Математика и Информатика

https://doi.org/10.53656/math2022-4-4-per

2022/4, стр. 365 - 378

PERSONAL DATA PROCESSING IN A DIGITAL EDUCATIONAL ENVIRONMENT

Evgeniya Nikolova
OrcID: 0000-0001-8313-1572
E-mail: evgeniyanikolova@gmail.com
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Laboratory of Digitization – Burgas
Burgas Free University
5 Demokratsiya Blvd.
8000 Burgas Bulgaria
Mariya Monova-Zheleva
OrcID: 0000-0001-8910-2502
E-mail: mariya@zhelev.com
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Laboratory of Digitization – Burgas
Burgas Free University
5 Demokratsiya Blvd.
8000 Burgas Bulgaria
Yanislav Zhelev
OrcID: 0000-0003-2783-5617
E-mail: yanislav@zhelev.com
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Laboratory of Digitization – Burgas
Burgas Free University
5 Demokratsiya Blvd.
8000 Burgas Bulgaria

Резюме: New technologies provide innovative spaces for cooperation and communication between employers and employees, citizens and structures, educators, and learners. Data protection issues have always been key to education providers, but the proliferation of online learning forms and formats poses new and unique challenges in this regard. When introducing a new technology that involves the collection of sensitive data, the General Data Protection Regulation (GDPR) of the European Parliament and the Council of the European Union requires the identification and mitigation of all risks that could lead to the misuse of personal data. The article discusses some critical points regarding the application of GDPR in online learning. The goal of this article is to investigate the vulnerabilities to personal data security during online learning and to identify methods that schools and universities may apply to ensure that personal data are kept private while students utilize online platforms to learn. For the purposes of the research, the published privacy, and data protection policies of all Bulgarian universities as well as papers on how universities could adapt to the new EU General Data Protection Regulation were revised and analysed. Best practices of some foreign universities in this regard were studied as well.

Ключови думи: Personal data, processing personal data; GDPR; online learning; online fraud detection

1. Introduction

Information technology and the Internet made collection of personal data much easier, which can lead to harassment, identity theft, or aiding and abetting the planning of criminal acts. In online learning, data is produced through the interaction between teachers, students, and platforms. By this reason the issues of data protection and confidentiality and literacy of the participants in the personal data protection are raised. The confidentiality and protection of personal data are closely linked. Confidentiality is related to authorized access to data and information with focus on implementing policies ensuring that users’ personal information is collected, shared, and used in appropriate ways. Laws, regulations, and policy documents have been formulated at organizational, national, and international levels to protect data as an aspect of confidentiality. As distance learning is an option in the schools and universities, staff, teachers/lecturers, and students need to be informed about the importance of data protection as well as to be provided with practices that they can apply to keep them protected.

The aim of this paper is to draw attention to the risk assessment of personal data security in the implementation of the GDPR in Bulgarian schools and universities. The first part introduces the GDPR related concepts, including the idea of personal data, its life cycle, several data models for managing personal data based on its unique qualities, and data protection principles. The second part focuses on the study of threats to personal data security and outlines the measures used by schools/higher education institutions to maintain the confidentiality of personal data in online learning.

2. Personal data and their lifecycle

Any information relating to an identified or identifiable living natural person, as well as individual data that, when combined, may lead to the identification of a specific person, is considered “personal data”. To clarify the methods of personal data protection, it is necessary to consider the issue of their classification. Personal data encompass many types of data that can measure and describe various aspects of an individual’s identity, characteristics, and behavior. There are different classifications of personal data. Summarizing the methods adopted for the classification of personal data, (R.H. Huang at all 2020) presents a classification of 12 categories, which are presented in Table 1.

Table 1. Categories of personal data

Basic informationName,age,placeofbirth,dateofbirth, gender,genderidentity,preferences,proclivities, personal photos, race, colour, national or ethnic originIdenticationGovernment-issuedidentication,driver’slicense,passport,healthIDs,Social Insurance Numbers, Social Security Numbers, PIN numbersBiometricsGenes, ngerprints, voice prints, palm prints, auricles, irises, facial featuresAuthenticatingPasswords,PIN,systemaccount,IPaddress,emailaddress,securityanswer, personal digital certicatesMedical and HealthPhysicalandmentalhealth, drugtest results,disabilities, familyor individualhealthhistory,healthrecords,bloodtype,DNAcode,medicalhistory,medical device logs, prescriptions, and health insurance coverageProfessionalJob titles,salary, workhistory,schoolattended, educationhistory,employeeles,employmenthistory,evaluations,references,interviews,employerdata, certications, disciplinary actionsFinancialCars,houses,apartments,personalpossessions,purchases,sales,credit,income,loanrecords,transactions,taxes,purchasesandspendinghabits,creditrecords,creditscores,creditstanding,creditcapacity,physicalassets,and virtual goods
CommunicationTelephone recordings,voicemail, emails,SMS,phone calls,IMand social,network post, physical address, telephone numberContactContactlists,friends,connections,acquaintances,associations,groupmembership, email addressBrowsing historyMediaproduced,consumed,andshared:in-text,audio,photo,video,andotherforms ofmedia;Real-worldandonlinecontext,activity, interests,andbehaviour: records of location, time, clicks, searches, browser histories andcalendardata,purchasesactivity,onlineshopping,socialnetworkproleinformation and the likeDeviceHardware serial number, software list, IP address, Mac address, browserngerprintLocationCountry, GPS coordinates, room number, longitude, and latitude

3. Regulations under international frameworks for personal data protection

3.1. Models for personal data management – taxonomy

Although personal data regulations vary considerably from country to country, it is still possible to identify three main approaches: one model based on open transfers and data processing, a second model based on conditional transfers and processing, and a third model based on limited transfers and processing (M. F. Ferracane, E. van der Marel 2021). These three data models have become a benchmark for many other countries in setting their rules for both cross-border transfers and internal processing of personal data. The main features of the existing data models are systematized in the table below (fig. 1).

