In this article, we propose a method for solving extremal problems in algebra and stereometry. The method is applied to problems that can be addressed using basic elementary inequalities and trigonometric equalities in triangles. These problems are reduced to finding the minimum or maximum value of a function. The aim of the method is to construct a systematic algorithm for solving extremal problems.
The article is devoted to the Twenty-second International Linguistics Olympiad held in 2025. It presents the results of the Bulgarian participants, a chart of the awards from past years, one of the problems from the qualification tests and its solution.
This article presents the results of a survey conducted among 50 computer science teachers in Bulgarian secondary schools, aiming to identify the main challenges in computer science education and to propose practical guidelines for addressing them. Through quantitative and qualitative analysis of the collected data, key problem areas were identified: outdated curriculum, insufficient technical resources, the need for professional development, and difficulties in motivating students. Seven recommendations for improvement are proposed, including curriculum reform, investment in technology, teacher training, the implementation of innovative teaching methods, and collaboration with the IT industry.
Dynamic Geometry Systems have advanced considerably in recent years, yet the lack of open-source solutions that can automatically verify answers and provide feedback on geometric construction exercises still limits their wider use in education. This challenge arises from the inherent difficulty of assessing complex geometric constructions with computational methods. GeoTry was designed to address this need as an interactive digital learning resource developed with GeoGebra. It supports both teachers and learners by automatically verifying geometric construction exercises and offering structured guidance based on Polya's problem-solving approach. We investigate how automatic verification can be meaningfully embedded in geometric construction exercises, focusing on the underlying design principles and technical considerations.
This work is dedicated to problems involving plane figures defined by points related to conic sections. The presented problems involve finding elements or areas of convex polygons whose vertices lie on ellipses, hyperbolas, and parabolas. Extremal area problems are also considered.
The study explores the possibilities of using the visual programming environment Kodu as an alternative to established block programming environments, offering propaedeutics to the youngest students. Within the context of Bulgarian curricula, the potential of using visual programming with Kodu as an effective tool for teaching Computer Modeling in the 3rd and 4th grade is confirmed. An analysis of the Semantic Web Rule Language (SWRL) is conducted, and an illustrative example of an ontology in Kodu is represented. The study shows that the visual programming rules in Kodu can be compared to SWRL rules. This conclusion provides valuable analytical opportunities to extend teaching practices into the field of artificial intelligence. In this way, the Kodu environment ensures a certain advantage because it can be used not only for the first steps in programming, but it can also serve as a propaedeutic for future training in working with artificial intelligence.
This article presents some possibilities for inserting an informal style of learning into a classical mathematics lesson. The concepts used are clarified - formal, nonformal and informal learning, and the characteristic communication style in informal learning. Some studies are presented, aimed at the benefits and possibilities of non-formal and informal learning for increasing mathematics achievement in formal learning. Various examples are given, in an informal style, which can be implemented in a classical mathematics lesson. In the article, the focus is on the application of elements (dialogues, texts, images and videos), atypical for a classical mathematics lesson and distinguished by an informal style of communication and presentation of information. The main idea of this approach is for the teacher to enter the surrounding environment in a non-formal way, as a bearer of a non-standard view of mathematics. The goal is for the student to have internal motivation to have fun with humor with a mathematical subtext. Specific examples are given and methodological guidelines for their application in formal learning are described. The aim of the study is to analyze the didactic potential of elements of the informal style in mathematics education. The study is guided by the hypothesis that the purposeful use of informal style (humor, visual memes and dialogic texts) is accepted by teachers and similar examples should be given in the preparation of future teachers. The article is based on the opinion of 230 mathematics teachers and observations on the work of future teachers. The study shows high acceptance of humorous and informal elements in the classroom.
Parsons puzzles are simplified code-construction tasks in which learners construct programs by selecting and rearranging predefined program segments. As a wellestablished pedagogical tool in programming education, they stimulate computational thinking and a deeper understanding of code by focusing on the structure and logic of the code rather than on syntax. Although many implementations of Parsons puzzles exist today, most of them offer only standard functionality and are locked to a single language. To overcome these limitations and expand the puzzle variability, this paper presents an improved architecture for Parsons puzzles, introducing new features designed to improve applicability across courses and languages. Several key additions have been integrated into the presented Parsons puzzles application: (1) Flexible support for languages with dissimilar syntax, (2) Multi-line puzzle blocks, (3) An improved version for handling nested constructions, (4) The introduction of the concept of mini-blocks for constructing more complex puzzles in realistic scenarios, (5) Support for “fill-in-the-blank” puzzles.
