| The Sharkovskii theorem as a tool for increasing the level of combinatorial thinking | 1$-$7 |
Abstract
The Sharkovskii theorem is an important mathematical result of the last century, which significantly helped the development of a modern mathematical field -- chaos theory. The Sharkovskii theorem is related to an interval on the real axis and a single-valued function. However, it was later generalized in various ways. This article will focus on the multivalued generalization of the Sharkovskii theorem, which has also found application in the theory of differential equations. We will show that the considered multivalued generalization of the Sharkovskii theorem is also suitable in mathematics education, especially for increasing the level of combinatorial thinking. We present some examples for pupils of different levels.
Keywords: Sharkovskii theorem; combinatorial thinking; chaos theory; concept of function; graph reading; numeracy.
MSC Subject Classification: 97F60, 97K20
MathEduc Subject Classification: F64, K24
| Cubic equations and geometric constructions | 8$-$13 |
Abstract
Examples are given of parametric families of equations of the third degree, for which all roots are expressed by square radicals. The problem of constructing a quadrilateral inscribed in a given semicircle by ruler and compass alone is discussed. It is shown that the problem of constructing an isosceles triangle if its three bisectors are given is equivalent to the problem of trisecting an angle. A connection was established between the problem of trisection of an angle and the problem of constructing a regular polygon.
Keywords: Cubic equation; solution by square radicals; Newton quadrilaterals; angle trisection; regular polygon.
MSC Subject Classification: 97H30, 97G40
MathEduc Subject Classification: H34, G44
| Evaluating open ai tools o1 and o3-mini in solving high school problems from Serbian national mathematics competition | 14$-$29 |
Abstract
Artificial intelligence (AI) tools are already being widely used in mathematics education in various ways and for different purposes. One important direction of their application lies in using AI tools for solving complex, competition-level mathematical problems. In this study, we contribute to this field by analyzing and evaluating the solutions to problems that high school students solved at the National Mathematics Competition in the Republic of Serbia, which were generated by the OpenAI tools o1 and o3-mini. The results indicate that not all solutions produced by these two tools are accurate and fully correct. However, based on the evaluated solutions, it can be concluded that, in most cases, the performance of these two tools would qualify them for one of the top three prizes in competition with the students. Experienced students, mathematics competitors who qualified for the national competition -- could benefit from reviewing the solutions to problems they were unable to solve independently, provided by o1 and o3-mini, as long as they approach these solutions critically and carefully analyze each aspect of the reasoning process.
Keywords: National mathematics competition; high-school mathematics problems; o1 tool, o3-mini tool, performance of AI tools.
MSC Subject Classification: 97P80
MathEduc Subject Classification: P84
| Application of different methods for solving linear difference equations with variable coefficients | 30$-$41 |
Abstract
Several methods for solving linear difference equations are considered: the operator method, the method of invariants, the method of generating functions, the $Z$-transform method, and the factorial series method. The application of these methods is demonstrated by solving higher-order linear difference equations with variable coefficients, in particular to solving an interesting problem from the Putnam student competition in 1990.
Keywords: Linear difference equation; operator method; invariant method; method of generating functions; $Z$-transform method; method of factorial series.
MSC Subject Classification: 97I30
MathEduc Subject Classification: I35, I75
| Balanced incomplete block designs for teaching combinatorics: Construction and applications | 42$-$73 |
Abstract
This article presents balanced incomplete block designs as an effective pedagogical tool for teaching combinatorics. Through their construction, analysis, and practical application, the article proposes an interdisciplinary approach that enables abstract content to be addressed in a contextualized and meaningful way. Specifically, it explores algebraic and matrix structures that model situations with structural regularity, fostering the development of logical and abstract thinking in the classroom. Additionally, key statistical concepts, such as experimental design and analysis, are introduced to provide an applied perspective on decision-making and data interpretation. These tools are presented through real-world examples that help students develop reasoning, modeling, and critical analysis skills, while also reinforcing the understanding of mathematics as a powerful tool for solving relevant and structured combinatorial problems.
Keywords: BIBD; balanced incomplete block design; combinatorial design; teaching combinatorics; Steiner triple system; resolvable design; Kramer-Mesner; Hadamard matrix; experimental design.
MSC Subject Classification: 97K20
MathEduc Subject Classification: K25