Figure 1. Main features of data models

3.2. The personal data processing principles under the GDPR

Personal data is processed in a variety of ways, including collection, recording, categorization, structuring, and storage. The General Data Protection Regulation (GDPR) of the European Parliament and the Council of the European Union, which was adopted on April 27, 2016, and took effect on May 25, 2018 (European Union 2016), governs the processing, storage, and use of personal data by third parties, such as individuals, businesses, and organizations. The GDPR supersedes the previous Data Protection Act (DPA) and strives to bring about a cultural shift in how we handle people’s data in the digital era. It champions data subjects’ rights by utilizing a variety of mechanisms, including recipient adequacy judgements, model agreements, and binding business obligations.

Seven main principles for processing personal data are specified in GDPR Article 5, which are presented in figure 2.

Figure 2. Principles for personal data processing

Everyone, as a user of the Internet, is a data subject. The GDPR acknowledges a slew of new privacy rights for data subjects, with the goal of giving people more control over the information they share with businesses. A person’s rights in connection to the collection, processing, and storage of personal data are outlined in the GDPR. These are the following: the right to be informed; the right of access; the right to rectification; the right to erasure; the right to restrict processing; the right to data portability; the right to object; rights in relation to automated processing and profiling.

3.3. GDPR in schools and higher education

Any information concerning a student’s identification, academics, medical issues, or anything else that is specific to that individual student and is collected, kept, and shared by schools/ universities or technology providers on behalf of the schools/ universities is considered student personal information. Name, address, contact information, date of birth, identity documents, assessment results, student curricular record, ethnic background, language, exclusion information, and attendance information are all included. It also includes data created or generated by students or teachers/professors using technology – email accounts, online bulletin boards, work completed using an educational application or app, and anything else created or generated by or about an individual student in a learning environment.

The process of the GDPR implementation in the schools could be considered in the following six steps:

1. Creating a framework for accountability and governance. In this phase, a unit/body is set up to monitor the implementation and compliance of the GDPR by the various groups.

2. This unit/body shall specify all the criteria and requirements for compliance with the GDPR that the various departments and administrative units and services must observe and comply with in carrying out their day-to-day operations.

3. Description of the data flows, including where and how it is kept and processed, who sees it, how much is exchanged, and how it is transported, should be provided as well.

4. Evaluation of the risks such as, i.e., the probability of data being leaked, lost, erased, or stolen while passing through the school/university system.

5. A gap analysis should be done to see if the data is encrypted or stored in plain text and whether the established data flows are in line with the specified criteria, requirements and plans for ensuring GDPR compliance.

These steps should be carried out cyclically over a period and allow for the timely elimination of identified weaknesses, as well as continuous improvement and optimization of the process.

Under new GDPR rules, higher education institutions need to have organized records of what personal data exists, as well as documentation explaining why it has been held, how it was collated, who has access to it and when it will be removed or anonymised. Its need to follow several data privacy and data security requirements, such as (A. Šidlauskas, T. Limba 2019):

– Ensuring data security practices are in place.

– Implementing privacy restrictions and personal data usage policies.

– Developing a personal data consent collection process.

– Identifying a data protection officer.

– Implementing appropriate measures to protect personal data.

– Adhering to the GDPR breach notification processes.

A. Šidlauskas and T. Limba (A. Šidlauskas, T. Limba 2019) offer nine stages for implementing GDPR in higher education in 2019:

1. Get your GDPR project ready to go. An examination of the data.

2. Create a Personal Data Policy and other high-level papers; update your institution’s privacy policy.

3. Make a list of all the actions that need to be completed. Validation of data.

4. Define a strategy for managing data subject rights, consent, and consents to send unsolicited, direct marketing communications, and data access, integrity, and deletion processes.

5. Conduct a risk assessment for data privacy.

6. Confidentiality in the sharing of personal information.

7. Update your institution’s data processor agreements by amending third-party contracts.

8. Protect personal and sensitive data by implementing security measures.

9. Define how to deal with data breaches; security breaches should be reported to the appropriate supervisory authority.

Every educational institution in Bulgaria has developed a Confidentiality and data protection policy, which clearly and in detail states what personal data are collected and from whom; how the data are stored and what they are used for; who are the employees involved in data collection, storage, and management processes as well as the responsibilities associated with these activities; and procedures for documenting violations related to personal data protection and security.

4. Online learning and personal data

Data protection in online learning is a direct function of the typical steps for using multi-type online learning tools given in Table 2 (R.H. Huang et al. 2020).