The following study addresses the issue of calculating with precision the volume of a truncated right circular cone while provided with limited information on the dimensions of the object itself (being given the relation between the radii, the vertical heights, or the slanted heights). The results include the proof of two theorem generalizations for the calculation of said volume with either of the three given elements via the principles of geometric similarity. It is shown that, due to the similarity between the full cone and the smaller removed cone, the volume of the truncated cone can be expressed using the difference of cubes of the corresponding linear dimensions. Revisiting the classical volume formula through the principles of geometric similarity, this work provides six alternative expressions that have both theoretical value and direct applications, especially in the field of education.
This paper proposes a fast, adaptive method for multilevel image segmentation based on Kapur's entropy. Implemented on CPU and GPU, the method achieves up to 40 & times; acceleration using CUDA and memory optimization. Segmentation quality is preserved while enabling real-time processing of large image batches. The experiments were carried out using the university's high-performance computing (HPC) mini-cluster, demonstrating its role as an educational and research platform. The approach is scalable and suited for scientific and educational use. It builds upon earlier work on multidimensional histogram analysis, applying entropy-driven modeling to image segmentation.
Teachers commonly consider the number of teaching hours in a particular subject to be insufficient for achieving sustainable results in the acquisition of knowledge and skills. It is assumed that increasing the number of hours should lead to an improvement in the quality of education. Thus, the question of the optimal amount of teaching time remains open. This study focuses on this methodological problem by analyzing the opinions of teachers from schools in the capital who teach two types of students simultaneously-those included in extended mathematics training and those who are taught only within the compulsory program. A comparative approach is used to examine the added value of extended training as perceived by teachers, as well as its impact on the quality of teaching and learning outcomes. The results contribute to a deeper understanding of the relationship between the quantitative parameters of teaching time and the qualitative dimensions of the educational process. In the European Schools attached to the institutions of the European Union, additional hours are organized in different ways. This study also examines these differences and seeks to answer the question of how to use additional teaching time most effectively.
Universities show a rapidly growing interest in using Artificial Intelligence tools. This paper presents LexaQuery, a new chatbot system for university information services that efficiently handles university information inquiries by combining computational linguistics with SQL database querying. Instead of relying on large, resource-intensive large language models (LLMs), LexaQuery uses rule-based natural language processing to translate student questions into structured SQL. The system has a three-tier architecture comprising language processing for query translation, knowledge extraction (from databases and web scraping), and a user interaction layer. Performance evaluation at the University of Plovdiv Paisii Hilendarski demonstrates that this hybrid approach provides significantly faster response times than neural network-based alternatives while maintaining satisfactory accuracy for domain-specific tasks. The paper discusses the system's advantages in terms of integration with existing university information systems, performance efficiency, explainability, and the ability to operate without extensive computational resources, as well as its linguistic flexibility and limitations in domain adaptation. This research contributes to developing practical, efficient chatbot systems for educational institutions with constrained technical infrastructure.
In this paper, the influence of the mathematical modelling approach in computer-supported collaborative learning (CSCL final) on students' knowledge of calculus contents, particularly the application of definite integral, is examined. The research was conducted with the final-grade students at the grammar school. Two groups of the students, the experimental and the control one, were observed. Both the groups learned in CSCL environment, while in the experimental group, mathematical modelling was applied. The work of the experimental group students during the whole mathematical modelling process was monitored and analyzed. Some examples of the students' solutions are described. After the learning process of the integral contents, the students' learning achievements were tested and compared. The results indicate that the use of mathematical modelling increases the students' interest in solving the definite integral application problems. It is proved that the students who applied modelling process had better results than the students who did not.
The present study is dedicated to describing a sample approach for integrating digital tools powered by artificial intelligence (AI) and project-based method in mathematics education. Numerous studies have demonstrated the significant potential of AI to revolutionize mathematics teaching and learning, while also highlighting the importance of addressing the associated challenges to ensure its effective and responsible implementation in the educational process. The project-based method enables students to collaborate, engage in discussion, think critically, and express their creativity. This paper presents a possible application of AI-based digital tools and the project-based method in teaching the topic of Trigonometry from the 11th grade mathematics curriculum. The results from a diagnostic study conducted with students from the Foreign Language High School “Antoine de Saint-Exupery” in Plovdiv are also analyzed.
This study aims to build and estimate a SARIMA model to predict the prices of the shares of a company listed on NYSE. With the help of Python, the daily closing prices of the stock for the period from 3.01.2023 to 17.03.2025 were taken. Also, SARIMA models were evaluated and code was compiled to select the most appropriate parameters. Finally, the daily forecasts for one month were calculated. The study found that SARIMA model can be considered reliable for forecasting stock prices and can complement any other analysis method. Future research could also include enhancing the study with external factors and improving the forecasts could be also done by testing of SARIMAX models.
