International Mathematical Olympiad Research Articles

The Book Review Column

Leslie Valiant, the father of computational learning theory, has written a new book, The Importance of Being Educable: A New Theory of Human Uniqueness (Princeton University Press, 2024), which proposes "educability" as a potentially more scientific theory of human cognition. Leslie Valiant, the father of computational learning theory, has written a new book, The Importance of Being Educable: A New Theory of Human Uniqueness (Princeton University Press, 2024), which proposes "educability" as a potentially more scientific theory of human cognition. Theorems of the 21st Century, Volume 1 by Bogdan Grechuk (Springer Nature Switzerland, 2019) deposits the first installment of important yet approachable results gleaned from the Annals of Mathematics in the first decade of the twenty-first century, from the point of view of an International Mathematical Olympiad gold and silver medalist.

Learning models and achievements of the most gifted students

The emergency of switching to an online learning model due to the COVID pandemic impacted the educational process in of schools world wide. We conducted an anonymous survey at the elite high schools in Serbia whose students won 45 medals at international Olympiads in mathematics, physics, and computer sciences in the last 4-year period (2019–2022). The impact of switching from in-person to online learning on the achievement of the gifted students was analyzed. The results of the research revealed a downward trend in the final grades and achievements in international Olympiads, which persists even after returning to the face-to-face educational model. This trend is confirmed by calculation of the Pearson correlation coefficient (−0.62) between school years and achievements in international mathematical Olympiads, international physics Olympiads, and international Olympiads in informatics. Gifted students’ satisfaction with online learning during the COVID pandemic was also analyzed using content analysis of open-ended questions. The surveyed gifted students showed a high level of self-criticism and maturity by expressing their opinion about online learning. In the group of gifted students, there is a higher level of agreement about harmful consequences of online learning on the level of their education and work habits than in other research conducted on similar topics. Moreover, the level of agreement is in a positive correlation with students’ achievements and final grades.

Career paths of the International Mathematics Olympiad (IMO) medalists

Each year, six exceptionally talented high school students from each country participate in the International Mathematics Olympiad (IMO). We thoroughly searched the internet to find the career paths of 2785 IMO medalists who participated in the competition from 1986 to 2005. We have found that the career paths of IMO medalists depend on their home country, despite their similar levels of mathematical talent. For example, the extent to which medalists studied abroad varies significantly among countries in the same geographical region and with comparable income levels. We also found that there is a high concentration in medalists’ foreign education. For example, MIT educated more foreign medalists than any country except for the United States both at the undergraduate and Ph.D. levels. Roughly half of the medalists ended up working abroad. Education is found to be an important channel for the migrant medalists. Although majority of the medalists were educated in mathematics, more than three quarters of their non-academic jobs are either software or economics-finance related jobs. Most medalists who became academics are specialized in mathematics. While the rate of becoming a mathematician academic is not dependent on the income level of the medalists' home country, those from higher-income countries tend to have higher academic performance.

New Math: How an AI proved dozens of geometry theorems from the International Mathematical Olympiad.

New Math: How an AI proved dozens of geometry theorems from the International Mathematical Olympiad.

Solving olympiad geometry without human demonstrations

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1–4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

Report on the 63rd Annual International Mathematical Olympiad

Summary We present the problems and solutions to the 63rd Annual International Mathematical Olympiad.

Math for olympiad: a didactic proposal for high school from the perspective of the international mathematical olympiad

The concept of the International Mathematics Olympiad (IMO) in learning mathematics in the classroom is something that can inspire students by structuring didactic proposals. This study aims to present a didactic proposal from the perspective of the International Mathematics Olympiad , using digital technology, particularly the GeoGebra software to be included in the mathematics Olympiad discipline. For theoretical studies, the analysis of the mathematical elements of the structured images in the GeoGebra software and the identification of their properties is included, through the structuring of the Olympic teaching situation, which allows the subject to seek solutions to the posed mathematical problems. olympics and validating the teaching of geometry. This study describes the arrangements made to build Olympic math objects that will be applied by math teachers. The research methodology is based on bibliographic reviews from authors such as Alves, Santiago, Almouloud and Brousseau. It is proven that the GeoGebra software helps in the elaboration of mathematical examples and problem solving situations.