Table 2. Typical steps for online learning

1) Preparingthe devices, network,andtools– Set up your device– Manage network connection on your device– Select and install learning tools– Browse the privacy policy
2)Preservingprivacywhensigning up/in on learning platforms– Whenregisteringonthe platform,useastrong passwordto create an account– Remember username / password3)Protectingprivacywhennavigatinglearning platforms– Enrolling in an online course– Utilizing personalized learning services– Using search services carefully– Recognizing location services– Backing up your data4)Stayingsafewhilelearningwithsocial networking– Using video conference tools with caution– Posting in the discussions and forums responsibly– Surng the Internet safely5)Clearingpersonaldataafternishinglearning online– Removing data traces in online learning– Deactivating your account

By registering in the e-learning platform the following personal data of the student will be processed: user account, last name and first name, email address at home school/university or email address of the guest account, registration file data (at what time you have access to which parts of the courses), user data (content, contribution, and activities of the user in the e-learning platform). The data in the user profiles (registration data) is stored until the user profile is deleted. Course participation data (user data) is stored until the course is deleted. The data from the registration files are deleted after the end of the process of use unless the legal provisions require longer storage. Students can delete user data and voluntary entries in their user profile at any time. Recipients of the students’ data are:

1. Only the staff responsible for the administration and management of the e-learning platform. They shall have access to all data stored in the system, including the log files, and may process this data solely to ensure the operation of the platform.

2. Professors shall only have access to the activities, contributions and data provided by their course participants.

Schools/universities shall provide appropriate technical and organizational measures to prohibit access to registration files or other data from the platform from which individual user profiles may be extracted. The following personal information will be processed by the school/university in online exams: name; student number; e-mail address; IP address of computer network to which computer is connected; image of student card (or identity card); screen recordings of what students see on their monitors; data about website visits during the exam; webcam and audio recording of students and the room in which they are sitting during the exam. There are available different systems that provide possibility for online proctoring but all of them provide several fundamental functionalities such as detecting and disabling computer functionalities as copy-paste as well as downloading, taking images of and recording both student and screen. Another important functionality is related to analyzing the gathered data to signal irregularities that may show fraud (Aarts et al. 2021).

A classification of methods used for online exams was proposed by O. L. Holden in 2021 (O. L. Holden 2021).

Online fraud detection. Exam monitoring is sometimes known as proctoring. Exams that are proctored are exams that you take while the proctoring software watches your computer’s desktop as well as video and audio from the webcam. The data collected by the control program are obtained for verification and evaluation. Video summation, web video recording, and live online proctoring are all common methods of online proctoring. Sensitive data collected when applying the method are name, video image (Basic information), irises, facial features (Biometrics), system account, email address (Authenticating). In this case, the teacher / professor should carefully review the privacy regulations and policies of the institution for access and storage. There are several approaches for implementing the online fraud detection as follows:

– Video Summation. Artificial intelligence is used in video summation software to detect fraudulent occurrences that may occur during the test. During the test, students are enrolled using their own webcam. The application will flag the video for future examination by a proctor if a fraud occurrence is discovered. These systems can produce keyframes (a collection of pictures derived from a video source) or video segments collected from a video source to depict a suspected fraud incident for human proctor identification in the future.

– Web Video Recording. In this case, the student is recorded on video throughout the exam for later viewing by the lecturer. Unlike video aggregation programs, web video programs do not have specific proctors who review all tagged instances and instead rely on review by the administrators and instructors themselves.

– Live Online Proctoring. The latest type of online proctoring uses the student’s webcam and microphone to allow a live proctor to observe students during an online exam. In case the school / university chooses to use online proctoring for online exams, the educational institution thus checks which student is taking the exam and can establish that the exam rules have been followed and that no fraud was committed during the exam. Many lecturers prefer to use this type of service as it is closest to a personal exam.

Table 3 provides information on compliance with the GDPR when using different approaches to detect online fraud during an exam.

Table 3. Online fraud detection approaches

MethodsGDPR implementationVideo SummationImages can also be considered personal data. Video recordsmay reveal individualaspects of thestudents whichare speciallyprotectedunderthelawsuchasrace,gender,religion,andhealthstatus.Iftheimageorfootagehasnotbeenexpresslytechnologicallyprocessedtocontributetotheidenticationofanindividual,itisnot constituted biometric data underArticle 9.Schools/universitiesastraininginstitutionsjustifytheprocessingof dataas “necessary”for theperformance oftheir contractswithstudentsunderArticle6(1b)oforas“necessary”forthelegitimateinterestsofstudentsinthetimelyassessmentandpreventionoffraud (under ofArticle 6 para 1f)).Personaldatashallberemovedafteritisconrmedthatastudenthasnotactedsuspiciously(Article5(1e)).Anysuchdecisionshouldbemadeinareasonabletime(forexample, latest30daysafter reviewing the examination).Educationalinstitutionshavetoobtaintherequisiteconsentofthecandidatesanditisrecommendablethistobeinsertedasastageoftheonlineproctoring.Moreover,analternativefortheexam should be proposed to the candidates as well.Web Video RecordingLive Online ProctoringTheonlineproctoring shouldfacilitate theexaminationprocessinline with the instructions of the educational institution.Acontractbetweentheeducationalinstitutionandtheonlineproctoring service outlines all actions and duties.After anexam has beenveried, thedata trailshould be asshortas feasible.Educationalinstitutionsmustbeabletodemonstratehowaparticular processing activity complies with the GDPR.

Knowledge-based authentication (KBA) method. This method requires students to ask multiple-choice questions based on their personal history to gain access to the exam. These questions are randomly generated from the initial profile setup questions or third-party information when the student starts the exam, and the answers are compared to verify his or her identity. Thе method cannot be used to monitor students’ behavior during the exam. Sensitive data collected when applying the method are video image (Basic information), system account, email address (Authenticating), education history, evaluations (Professional). To catch up with technological developments and the prevalence a data driven business models, a new General Data Protection Regulation (GDPR) will enter into force in May 2022. The compliance should be demonstrated through detailed documentation of all steps that would lead to a lawful data processing. A privacy impact assessment is required component and the results of this assessment should be taken into account from the very beginning of the method designing.