The paper proposes a comprehensive student academic performance prediction approach by integrating machine learning with metaheuristic optimization. Initial models (Logistic Regression, Decision Tree, Random Forest, MLP) were refined using boosting techniques (Gradient Boosting, XGBoost, LightGBM), with XGBoost achieving 95.59% accuracy. Eight modern optimization algorithms were applied for feature selection to enhance model efficiency and interpretability, with the Grey Wolf Optimizer and the Heap-Based Optimizer outperforming others in key metrics. Support Vector Machine algorithms applied after feature selection strengthened the predictive capability of the selected feature subsets. The research outcomes demonstrate that uniting boosting approaches with feature selection algorithms enables the creation of reliable and scalable predictive models that detect student success and failure earlier.
Computer models of tasks related to percentage are presented. The files are created with the dynamic software GeoGebra and are provided in the Virtual Mathematics Laboratory, developed by the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. The goal is to create conditions for the development of the visual estimation of a percentage, which also supports the understanding of the concept. The computer models contain rectangles and circles. Help and feedback are provided. Options for obtaining a new example and feedback are described. Emphasis is placed on the analogy of the tasks in the four presented topics, each of which contains four tasks. An assessment of the resources is presented, obtained from an anonymous survey with teachers from different subject areas and teaching at different educational levels. The assessment is based on the criteria of easy technical orientation, design, usefulness, entertainment, motivation to solve. The simultaneous development of digital and mathematical competence when working with these resources is commented on, as well as the possibility of their use in STEM centers. Ideas for expanding the resources for checking and developing the percentage calculator in several directions are described.
The widespread use of information and communication technologies (ICT) offers new opportunities in many topics of mathematics education. As science and technology are constantly evolving, information technology is becoming increasingly intertwined with education. Modeling, simulation and visualization are already proven methods in teaching subjects such as physics, chemistry or engineering. These methods can help students see connections more clearly and develop their creative thinking. This paper aims to further explore this direction in the field of mathematics education, with focus on differential equations. We chose spreadsheets as our tool to calculate and visualize the processes described by differential equations. We demonstrate a wide range of applications of differential equations through real-life examples, such as in modeling physical, biological, and economic processes. This method provides students a better understanding of the practical usefulness and applicability of these equations. The study thus shows how integrating ICT into mathematics education can help students gain a deeper understanding of the underlying mathematical concepts and improve their mathematical thinking and problem-solving skills. ICT tools enable teachers to use interactive and engaging teaching methods, resulting in an exciting and practical education for students. This paper outlines the potential of ICT in mathematics education, with a focus on the use of spreadsheets for modelling and visualization. It highlights the benefits of integrating technology into the classroom to enhance student learning and engagement.
A variant for interactive presentation of content with the inclusion of augmented reality using dynamic software GeoGebra is described. A concretization is made with the Pythagorean Tree fractal. The goal is to support the understanding and memorization of the relevant content by providing conditions for research and experience, as well as for gaining experience in using augmented reality as a technology and motivating teachers to implement it in school. Models are described with which the growth of the fractal is observed through animation, as well as changes depending on several parameters. Basic ideas for creating the relevant compositions with dynamic software GeoGebra are described. The goal is to facilitate the use of this resource when working in STEAM centers. An expert assessment of the proposed resources and methodology is presented, as well as the results of a pilot study. Possibilities for continuing research on the topic are discussed both in terms of interpretations related to the fractal (for example, using different angles in the steps, different coloring options, using figures other than regular polygons, etc.), and in terms of software products for creation and technological tools for impact (for example, virtual reality, mapping, etc.). The possibility of using such resources in science museums is also emphasized, which will improve the implementation of their educational function.
Under stereographic projection, the projection point is colli-near with the Lemoine points of the projection and the projected triangles or with the centers of their Apollonian circles (Natchev 2025). In the current paper, we generalize the discovered property of stereographic projection by proving that it applies to every point on the plane expressed in the barycentric form (delta a(2) : epsilon b(2) : rho c(2)) where delta, epsilon, rho is an element of R. For the particular case, we investigate the points on the symmedians of the triangle and the tangents to the circumcircle at the vertices, where we derive collinearity of two more notable points of a triangle, namely the feet of the symmedians and the vertices of the tangential triangle. By setting the planimetric equivalent of the newly found facts, we connect the configurations they give rise to with Olympiad geometry.
