1. History of the International Mathematics Olympiad

The International Mathematical Olympiad is a competition for students aged under 20 years who are not yet attending university. It was first held in Romania in 1959 between the Soviet Union and nine Eastern European countries. The number of participating countries has increased every year, reaching 27 in 1981 and 54 in 1990. At the 56th IMO, which took place in Chiang Mai, Thailand in 2015, 104 countries and regions participated. The United Kingdom and France first participated in 1967, the United States in 1974, (West) Germany in 1977, China in 1985, and the Republic of Korea in 1988.

Japan’s debut in the IMO was in 1990, making it the latest of the developed nations to join the competition. Rumor has it that one reason why it took so long for Japan to participate was that, due to the Japanese Education Ministry’s policy of yutori kyoiku (relaxed education) at the time, it may not have been very enthusiastic about the education of particularly gifted children. Today, there has been a complete turnaround in the situation, and Japan currently takes part in seven science-related Olympiads (mathematics, informatics, physics, chemistry, biology, earth science, and geography). The teams sent to these Olympiads all receive financial assistance from the Japan Science and Technology Agency (JST). The International Mathematical Olympiad was the first such competition that Japan took part in, so it could be described as the pioneer in science Olympiads in Japan.

I have been involved in the Japan Mathematical Olympiad for about ten years, as a Counsellor, Director and Managing Director. In 2010, I attended IMO 2010 Kazakhstan as a Director, and I have acted as Leader of the Japanese Team since IMO 2013 Columbia. This article will describe some of my impressions of the IMO, mainly from my perspective as Team Leader.

2. Japan Mathematical Olympiad

Qualifying rounds for both the Japan Mathematical Olympiad (JMO) and the Japan Junior Mathematical Olympiad (JJMO; a mathematical Olympiad for students in middle school and younger) take place on Coming-of-Age Day (public holiday in mid-January). In both contests, participants must solve 12 problems in three hours. This year, the JMO qualifying round attracted approximately 3,500 entries and the JJMO approximately 2,200. 175 entrants passed the qualifiers in the JMO and 92 in the JJMO. These 267 participants went on to participate in the main selection rounds on National Foundation Day (February 11). Gold, silver and bronze medals and high achievement certificates are awarded to about twenty participants in the JMO (gold medal winners also receive the Kawai Trophy) and about ten entrants in the JJMO. A final selection round is held over four days at the end of March, and the top six participants are selected for the team that will represent Japan at the International Mathematical Olympiad (IMO). At the IMO, which takes place in July every year, 1/12 of entrants are awarded a gold medal, 2/12 a silver medal, and 3/12 a bronze medal. Participants must solve a total of six problems worth a total of 42 points over two days, and the country rankings are announced based on the total scores of the six members of each national team. Japan’s results over the past three years are as follows.
2013 (54th IMO, Columbia): no gold, six silver, no bronze, 11th place (97 countries, 560 participants);
2014 (55th IMO, South Africa): four gold, one silver, one bronze, 5th place (101 countries, 528 participants);
2015 (56th IMO, Thailand): no gold, three silver, three bronze, 22nd place (104 countries, 577 participants (525 boys, 52 girls)

3. The Olympiad (from the Team Leader’s Perspective)

People often say to me that, as Team Leader, I am lucky because I can go sightseeing while the contestants are taking the tests, but that is absolutely not the case. We are actually quite busy. We depart for the IMO three or four days in advance of the contestants to help select the six problems— three for each of the two days—from the shortlist of several dozen problems already chosen by the Problem Selection Committee.

The selected problems are well balanced between the four areas of Algebra (A), Geometry (G), Combinatorics (C), and Number Theory (N), and are presented so that their order in increasing difficulty is Q1, Q2, Q3 (Q4, Q5, Q6 on the second day). When choosing the problems, we must consider factors such as whether there have been similar problems used in the past and whether a problem has been used in any of the national contests. This process is conducted in a completely democratic manner, with each participating country being given one vote. All decisions are made by consultation and vote at meetings of the IMO Jury. Therefore, when deciding on the six problems and their marking criteria, we start by solving the problems ourselves to gain a feel for them, which takes about four days. Once the problems have been set, the representatives from the English-speaking nations rewrite them into correct English, and they then need to be translated into all of the languages of the participating countries. The translated problems are all displayed and we each check the others’ translations to ensure that nothing has been added that would give any team an advantage.

The sight of those problems translated into the languages of more than 100 countries—English, French, Germany, Russian, Spanish, Korean, Chinese, Khmer, Arabic, and so on—all lined up together is certainly an impressive one to behold.

The tests take place over four and a half hours, from 9:00 a.m. to 1.30 p.m. on both the first and second days. Contestants may ask questions from 9:30 a.m. To ensure that the team asking the question is not given an unfair advantage, the questions are shown on a display where everyone can see them and answers are given. Marking begins as soon as the tests are over. The solutions are marked with the cooperation of a university student (Observer A), who is a past IMO medalist, but it does not end there. It then enters a process of coordination (marking negotiation). Even if the host coordinators are unable to understand Japanese, they mark the papers from a copy, and check to see that the results are the same as the results of our own marking for all six problems for all six contestants. This process takes at least two or three days to complete, and, once we are mutually satisfied with the results, we sign them and the contestants’ final scores are determined. After that, the cut-off marks for gold, silver and bronze medals are decided at a final meeting of the IMO Jury, and the medals are awarded at the closing ceremony on the following day.