Biometrics. Biometric data that do not need physical contact with a scanner, such as height, weight, age, gender, eye color, and ethnicity, or biometric traits that do require direct physical contact with a scanner, such as a fingerprint, are another means of verification. To identify a student, this approach compares a previously registered biometric sample with freshly recorded biometric data. Biometric features should be consistent and unchanging, and the technique for gathering them should be inconspicuous and carried out by instruments that need little or no interaction. Because integrating two or more traits enhances recognition accuracy, multimodal biometric systems employ numerous biometric features and technologies at the same time to validate the user’s identity.

Assess the security risks of personal data protection

An important point of GDPR is a risk assessment. In 2016, the European Union Agency for Cybersecurity – ENISA (ENISA 2016) published a set of guidelines for organizations acting as data controllers or processors to help them assess security risks and take personal data protection measures accordingly. The proposed risk assessment process is implemented in four steps:

– Definition of the processing operation and its context.

– Understanding and evaluation of impact.

– Definition of possible threats and evaluation of their likelihood.

– Risk assessment by combining the probabilities of threats and impacts.

The first step of this process defines the data processing operation in the context of risk assessment with the following set of questions:

1. What is the personal data processing operation?

2. What are the types of personal data that are processed?

3. What is the purpose of the processing?

4. What are the means used to process personal data?

5. Where is the processing of personal data carried out?

6. What are the categories of data subjects?

7. Who are the recipients of the data?

In the second step, the data controller must assess the impact of the loss of confidentiality, integrity, and availability. Four levels of impact are considered:

– Low – Individuals may encounter several minor inconveniences that they will overcome without difficulty;

– Medium – Individuals may face significant inconveniences that they will be able to overcome despite some difficulties;

– High – Individuals may face significant consequences that they must be able to overcome, albeit with serious difficulties;

– Very high – Persons who may face significant or even irreversible consequences that they cannot overcome.

In the third step, the data controller identifies possible threats and assesses their likelihood, using a set of questions that address four main dimensions of this environment:

– Network and technical resources (hardware and software);

– Processes / procedures related to the data processing operation;

– Different countries and people involved in the processing operation;

– Business sector and scale of processing.

Developed questionnaires are presented in (ENISA 2017), which organizations can use in this step.

Through this approach, the level of probability of occurrence of a threat can be determined for each of the areas of assessment as:

– Low: the threat is unlikely to materialize;

– Medium: there is a reasonable chance that the threat will materialize;

– High: the threat is likely to materialize.

The final risk assessment is determined in the fourth step summarizing the result of steps two and three.

After assessing the level of risk, in step five the organization can proceed to select appropriate security measures to protect personal data. The ENISA guidelines address two categories of measures: organizational and technical and provide a list of proposed risk level measures.

Based on the 2016 guidelines, in 2017 ENISA prepares the reports that focuses mainly on the electronic processing of personal data by organizations, which is based on IT networks and systems, as well as new digital technologies (e.g., cloud computing, mobile devices, etc.). Each of they present an assessment of security risks in the processing of personal data by a schools and university that of fers a platform for e-learning and course management hosted internally on a web server. The summarized results of the four main steps of the evaluation process in these reports show that the overall risk for these two specific cases is considered medium. This methodology can be successfully used by schools and universities to self-assess risks and adopt security measures in accordance with the GDPR.

Considerations and recommendations to ensure that online training meets the requirements for privacy and personal data protection

Тo ensure that online learning meets the requirements for personal data protection and learners’ privacy rights the following recommendations could be made:

1. At the institutional level

1.1. Educational institutions need to identify the legal basis for the processing of personal data when using online learning platforms. The legal basis for the processing of personal data when conducting online lessons or exams may be the provision of the educational service itself.

1.2. When choosing or adapting an online learning software platform, it is necessary to ensure compliance with the country’s privacy laws and to ensure that no more personal data is collected than necessary. The educational institution may require students, teachers and staff to use only the institution’s platform as well as institutional email addresses.

1.3 It is recommended that the educational institution conduct a risk assessment annually, which will help it to evaluate and mitigate the various risks associated with the conducting of online sessions – live streaming and/or recording. The procedure proposed by ENISA can be used for the purpose.

1.4 Rules and regulations for working in the institutional online platform should be elaborated and published on the website of the institution. Teaching and administrative staff should be trained to act in compliance with these rules.

1.5 Rules for storage, access, control and preservation of the virtual educational activities’ records should be regulated.

2. At the level of teaching staff

2.1. At the beginning of each online course, learners should be informed that they are obliged to strictly follow the rules introduced by the institution for working in an online environment.

2.2. The explicit consent of all participants for recording virtual educational activity as a part of the online course is mandatory.

2.3. The written consent of the trainees to keep their cameras and microphones included during the examination procedure included in the online course must be obtained through a standard form prepared by the educational institution.

2.4. The results from the conducted online examinations should be communicated by the lecturer to the concrete student personally in a form of bilateral communication.

Conclusion

Successful implementation of the new GDPR rules in the educational institutions requires balance of system, process and privacy resources, as well as proper methodology performed by a team. The article presents the main steps of the process for the application of this regulation, the main points of its application in the various methods of online exams, as well as a four-step method for risk assessment.

