A property of a tetrahedron with a right trihedral angle is considered. Several approaches to its proof are presented. For this purpose, auxiliary statements are discussed, which also have independent significance.

Анализиран е приносът на разширената подготовка по математика в прогимназиален етап върху резултатите от Националното външно оценяване (НВО) след VII клас. Получените резултати не показват статистически значима положителна връзка между участието в разширена подготовка по математика и резултатите на учениците при Националното външно оценяване.

Статията разглежда представянето на седмокласниците на Националното външно оценяване по математика през 2025 г., като цели да очертае някои основни изводи, да идентифицира предизвикателства и да предложи насоки за подобряване на учебния процес.

This article examines the application of metacognitive strategies in the instruction of rational inequalities for \(9^{\text {th }}\)-grade students. The study presents a pedagogical model that integrates artificial intelligence (AI) tools – specifically ChatGPT and Desmos – to support students’ self-reflection and selfregulation. Conducted as a qualitative classroom-based observation over 12 lessons, the study explores how these digital tools may act as cognitive supports within mathematical problem-solving. The focus is on describing the dynamics of student engagement and reflective thinking in an authentic classroom environment rather than on providing controlled empirical measurement of the outcomes.

The article examines the application of business processes in the design and implementation of corporate strategy for employee’s professional development. A training approach using processes in an organization is presented, which allows for the definition and unambiguous formalization of a learning goal and the learning path for its achievement. A model of a corporate hierarchy is used, represented by a tree structure of roles, implemented in cloud space. Each role is attached to the business processes in which it participates. In the proposed approach, the training of an employee is initiated by a manager with leadership duties, who defines the structure of the business process in accordance with corporate strategy. A simulation of corporate training was conducted with students of Plovdiv University. The students are divided into teams, each student simulating the activities of an employee with a specific role in the organization.

Virtual players, also known as non-playable characters (NPCs), are utilized in educational video games to create an engaging gameplay experience that leads to improved learning outcomes and enhanced skills and competencies. To foster motivation and engagement among real game players, such NPCs often exhibit dynamically adapted behavior tailored to the player's specific attitudes, outcomes, and emotions. This literature review examines existing approaches and techniques for user-centered adaptation of NPC behavior in educational video games. To create more effective, plausible, and motivating NPCs, the authors seek to identify the dimensions of NPC behavior that convey emotional state, as well as the player characteristics modeled to support user-centric adaptation of NPC behavior. Next, the paper outlines the techniques implied for the implementation and validation of such adaptation, followed by modern trends and research challenges in NPC behavior adaptation, in the epoch of a massive invasion of AI. The findings promise to help specialists design, implement, and validate more plausible and intriguing NPCs, with enhanced abilities to motivate and engage real players of educational video games.

ChatGPT is a part of the artificial intelligence developed by OpenAI, which was officially launched in 2022. It is a tool that enables interactive conversations and provides explanations, guidance and information across various fields related to everyday life, education and professional work. On the other hand, differential equations are deeply connected to nature and the laws of physics, playing an essential role in the mathematical modelling of real-world phenomena. Therefore, in this paper we analyse ChatGPT’s capability to solve first-order ordinary differential equations, with a focus on obtaining accurate analytical solutions. Our analysis includes the following types of equations: equations with separable variables, linear differential equations, Bernoulli differential equations, and other nonlinear differential equations. Our goal is to assess the accuracy and applicability of ChatGPT in solving this type of mathematical problem through the analysis of various examples.