Analytics Framework for Comparing National Performance Achievements in International Mathematical Olympiads

This study measures and analyzes the performances of the participating nations at the International Mathematical Olympiad (IMO). A country’s success at the IMO is evaluated by the number of gold, silver, and bronze medals won. To measure and compare the performances of the countries, the data envelopment analysis approach is employed and all countries that have won at least one medal have been included in our dataset. As measures for the inputs, two macro-level determinants that can support the effort made by a country are considered: population size and gross domestic product (GDP) per capita. Our findings indicate a mean level of technical efficiency for the nations analyzed. The results suggest that, on average, the inefficient countries in the sample could achieve improved results by increasing the number of outputs produced while maintaining the same level of effort spent.

Why U.S. immigration matters for the global advancement of science

While the impact of U.S. immigration on innovation in the U.S. has been well-studied, less attention has been paid to the relationship between U.S. immigration and global innovation. This paper fills this gap using novel data on a set of highly talented teenagers— participants in the International Mathematical Olympiad (IMO)—and presents three results. First, migrants to the U.S. are up to six times more productive than migrants to other countries, even after accounting for talent during one's teenage years. Second, most of the productivity difference is associated with academics being more productive in the U.S. compared to those who stay home (with differences in entry rates into academia mattering less). Third, there is a large gap between aspirations to move to the U.S. and actual migration: about 2 in 3 of the world's most talented youth from developing countries would like to migrate to the U.S. for their undergraduate studies, but only 1 in 4 do, with financing constraints being a key barrier. A back of the envelope calculation suggests that removing constraints on immigration could increase the global scientific output of future cohorts by up to 50 %.

Outreach by the International Mathematical Olympiad to the mathematics education community

Outreach by the International Mathematical Olympiad to the mathematics education community

Mathematical competitions in Africa: their prevalence and relevance to students and teachers.

Mathematical competitions feature in most developed countries as a part of the secondary school experience, but to a lesser extent in developing countries. In this paper we investigate how widespread these competitions are in Africa, both historically as well as currently, focussing both on national Olympiads and on the participation of African countries in competitions beyond their borders, such as the continent-wide Pan African Mathematical Olympiad and the global International Mathematical Olympiad. Since we could not find a consolidated source of information on African competitions, we include many details and links, hoping that this paper will serve as a valuable starting point for future investigators. Next, we present some survey findings on the attitudes towards and experiences of mathematical competitions, both from the perspective of teachers in secondary schools as well as from secondary school students, and whether these experiences have served to improve or degrade their perceptions of mathematics in general. Among other results, we find that teachers view their students’ participation in mathematics competitions as being important, with the biggest barrier to participation being that there are not enough competitions at appropriate levels available, and that students enjoy mathematics competitions and are more likely to pursue a STEM career because of them. Finally, we present some suggestions on how this research may be taken further, and on how the state of mathematical competition culture in Africa may be improved.

Report on the 62nd Annual International Mathematical Olympiad

Summary We present the problems and solutions to the 62nd International Mathematical Olympiad.

The Chinese experience at the International Mathematical Olympiad

The Chinese experience at the International Mathematical Olympiad

ОБОБЩАВАНЕ НА ОЛИМПИЙСКИ ЗАДАЧИ ЧРЕЗ ДИНАМИЧЕН СОФТУЕР

Various meaningful generalizations could be obtained after gaining insight into some mathematical assertions from different points of view. Generalizations of two problems from International Mathematical Olympiad papers are shown when the details are combined with the opportunities of the program software „The Geometer’s Sketchpad “ (GSP).

The mathematical competence of Olympiad participants as an indicator of the quality of mathematical training level

Introduction. The relevance of the research on the formation and assessment of the level of development of competencies in the mathematical educational field is due to the fact that the subject competence of the participant of the Olympiad predetermines his victory in the competition. And the Mathematical Olympiad, as a form of education, has potential not only for the formation, development, but also for determining the level of mathematical competence of its participants. The research problem is to justify the didactic potential of the mathematical Olympiad as a tool for determining the level of mathematical competence of school students. Aim of the study: to theoretically substantiate, develop and specify the content, indicators of mathematical competence of mathematical Olympiad participants Methodology and research methods. The methodological basis of the study is determined by: the activity and competence-based approaches to teaching; a retrospective analysis of psychological and pedagogical studies affecting the formation and development of key and subject competencies of schoolchildren; an analysis of the content of mathematical Olympiads; an analysis of the results of Kyrgyz Republic schoolchildren in international mathematical Olympiads; the study and generalization of the experience of juries of Olympiads. Results. The subject Olympiad forms a competence-based educational environment in which the levels of formation of key and subject competencies of its participants are most fully displayed. The competence-based approach to training in the Olympiad environment is characterized by the formulation of objectives from the point of view of the activity approach to the formed competence. Subject competence is leading in determining the quality of the student’s Olympiad activity. The mathematical Olympiad is one of the effective forms of both the formation and development, and the determination of the level of mathematical competence of its participants. The introduction of presented system for preparing schoolchildren for mathematical Olympiads, using model’s formation of mathematical, informational competencies in the experimental group, led to an increase in students' knowledge of the theory and practice of solving Olympiad problems in mathematics by 12,95%; the qualitative indicator of knowledge of the school curriculum in mathematics increased by 15,25%. The index of absolute indicators in the experimental groups in the theory of Olympiad mathematics was 53,12%, in the methods of solving Olympiad problems – 55,38%. In control groups, these indicators were 41,23% and 42,36%, respectively. The qualitative indicator of exam results of schoolchildren of the Olympic reserve turned out to be 19% higher. The results of the questionnaire survey participants of the Olympiads confirm the expediency of using technology for the development of critical thinking, the project method of teaching, ICT in the process of preparing for the Olympiads, contributing to the emergence of motivation to study an extracurricular course in mathematics in 68% of students, 42% of students showed a desire to participate in the Olympiads. The results obtained confirm our conclusions that the development of the mathematical competence of the participants of the Olympiads is successfully realized only in a situation of continuous Olympiad activity.

Romania helps Uganda on its way to the International Mathematical Olympiad

Romania helps Uganda on its way to the International Mathematical Olympiad

Romania helps Uganda on its way to the International Mathematical Olympiad

Romania helps Uganda on its way to the International Mathematical Olympiad

Report on the 61st Annual International Mathematical Olympiad

Summary We present the problems and solutions to the 61st Annual International Mathematical Olympiad.

Creativity as a function of problem-solving expertise: posing new problems through investigations

This study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through-investigations (PPI) in a dynamic geometry environment (DGE). The first type of expertise—MO—involved being a candidate or a member of the Israeli International Mathematical Olympiad team. The second type—MM—was comprised of mathematics majors who excelled in university mathematics. We conducted individual interviews with eight MO participants who were asked to perform PPI in geometry, without previous experience in performing a task of this kind. Eleven MMs tackled the same PPI task during a mathematics test at the end of a 52-h course that integrated PPI. To characterize connections between creativity and expertise, we analyzed participants’ performance on the PPI tasks according to proof skills (i.e., auxiliary constructions, the complexity of posed tasks, and correctness of their proofs) and creativity components (i.e., fluency, flexibility and originality of the discovered properties). Our findings demonstrate significant differences between PPI by MO participants and by MM participants as reflected in the more creative performance and more successful proving processes demonstrated by MO participants. We argue that problem posing and problem solving are inseparable when MO experts are engaged in PPI.

A New Proof of the Ionescu–Weitzenböck Inequality

Let a≤b≤c be the sides of a triangle, and let T be its area. Prove: a2+b2+c2≥43T. In what case does equality hold?This problem appeared in the third International Mathematical Olympiad (IMO) in 196...