Acknowledgement

In this paper are presented some results obtained in the framework of more comprehensive research conducted under the Erasmus+ project EDucational University GATeway to enhance innovative E-learning capabilities, resilience, and new best practices (EDU-GATE) № 2020-1-IT02-KA226-HE-095538.

REFERENCES

AARTS, E., FLEUREN, H., SITSKOORN, M., WILTHAGEN, T., 2021. The New Common (How the COVID-19 Pandemic is Transforming Society). Springer, Cham. https://doi.org/10.1007/978-3-030-65355-2.

Edu-Gate: ENISA, Guidelines for SMEs on the security of personal data processing, 2016, https://www.enisa.europa.eu/publications/guidelinesfor-smes-on-the-security-of-personal-data-processing.

ENISA, Handbook on Security of Personal Data Processing, 2017, https:// www.enisa.europa.eu/publications/handbook-on-security-of-personaldata-processing. European Union, Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (GDPR), O.J. (L 119) 32, European Union, Brussels, 2016, http://data.europa.eu/eli/reg/2016/679/oj.

FERRACANE, M. F. E van der Marel, Regulating Personal Data: Data Models and Digital Services Trade, World Bank Policy Research Working Paper No. 9596, World Development Report 2021, March 2021, https://openknowledge.worldbank.org/bitstream/handle/10986/35308/ Regulating-Personal-Data-Data-Models-and-Digital-Services-Trade. pdf.

HOLDEN O. L., M. E. NORRIS, V. A. Kuhlmeier, Academic Integrity in Online Assessment: A Research Review, 14 July 2021, https://www. frontiersin.org/articles/10.3389/feduc.2021.639814/full.

HUANG R.H., LIU, D.J., ZHU, L.X., CHEN, H.Y., YANG, J.F., TLILI, A., FANG, H.G., WANG, S.F., 2020. Personal Data and Privacy Protection in Online Learning: Guidance for Students, Teachers and Parents. Beijing: Smart Learning Institute of Beijing Normal University.

SWAMINATHAN N. What Is Columbia Doing With Your Data?, THE EYE | FEATURES, March 01, 2019, https://www.columbiaspectator. com/the-eye/2019/03/01/what-is-columbia-doing-with-your-data/.

SHAW M., M. HALKILAHTI, M. REISSMAN, S. RUIZ RUIZ, J. VOUTILAINEN. Futures Personal Data Can Build: Scenarios for 2030, In book: COOLEST STUDENT PAPERS AT FINLAND FUTURES RESEARCH CENTRE 2017 – 2018, Publisher: Finland Futures Research Centre University of Turku, 2018,

https://www.researchgate.net/publication/336923184_Futures_Personal_ Data_Can_Build_Scenarios_for_2030.

ŠIDLAUSKAS A., T. LIMBA. General data protection regulation implementation in higher education institutions, Proceedings of EDULEARN19 Conference 1st-3rd July 2019, Palma, Mallorca, Spain, 2040 – 2047.

2025 година
Книжка 6
ENHANCING STUDENT MOTIVATION AND ACHIEVEMENT THROUGH DIGITAL MIND MAPPING

Mikloš Kovač, Mirjana Brdar, Goran Radojev, Radivoje Stojković

OPTIMIZATION VS BOOSTING: COMPARISON OF STRATEGIES ON EDUCATIONAL DATASETS TO EXPLORE LOW-PERFORMING AT-RISK AND DROPOUT STUDENTS

Ranjit Paul, Asmaa Mohamed, Peren Jerfi Canatalay, Ashima Kukkar, Sadiq Hussain, Arun K. Baruah, Jiten Hazarika, Silvia Gaftandzhieva, Esraa A. Mahareek, Abeer S. Desuky, Rositsa Doneva

ARTIFICIAL INTELLIGENCE AS A TOOL FOR PEDAGOGICAL INNOVATIONS IN MATHEMATICS EDUCATION

Stanka Hadzhikoleva, Maria Borisova, , Borislava Kirilova

Книжка 4
Книжка 3
МОДЕЛИ НА ВЕРОЯТНОСТНИ ПРОСТРАНСТВА В ОЛИМПИАДНИ ЗАДАЧИ

Драгомир Грозев, Станислав Харизанов

Книжка 1
A NOTE ON A GENERALIZED DYNAMICAL SYSTEM OCCURS IN MODELLING “THE BATTLE OF THE SEXES”: CHAOS IN SOCIOBIOLOGY

Nikolay Kyurkchiev, Anton Iliev, Vesselin Kyurkchiev, Angel Golev, Todorka Terzieva, Asen Rahnev

EDUCATIONAL RESOURCES FOR STUDYING MIDSEGMENTS OF TRIANGLE AND TRAPEZOID

Toni Chehlarova1), Neda Chehlarova2), Georgi Gachev

2024 година
Книжка 6
ВЪЗМОЖНОСТИ ЗА ИЗГРАЖДАНЕ НА МЕЖДУПРЕДМЕТНИ ВРЪЗКИ МАТЕМАТИКА – ИНФОРМАТИКА

Елена Каращранова, Ирена Атанасова, Надежда Борисова

Книжка 5
FRAMEWORK FOR DESIGNING VISUALLY ORIENTATED TOOLS TO SUPPORT PROJECT MANAGEMENT

Dalibor Milev, Nadezhda Borisova, Elena Karashtranova

3D ОБРАЗОВАТЕЛЕН ПОДХОД В ОБУЧЕНИЕТО ПО СТЕРЕОМЕТРИЯ

Пеньо Лебамовски, Марияна Николова

Книжка 4
DYNAMICS OF A NEW CLASS OF OSCILLATORS: MELNIKOV’S APPROACH, POSSIBLE APPLICATION TO ANTENNA ARRAY THEORY

Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev, Asen Rahnev

Книжка 3
РАЗСТОЯНИЯ МЕЖДУ ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ И НЕРАВЕНСТВА В ИЗПЪКНАЛ ЧЕТИРИЪГЪЛНИК

Йордан Табов, Станислав Стефанов, Красимир Кънчев, Хаим Хаимов

USING AI TO IMPROVE ANSWER EVALUATION IN AUTOMATED EXAMS

Georgi Cholakov, Asya Stoyanova-Doycheva

Книжка 2
ON INTEGRATION OF STEM MODULES IN MATHEMATICS EDUCATION

Elena Karashtranova, Aharon Goldreich, Nadezhda Borisova

Книжка 1
STUDENT SATISFACTION WITH THE QUALITY OF A BLENDED LEARNING COURSE

Silvia Gaftandzhieva, Rositsa Doneva, Sadiq Hussain, Ashis Talukder, Gunadeep Chetia, Nisha Gohain

MODERN ROAD SAFETY TRAINING USING GAME-BASED TOOLS

Stefan Stavrev, Ivelina Velcheva

ARTIFICIAL INTELLIGENCE FOR GOOD AND BAD IN CYBER AND INFORMATION SECURITY

Nikolay Kasakliev, Elena Somova, Margarita Gocheva

2023 година
Книжка 6
QUALITY OF BLENDED LEARNING COURSES: STUDENTS’ PERSPECTIVE

Silvia Gaftandzhieva, Rositsa Doneva, Sadiq Hussain, Ashis Talukder, Gunadeep Chetia, Nisha Gohain

МОДЕЛ НА ЛЕОНТИЕВ С MS EXCEL

Велика Кунева, Мариян Милев

Книжка 5
AREAS ASSOCIATED TO A QUADRILATERAL

Oleg Mushkarov, Nikolai Nikolov

ON THE DYNAMICS OF A ClASS OF THIRD-ORDER POLYNOMIAL DIFFERENCE EQUATIONS WITH INFINITE NUMBER OF PERIOD-THREE SOLUTIONS

Jasmin Bektešević, Vahidin Hadžiabdić, Midhat Mehuljić, Sadjit Metović, Haris Lulić

СИСТЕМА ЗА ИЗВЛИЧАНЕ И ВИЗУАЛИЗАЦИЯ НА ДАННИ ОТ ИНТЕРНЕТ

Георги Чолаков, Емил Дойчев, Светла Коева

Книжка 4
MULTIPLE REPRESENTATIONS OF FUNCTIONS IN THE FRAME OF DISTANCE LEARNING

Radoslav Božić, Hajnalka Peics, Aleksandar Milenković

INTEGRATED LESSONS IN CALCULUS USING SOFTWARE

Pohoriliak Oleksandr, Olga Syniavska, Anna Slyvka-Tylyshchak, Antonina Tegza, Alexander Tylyshchak

Книжка 3
ПРИЛОЖЕНИЕ НА ЕЛЕМЕНТИ ОТ ГЕОМЕТРИЯТА НА ЧЕТИРИЪГЪЛНИКА ЗА РЕШАВАНЕ НА НЕСТАНДАРТНИ ЗАДАЧИ

Йордан Табов, Веселин Ненков, Асен Велчев, Станислав Стефанов

Книжка 2
Книжка 1
НОВА ФОРМУЛА ЗА ЛИЦЕ НА ЧЕТИРИЪГЪЛНИК (ЧЕТИВО ЗА VII КЛАС)

Йордан Табов, Асен Велчев, Станислав Стефанов, Хаим Хаимов

2022 година
Книжка 6
MOBILE GAME-BASED MATH LEARNING FOR PRIMARY SCHOOL

Margarita Gocheva, Nikolay Kasakliev, Elena Somova

Книжка 5
SECURITY ANALYSIS ON CONTENT MANAGEMENT SYSTEMS

Lilyana Petkova, Vasilisa Pavlova

MONITORING OF STUDENT ENROLMENT CAMPAIGN THROUGH DATA ANALYTICS TOOLS

Silvia Gaftandzhieva, Rositsa Doneva, Milen Bliznakov

TYPES OF SOLUTIONS IN THE DIDACTIC GAME “LOGIC MONSTERS”

Nataliya Hristova Pavlova, Michaela Savova Toncheva

Книжка 4
PERSONAL DATA PROCESSING IN A DIGITAL EDUCATIONAL ENVIRONMENT

Evgeniya Nikolova, Mariya Monova-Zheleva, Yanislav Zhelev

Книжка 3
Книжка 2
STEM ROBOTICS IN PRIMARY SCHOOL

Tsanko Mihov, Gencho Stoitsov, Ivan Dimitrov

A METAGRAPH MODEL OF CYBER PROTECTION OF AN INFORMATION SYSTEM

Emiliya Koleva, Evgeni Andreev, Mariya Nikolova

Книжка 1
CONVOLUTIONAL NEURAL NETWORKS IN THE TASK OF IMAGE CLASSIFICATION

Larisa Zelenina, Liudmila Khaimina, Evgenii Khaimin, D. Khripunov, Inga Zashikhina

INNOVATIVE PROPOSALS FOR DATABASE STORAGE AND MANAGEMENT

Yulian Ivanov Petkov, Alexandre Ivanov Chikalanov

APPLICATION OF MATHEMATICAL MODELS IN GRAPHIC DESIGN

Ivaylo Staribratov, Nikol Manolova

РЕШЕНИЯ НА КОНКУРСНИ ЗАДАЧИ БРОЙ 6, 2021 Г.

Задача 1. Дадени са различни естествени числа, всяко от които има прос- ти делители, не по-големи от . Докажете, че произведението на някои три от тези числа е точен куб. Решение: числата са представим във вида . Нека разгледаме квадрат

2021 година
Книжка 6
E-LEARNING DURING COVID-19 PANDEMIC: AN EMPIRICAL RESEARCH

Margarita Gocheva, Nikolay Kasakliev, Elena Somova

Книжка 5
ПОДГОТОВКА ЗА XXV МЛАДЕЖКА БАЛКАНИАДА ПО МАТЕМАТИКА 2021

Ивайло Кортезов, Емил Карлов, Мирослав Маринов

EXCEL’S CALCULATION OF BASIC ASSETS AMORTISATION VALUES

Vehbi Ramaj, Sead Rešić, Anes Z. Hadžiomerović

EDUCATIONAL ENVIRONMENT AS A FORM FOR DEVELOPMENT OF MATH TEACHERS METHODOLOGICAL COMPETENCE

Olha Matiash, Liubov Mykhailenko, Vasyl Shvets, Oleksandr Shkolnyi

Книжка 4
LEARNING ANALYTICS TOOL FOR BULGARIAN SCHOOL EDUCATION

Silvia Gaftandzhieva, Rositsa Doneva, George Pashev, Mariya Docheva

Книжка 3
THE PROBLEM OF IMAGES’ CLASSIFICATION: NEURAL NETWORKS

Larisa Zelenina, Liudmila Khaimina, Evgenii Khaimin, D. Khripunov, Inga Zashikhina

MIDLINES OF QUADRILATERAL

Sead Rešić, Maid Omerović, Anes Z. Hadžiomerović, Ahmed Palić

ВИРТУАЛЕН ЧАС ПО МАТЕМАТИКА

Севдалина Георгиева

Книжка 2
MOBILE MATH GAME PROTOTYPE ON THE BASE OF TEMPLATES FOR PRIMARY SCHOOL

Margarita Gocheva, Elena Somova, Nikolay Kasakliev, Vladimira Angelova

КОНКУРСНИ ЗАДАЧИ БРОЙ 2/2021 Г.

Краен срок за изпращане на решения: 0 юни 0 г.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 1, 2021

Краен срок за изпращане на решения: 0 юни 0 г.

Книжка 1
СЕДЕМНАДЕСЕТА ЖАУТИКОВСКА ОЛИМПИАДА ПО МАТЕМАТИКА, ИНФОРМАТИКА И ФИЗИКА АЛМАТИ, 7-12 ЯНУАРИ 2021

Диян Димитров, Светлин Лалов, Стефан Хаджистойков, Елена Киселова

ОНЛАЙН СЪСТЕЗАНИЕ „VIVA МАТЕМАТИКА С КОМПЮТЪР“

Петър Кендеров, Тони Чехларова, Георги Гачев

2020 година
Книжка 6
ABSTRACT DATA TYPES

Lasko M. Laskov

Книжка 5
GAMIFICATION IN CLOUD-BASED COLLABORATIVE LEARNING

Denitza Charkova, Elena Somova, Maria Gachkova

NEURAL NETWORKS IN A CHARACTER RECOGNITION MOBILE APPLICATION

L.I. Zelenina, L.E. Khaimina, E.S. Khaimin, D.I. Antufiev, I.M. Zashikhina

APPLICATIONS OF ANAGLIFIC IMAGES IN MATHEMATICAL TRAINING

Krasimir Harizanov, Stanislava Ivanova

МЕТОД НА ДЕЦАТА В БЛОКА

Ивайло Кортезов

Книжка 4
TECHNOLOGIES AND TOOLS FOR CREATING ADAPTIVE E-LEARNING CONTENT

Todorka Terzieva, Valya Arnaudova, Asen Rahnev, Vanya Ivanova

Книжка 3
MATHEMATICAL MODELLING IN LEARNING OUTCOMES ASSESSMENT (BINARY MODEL FOR THE ASSESSMMENT OF STUDENT’S COMPETENCES FORMATION)

L. E. Khaimina, E. A. Demenkova, M. E. Demenkov, E. S. Khaimin, L. I. Zelenina, I. M. Zashikhina

PROBLEMS 2 AND 5 ON THE IMO’2019 PAPER

Sava Grozdev, Veselin Nenkov

Книжка 2
ЗА ВЕКТОРНОТО ПРОСТРАНСТВО НА МАГИЧЕСКИТЕ КВАДРАТИ ОТ ТРЕТИ РЕД (В ЗАНИМАТЕЛНАТА МАТЕМАТИКА)

Здравко Лалчев, Маргарита Върбанова, Мирослав Стоимиров, Ирина Вутова

КОНКУРЕНТНИ ПЕРПЕНДИКУЛЯРИ, ОПРЕДЕЛЕНИ ОТ ПРАВИЛНИ МНОГОЪГЪЛНИЦИ

Йоана Христова, Геновева Маринова, Никола Кушев, Светослав Апостолов, Цветомир Иванов