4. Japan’s Results

The strongest countries are China, United States, Russia and South Korea. China, in particular, has received the highest team score 16 times in the past twenty years.

In recent years, Japan’s best results have been second place at the 2009 IMO in Germany, seventh place in Kazakhstan in 2010, and fifth place in South Africa last year. Apart from these, it has placed around tenth in most years, including eighth place in 2003 when Japan was the host country.

　

Under IMO regulations, Japan actually appears to have some small disadvantages. Perhaps surprisingly, one of those disadvantages is the fact that Japan’s school/university year starts in April. The IMO is always held in July. In many of the other participating countries, university usually begins in September, so there are students in those countries who, although having already graduated from high school by July, are still able to participate in the contest because they are under 20 years of age and are not yet in university. In Japan’s case, however, 18-year-olds enter university in April, so naturally, they are unable to compete in July. In that respect, it must be said that Japan bears a handicap of one year of age compared to other countries.

Another disadvantage for Japan is that, in July, students are right in the middle of their high school year. On the other hand, in the United States, for example, which is one of the strongest countries in the competition, the school year ends in May, so they can spend all of June training for the Mathematical Olympiad. Indeed, I have heard that they hold a four-week training camp in Colorado. In South Korea, once the team has been selected in April, the contestants gather at Seoul National University every Saturday and Sunday for training. This is apparently possible because past contestants have all come from Seoul (and many of them from a specific middle school and high school). After selection at the end of March, Japan’s team meets only once, during the Golden Week string of public holidays in May, at the Olympic Center in Tokyo’s Yoyogi Park for a 2-night, 3-day training camp. Apart from this, they solve practice problems at home and e-mail them in for marking. Compared to the giants of the IMO, this may not be enough. Even so, Japan has produced a large number of high-achieving contestants, including one student who obtained a perfect score at IMO Germany. Most of them do their own training, using, for example, a website called Art of Problem Solving (AOPS), in which past problems from mathematics competitions from various countries are posted and discussed.

5. Girls’ Mathematical Olympiad

On a global level, participation by girls in the IMO is only around 10% that of boys (with some exceptions, such as the Ukraine, which usually has two or three girls in its six-member team every year). In the 25 years of Japan’s participation in the IMO, only two girls have ever taken part. In recent years, in a bid to encourage more girls to participate in the IMO, particularly in Europe, the European Girls’ Mathematical Olympiad (EGMO) began in 2012. Japan participated in the EGMO for the first time in 2014, when entry was opened to the country. I acted as team leader for the Japanese national team to EGMO 2015 in Belarus. The promotion of STEM education among girls is an important issue in Japan as well, so it is hoped that this initiative will provide some impetus. In fact, participation by girls is gradually increasing.

6. The Future

The host countries for the IMO for the next few years have already been decided. It will be held in Hong Kong in 2016, Brazil in 2017, Romania in 2018, and the United Kingdom in 2019. The Olympic Games will take place in Tokyo in 2020, and Japan will also host a number of science Olympiads in the years before and after that, starting with the International Olympiad in Informatics in 2018. Japan last hosted the IMO in 2003. Hosting tends to be on a cycle of around twenty years, which means that, although nothing has been decided, the International Mathematical Olympiad will likely take place in Japan around 2023-2025.

I look forward to your cooperation at that time.

Takahiko Fujita
Professor, Industrial and Systems Engineering Department, Faculty of Science and Engineering, Chuo University
Managing Director, Mathematical Olympiad Foundation of Japan
Leader of Japanese Team to 56th International Mathematical Olympiad, Thailand

Professor Fujita was born in Hyogo Prefecture in 1955, graduating from the Faculty of Science, Kyoto University in 1978. In 1980, he obtained his Master’s degree in mathematics from Kyoto University’s Graduate School of Science. In 1981, he left the doctoral program in the Division of Mathematics, Graduate School of Science, Kyoto University. He later obtained his Ph.D. (Science) from Kyoto University.
In 1981, he became an Assistant in the Department of Mathematics, Faculty of Science, Kyoto University. Subsequently, he was appointed assistant professor and later full professor at Hitotsubashi University. After serving as Professor of the Graduate School of Commerce and Management, Hitotsubashi University, he took up his current position in 2011.
His current areas of research are probability theory and its application in financial engineering, actuarial science, and mathematics education. Also, in his current position in the Industrial and Systems Engineering Department, he is training students to become actuaries (a specialist occupation in actuarial science that requires passing a qualification examination in mathematics). He is also actively involved in mathematics education as editor-in-chief at Keirinkan, a specialist publisher of high school mathematics textbooks.
Professor Fujita is the Managing Director of the Japan Mathematical Olympiad Foundation. His major publications include Introduction to Probability Analysis in Finance [Fainansu no Kakuritsu Kaiseki Nyumon] (Kodansha, 2003), Conquering Weaknesses—Probability Statistics for College Students (Tokyo Tosho, 2010)Conquering Difficult Problems—Understanding by Solving with Galois Theory [Nanmon Kokufuku—Toite Wakaru Garoa Riron] (Tokyo Tosho, 2012).