the competition, and a deputy leader, who looks after the contestants.

The IMO competition lasts two days. On each day students are given four and a half hours to solve three problems, for a total of six problems. The first problem is usually the easiest on each day and the last problem the hardest, though there have been many notable exceptions. (IMO96-5 is one of the most difficult problems from all the Olympiads, having been fully solved by only six students out of several hundred!) Each problem is worth 7 points, making 42 points the maximum possible score. The number of points obtained by a contestant on each problem is the result of intense negotiations and, ultimately, agreement among the problem coordinators, assigned by the host country, and the team leader and deputy, who defend the interests of their contestants. This system ensures a relatively objective grade that is seldom off by more than two or three points.


Though countries naturally compare each other’s scores, only individual prizes, namely medals and honorable mentions, are awarded on the IMO. Fewer than one twelfth of participants are awarded the gold medal, fewer than one fourth are awarded the gold or silver medal, and fewer than one half are awarded the gold, silver or bronze medal. Among the students not awarded a medal, those who score 7 points on at least one problem are awarded an honorable mention. This system of determining awards works rather well. It ensures, on the one hand, strict criteria and appropriate recognition for each level of performance, giving every contestant something to strive for. On the other hand, it also ensures a good degree of generosity that does not greatly depend on the variable difficulty of the problems proposed.

Problem selection procedure

The selection of the problems consists of several steps. Participant countries send their proposals, which are supposed to be novel, to the IMO organizers. The organizing country does not propose problems. From the received proposals (the so-called longlisted problems), the Problem Committee selects a shorter list (the so-called shortlisted problems), which is presented to the IMO Jury, consisting of all the team leaders. From the short-listed problems the Jury chooses 6 problems for the IMO.

Apart from its mathematical and competitive side, the IMO is also a very large social event. After their work is done, the students have three days to enjoy the events and excursions organized by the host country, as well as to interact and socialize with IMO participants from around the world. All this makes for a truly memorable experience.

(This text was excerpted from "The IMO Compendium.")

Having read this general story, you may wish to read some texts about particular IMOs. For this purpose, we have provided the following two texts: