Introduction

Many data and studies are available on mathematics competitions.  In many cases, the main features of the competition, topics, results, statistics, comments, can be found in brochures or books published by the organizers.  Mathematics Competitions, the journal the World Federation of National Mathematics Competitions (WFNMC), is an invaluable source of information on what happens in the world.  Several ICMI Studies, in particular those on The Popularization of Mathematics and Assessments in Mathematics Education, are related to the subject of competitions.

It may be time for a new Study on this specific question: what are the key problems raised by mathematics competitions?  For example, how are competitions related to exams and teaching?  What is their social impact?  How are mathematics as a science and research mathematicians involved?  How different and how similar are experience and actions in different parts of the world?

I was invited to give my point of view, on the basis of my own experience and interests, for this ICMI Bulletin.  My ideas were expressed already in a talk, in French, at the Seventh Southeast Asian Conference on Mathematics Education (SEACME-7) in Hanoi (1996).  I shall organize them in the same way: 1) competitions and society, two examples, 2) scientific competition, different forms, 3) mathematics competitions, a few examples, 4) some questions.

All competitions have a common feature: a strong involvement of individuals, participants or supporters.  During the time of the competition, nothing is more important in the world.  Competitions generate a specific tension, both on individuals and societies. Such a tension can have two opposite effects, sometimes simultaneously: enthusiasm, and rejection.

The main reference for the history of sports is the Olympic games in ancient Greece.  Apparently they gave rise to a universal enthusiasm.  On the other hand, the circus games in Rome had two aspects; they were highly praised by their public, and more and more rejected as inhuman.  The gigantic sports competitions of our time have a twofold aspect: passionate support from millions of people, and sudden distrust and disgust when the system grinds.

The world economy is based on economic competition.  Competition is useful in order to obtain better products and better services for a lower price and the benefit of more people. However, a strange aberration appeared recently: the reduction of economy to economic competitiveness. "Economic competitiveness" is a magic term; for example, the Maastricht treaty, on which the European Community is established, states that the aim of scientific research is to provide the scientific grounds of economic competitiveness. But economic competitiveness depends on the rules of the game and can produce the better or the worse, as a function of these rules; it cannot change the rules by itself.  On the contrary, economics as a science should study these rules and investigate the consequences of possible changes of the rules.  If the distinction is not clear, disappointment or distrust towards economic competitiveness can engender negative attitudes towards economy and economics at the very moment when we need them most.

The common feature of all competitions is the existence of rules.  Who decides the rules, why and how? Where do they lead? How should we change them according to the directions we want? These general questions appear for all types of competitions. Moreover many competitions have much in common with sports and with economy.  It was my reason to begin with these two examples.

Science is a field of competition, in various forms.  Nowadays people compete and rush for getting published, getting grants and positions and honors.  There were other forms in the past, especially in mathematics: challenges, controversies, open questions, prizes.

Challenge is one of the first form of scientific communication.  "Here is what I am able to do. Who can do the same?" In the 17th century Father Mersenne exchanged letters with all scientists of the time, Galileo, Hobbes, Fermat, Descartes, and he carried the challenges among them.  Sometimes the confrontation was direct, as between Fermat and Descartes on the construction of tangents to a curve.

Controversies played an important role in the 17th and 18th centuries.  The struggle between Newton and Leibniz on the foundations of calculus was not very productive, except that it contributed in expanding scientific ideas and knowledge through the learned part of the society in Europe.  The most interesting scientific controversy of the 18th century is the famous debate on vibrating strings that involved d'Alembert, Euler, Daniel Bernoulli and Lagrange; therefrom came the most fundamental notions on partial differential equations and representations of functions.

The Paris Academy of Sciences organized scientific competitions in the form of open questions, examination of the contributions sent by different authors, and prizes for the best.  The effect was twofold: to select the best contributor as a possible future member of the Academy, and to point out what could appear as the most important scientific questions of the time.  Examples are Joseph Fourier in 1811 on propagation of heat, and Sophie Germain in 1816 on vibrations of elastic leaves.  Fourier became an academician. Sophie Germain had to be satisfied with the prize.

Competitions of this type expressed a scientific policy.  They lost their importance during the 19th century and disappeared in the 20th.  Scientific policy uses other ways.

However, open problems kept their role in mathematics as an incentive for research.  The most spectacular example is the solution by Andrew Wiles in 1994 of the long-standing problem, how to prove the ?Fermat Theorem?.  The solutions of the Hilbert Problems, proposed at the Paris Congress of Mathematicians in 1900, are landmarks of the century.  The ?Scottish Book? where, after 1930, Banach and his friends wrote regularly what they found interesting to investigate became an important source of inspiration.  In the 50's many mathematical journals had a problem section.  Some mathematicians, like Littlewood or Erdös, are famous for their problems, but the problems, in their case, are kinds of by-products of an intense mathematical activity.  The importance of open problems is easy to check in the current mathematical literature.

Problems are changing.  Historical problems like the Riemann Hypothesis keep their value.  But other questions arise from other sciences and techniques.  They are less formalized and more in the spirit of the 18th century.

Prizes are changing too.  The Fields Medal and the Nevanlinna Prize mimic the Nobel Prize.  New scientific prizes were created.  Of course, competition still exists, but it is not open any more.

It would be worth describing and comparing extreme cases, like the Argentinian system of successive selections on a very large basis, and the Schweitzer competition in Hungary, open to university students who are willing to spend a week on a series of pretty difficult problems.

On the other hand, it would be interesting to list all mathematical competitions that exist in a given country, and France is very rich in this respect, with elite competitions and competitions for all, rallies, games, competitions between clubs, etc.  All actions of this type are coordinated now by a recently created association: Animath.

Competitive exams with a strong mathematical component used to be a French specialty.  The oldest is the entrance examination to École Polytechnique, in which Évariste Galois failed.  It was the model of competitive exams for entering all scientific "Grandes écoles", including Écoles Normales Supérieures.  Such exams are prepared during two years after the baccalauréat in special classes of a few high schools, called "classes préparatoires".  The best students in mathematics usually enter these "classes préparatoires" rather than the corresponding first cycle of universities.  This split is a serious social and scientific problem.

The traditional style of mathematical problems proposed in the French competitive exams is a series of articulated questions, constructed in the manner of a scientific article.  Fifty years ago, when I passed the ?agrégation?, the only competitive exam at that time in order to become a high school teacher, the written part of the examination consisted of five parts: elementary mathematics (7 hours), special mathematics (7 hours), numerical analysis (4 hours), calculus (7 hours) and rational mechanics (7 hours), all that in a week.  It required a pretty good physical condition.

Let me turn to the International Mathematical Olympiads.  They were created after the war on the initiative of Romania, and at first they were restricted to the Eastern socialist countries of that time.  They extended progressively: 45 countries participated in 1988, 75 in 1996.  There is a new location each year, and once, in the early 80?s, ICMI was involved in the setting up of a Site Committee for the IMO (later renamed the IMO Advisory Board).  Since then the system has been running smoothly.  Obviously all countries are interested, not so much but in the same way as it is the case for Olympic games.  Each country is represented by a small team. Romania can be proud of its results, France not so much.  The time allowed is short: in 1996, in Mumbai, India, twice  hours, for six different and difficult short questions.  Obviously French young people are not well prepared for this type of competition, and it is a field of action now for Animath.

In all international competitions, in particular in sports, national feelings are concerned, and rules have to be clear and respected.  It is the case indeed with mathematics competitions and the results are meaningful.  The IMO occupy a particular situation compared to competitions in sports, because they are related to a mental activity and a scientific field.  They recognize a different scale of values, a different type of excellence.

The Australian Mathematics Competition was created in the 70's by the late Peter O'Halloran and it has a completely different character.  It may be the most popular mathematical competition in the world, with 500 000 participants every year, representing about one in every three students enrolled in secondary schools.  The principle is that every student can find a challenge somewhere, the first questions being very easy and the last pretty difficult.  For such a number of participants, the only possibility is questions with multiple choice.  When I got in touch with this competition, in 1983, I had a strong prejudice against multiple choice: crossing boxes was not the way I felt possible to express oneself in mathematics.  Then I saw the results and was convinced that it was an original and relevant manner to detect mathematical aptitudes and talents, in particular among young people who usually express themselves in a poor way.  The same is true for the French version of the Australian Competition, Kangourou, organized in a different way but on the same principle.

When ICMI organized the Study on The Popularization of Mathematics in 1989, many of us discovered the Leeds competition.  It is not a competition among individuals, but among schools of the Leeds area.  Three research topics are proposed by professors of the University of Leeds at the beginning of the year.  Students organize themselves in teams, work during several months, and produce at the end a series of documents: papers, posters, films, computer programs.  Sometimes the solution is achieved, sometimes no complete solution can be given.  I saw both cases as a member of the jury: a question in cryptology needed some algebra and had a complete solution, discovered by a few teams; a question of tiling in the plane allowed many variations; and a question on population dynamics led to the beautiful fractal drawings as usual in non-linear dynamics.  This competition consists really in research work, very much similar to the work of professional mathematicians, and I was impressed by the professional quality of a few realizations. On the other hand, only elite schools are concerned.  Is it possible to enlarge the social range and keep the same quality?  The experience of "MATh.en.JEANS" in France is of the same type with more kinds of schools, so that it seems that the answer should be positive.

My last example does not exist anymore but it is worth thinking about.  In the 60's, the Hungarian TV organized TV contests in science under the title "Ki miben tudós?" (who is learned in what?).  Mathematicians took part only in 1964 and 1966.  It was a complex machinery, with a selection on a national basis of 8 young people at the level of end of secondary schools, then a three step competition in order to reduce their number to 4, then 2, then 1, the winner.  Each step consisted of a written work (45 minutes) and a series of oral questions to be solved in 2 or 3 minutes.  The TV broadcast the beginning of the written work (and the public could work as well), then, after two hours, the conclusions of the jury, then the oral part.  The jury was really professional (for example, George Alexits, Paul Turán, George Hajós and Alfred Rényi) and most of the 16 selected young people became professional mathematicians.  Here is an example of topic (final step, 1966): prove that the greatest common divisor of a+b and the lowest common multiple of a and b is the greatest common divisor of a and b.  Questions and answers impressed the public.  Apparently, this short-lived contest was a TV hit.

I shall consider three questions only:
1) the relation of competitions with exams and education,
2) their social impact,
3) their relation to living science.

1) The relation of mathematics competitions with exams and education
What is the relation between education, exams, and competitions?  The aim of education is to open the mind and teach methods, concepts and knowledge.  Exams indicate whether and how far this aim is achieved.  There is a feedback.  If the emphasis of exams is on factual knowledge, or on problem solving, teachers and teaching will go in that direction.  If proofs are absent from exams, proofs will disappear from teaching.

Exams have another role: when passed, they qualify the student for a social recognition.  Therefore, they tend to be as objective, neutral and indisputable as possible.  Mathematics is good for that purpose when only results and performance are tested.  Actually, most math exams test performance rather than work and opening of the mind.  Since there is a discrepancy between the action (working, learning, thinking) and the result (performance), exams may fail to be an incentive for working.  How to evaluate working as such?  We all know how important and difficult this is.

Anyway, in spite of the feedback, the role of ordinary exams is not to determine the form and content of teaching and curricula.

But the situation is reversed when we consider competitive exams and their preparation.  Then the aim of teaching is just helping the student to pass the exam.  It is exactly the case in the French "classes préparatoires".  The program and rules of the exam
define the program and rules of the class.  During a long period in France, the entrance examination to École Polytechnique played a decisive role in the training of the best students.  The evaluation of teachers in these ?classes préparatoires? is simple: better results = better teacher.

More or less this happens in the whole world.  The competitive exams for getting positions in life have an enormous impact on the teaching of mathematics.  We should be aware of the corresponding dangers.

The impact of non-professional mathematics competitions is not the same.  It can be highly beneficial in introducing new ideas and new topics.  On the other hand, the effect of new ideas is not necessarily beneficial.  I used to give the following example.  At the 1996 Kangourou competition, there was a nice question, among the most difficult ones: a field with the shape of a quadrilateral is divided by the diagonals into four triangles; three of these triangles have areas 3 000 m², 4 000 m² and 5 000 m² respectively when we turn around their common point; is the area of the fourth triangle less than 3 000 m², or between 3 000 m² and 4 000 m², or between 4 000 m² and 5 000 m², or larger than 5 000 m²?  It is not so easy to draw a figure, but every student knows what is needed for the conclusion.  It is a very good question.  Now suppose that a similar question is asked in 1997 or 1998 (fortunately, it was not the case).  Then every teacher will teach that when dividing a convex quadrilateral by means of the diagonals, the areas of the triangles that are obtained, A, B, C, D, when you turn around their common point, satisfy AC=BD.  The clever question will become a standard exercise.

In short, competitive exams have a direct effect on education in mathematics, and mathematics competitions an indirect effect.  Both deserve attention.

What is, a part from education, the impact of competitions on individuals and society?

At first sight, competition goes along with selection of the best, and selection of the best has an elitist flavor.  To associate mathematics competitions with selection, elite, inequalities, seems obvious, and mathematics as a tool for selection has a very negative image.  I think that it is necessary to look at the social impact more carefully.

Let me begin with my first example: the entrance examination to École Polytechnique or École Normale Supérieure, and more generally profession-oriented mathematics competitions.  Nowadays it is true that the selection they operate coincides more or less with a social selection: children of intellectuals are overrepresented, children from poor families are almost absent.  The situation was different a century ago: École Normale and École Polytechnique had many students of popular origin, and still kept the role they played when they were founded, two centuries ago, for social promotion as well as higher education.  The selective role of mathematics in exams and competitions may counterbalance as well as reinforce the social selection.

Let me insist on the counterbalance function.  In France at least, failure at school is directly related with a poor written and oral expression, a poor use of the French language and a poor intellectual environment; moreover, unemployment and lack of perspective cause a general lack of motivation.  Large popular competitions, like the Australian Mathematics Competition or Kangourou, can reveal hidden aptitudes and talents, as I already said, and they stimulate a large number of children and young people.  When a mathematical activity looks like a game and when the children like the game, the parents cannot consider mathematics as tyrannous and inhuman anymore.

It is important to have various types of mathematics competitions, with different rules and contents.  In that way they can be of interest to various kinds of people.  Some of them can raise interest among adult amateurs; it may even be a way to connect amateurs with living mathematics, if living mathematics is present in the subjects of the competition (as in Leeds).  Others can engender excitement or enthusiasm in the general public: I mentioned the success of the Budapest TV competition with the TV audience.  Some others would contribute to the self-respect of people in poor countries, as is the case for the International Olympiads.

Therefore mathematics competitions play a positive role in the popularization of mathematics, all the more efficient because young people are involved.  Young people have a special impact - it may be one reason for the success of the Budapest TV show. Without using the competitors as TV stars, is it not possible to make them more visible?

Just a word on the effect on competitors themselves.  When the competition appears as a pleasant game, it is excellent.  When in addition it is successful, it may arouse or confirm vocations.  Many laureates of IMO are important mathematicians.  The Hungarian competitions revealed a number of first-class mathematicians.  The Australian Mathematics Competition of 1983 had an unpredictable outcome: the best candidate, who obtained full mark, obtained only poor results at school, because he spent his life on a boat on the Pacific Ocean, with his father and the books he liked; he is now a bright computer scientist.

Of course, not every new subject in mathematics can be a subject for math competitions.  But some are, and I already gave examples à propos the Leeds competition.  It is a Hungarian tradition to convert research problems for professionals into research problems for students, and the most famous mathematicians, like Paul Erdös, were involved in this activity; traces can be found in the "Matematikai Lapok".  A problem of the IMO that I remember used to be an open question: given a finite set of points in the plane, such that every line containing two of these points contains another, prove that they are carried by one line.  Though it is a beautiful problem, I am not sure that it was a good question for young people; the answer is elementary, but much easier to find if you are a professional mathematician.  I also remember a long problem given in mechanics at the "agrégation de mathématiques" a long time ago: it was the complete theory of the bicycle.  A long problem that I gave to the entrance examination at the École Normale was the newly published proof by Thøger Bang of the Kolmogorov inequalities on the supremum norms of the successive derivatives of a function on the line.  The problems given at the Schweitzer competition that I mentioned at the beginning are proposed and selected by highly qualified professional mathematicians.  It is most useful to involve active research mathematicians in the proposal and selection of problems.  On the other hand, it seems to me very necessary to involve them also in the marking job, so that they can measure the impact of their proposals or decisions.

At a higher level, say, national mathematical olympiads or IMO, it is desirable to associate research mathematicians and experienced teachers for discussing the subjects.  The juries have a very important function if they try to insure the necessary renewal of the matter of the competition through what can be grasped from new trends in mathematics.  The juries may be focal points in the popularization of mathematics.

The aim of this paper was to give matter for future discussions.  Mathematics competitions are developing all around the world and that is excellent.  They can be more and more in contact with living mathematics.  Each of them needs strict rules and perfect observance of the rules.  The rules are diverse and more and more formulas should be experienced.  Competitions in mathematics are flexible, more than in sports, much more than in economic matters.  They may contribute to make mathematics more human and more popular.  They already constitute an incentive for the teaching and learning of mathematics.  Is it not time to emphasize their importance among our colleagues as well as in the general public?


Jean-Pierre Kahane
Mathématique, Université de Paris-Sud
F-91405 Orsay  Cédex  FRANCE
Jean-Pierre.Kahane@math.u-psud.fr