A NEW PROOF OF THE FEUERBACH THEOREM

Sava Grozdev, Hiroshi Okumura, Deko Dekov

PROBLEM 3 ON THE IMO’2019 PAPER

Sava Grozdev, Veselin Nenkov

Книжка 1
GENDER ISSUES IN VIRTUAL TRAINING FOR MATHEMATICAL KANGAROO CONTEST

Mark Applebaum, Erga Heller, Lior Solomovich, Judith Zamir

KLAMKIN’S INEQUALITY AND ITS APPLICATION

Šefket Arslanagić, Daniela Zubović

НЯКОЛКО ПРИЛОЖЕНИЯ НА ВЪРТЯЩАТА ХОМОТЕТИЯ

Сава Гроздев, Веселин Ненков

2019 година
Книжка 6
DISCRETE MATHEMATICS AND PROGRAMMING – TEACHING AND LEARNING APPROACHES

Mariyana Raykova, Hristina Kostadinova, Stoyan Boev

CONVERTER FROM MOODLE LESSONS TO INTERACTIVE EPUB EBOOKS

Martin Takev, Elena Somova, Miguel Rodríguez-Artacho

ЦИКЛОИДА

Аяпбергенов Азамат, Бокаева Молдир, Чурымбаев Бекнур, Калдыбек Жансуйген

КАРДИОИДА

Евгений Воронцов, Никита Платонов

БОЛГАРСКАЯ ОЛИМПИАДА ПО ФИНАНСОВОЙ И АКТУАРНОЙ МАТЕМАТИКЕ В РОССИИ

Росен Николаев, Сава Гроздев, Богдана Конева, Нина Патронова, Мария Шабанова

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намерят всички полиноми, които за всяка реална стойност на удовлетворяват равенството Татяна Маджарова, Варна Задача 2. Правоъгълният триъгълник има остри ъгли и , а центърът на вписаната му окръжност е . Точката , лежаща в , е такава, че и . Симетралите

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 1, 2019

Задача 1. Да се намерят всички цели числа , за които

Книжка 5
ДЪЛБОКО КОПИЕ В C++ И JAVA

Христина Костадинова, Марияна Райкова

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намери безкрайно множество от двойки положителни ра- ционални числа Милен Найденов, Варна

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 6, 2018

Задача 1. Точката е левият долен връх на безкрайна шахматна дъска. Една муха тръгва от и се движи само по страните на квадратчетата. Нека е общ връх на някои квадратчета. Казва- ме, че мухата изминава пътя между и , ако се движи само надясно и нагоре. Ако точките и са противоположни върхове на правоъгълник , да се намери броят на пътищата, свърз- ващи точките и , по които мухата може да мине, когато: а) и ; б) и ; в) и

Книжка 4
THE REARRANGEMENT INEQUALITY

Šefket Arslanagić

АСТРОИДА

Борислав Борисов, Деян Димитров, Николай Нинов, Теодор Христов

COMPUTER PROGRAMMING IN MATHEMATICS EDUCATION

Marin Marinov, Lasko Laskov

CREATING INTERACTIVE AND TRACEABLE EPUB LEARNING CONTENT FROM MOODLE COURSES

Martin Takev, Miguel Rodríguez-Artacho, Elena Somova

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се реши уравнението . Христо Лесов, Казанлък Задача 2. Да се докаже, че в четириъгълник с перпендикулярни диагонали съществува точка , за която са изпълнени равенствата , , , . Хаим Хаимов, Варна Задача 3. В правилен 13-ъгълник по произволен начин са избрани два диа- гонала. Каква е вероятността избраните диагонали да не се пресичат? Сава Гроздев, София, и Веселин Ненков, Бели Осъм

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 5, 2018

Задача 1. Ако и са съвършени числа, за които целите части на числата и са равни и различни от нула, да се намери .

Книжка 3
RESULTS OF THE FIRST WEEK OF CYBERSECURITY IN ARKHANGELSK REGION

Olga Troitskaya, Olga Bezumova, Elena Lytkina, Tatyana Shirikova

DIDACTIC POTENTIAL OF REMOTE CONTESTS IN COMPUTER SCIENCE

Natalia Sofronova, Anatoliy Belchusov

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Краен срок за изпращане на решения 30 ноември 2019 г.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 4, 2018

Задача 1. Да се намерят всички тройки естествени числа е изпълнено равенството: а)

Книжка 2
ЕЛЕКТРОНЕН УЧЕБНИК ПО ОБЗОРНИ ЛЕКЦИИ ЗА ДЪРЖАВЕН ИЗПИТ В СРЕДАТА DISPEL

Асен Рахнев, Боян Златанов, Евгения Ангелова, Ивайло Старибратов, Валя Арнаудова, Слав Чолаков

ГЕОМЕТРИЧНИ МЕСТА, ПОРОДЕНИ ОТ РАВНОСТРАННИ ТРИЪГЪЛНИЦИ С ВЪРХОВЕ ВЪРХУ ОКРЪЖНОСТ

Борислав Борисов, Деян Димитров, Николай Нинов, Теодор Христов

ЕКСТРЕМАЛНИ СВОЙСТВА НА ТОЧКАТА НА ЛЕМОАН В ЧЕТИРИЪГЪЛНИК

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

A TRIANGLE AND A TRAPEZOID WITH A COMMON CONIC

Sava Grozdev, Veselin Nenkov

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Христо Лесов, Казанлък Задача 2. Окръжност с диаметър и правоъгълник с диагонал имат общ център. Да се докаже, че за произволна точка M от е изпълне- но равенството . Милен Найденов, Варна Задача 3. В изпъкналия четириъгълник са изпълнени равенства- та и . Точката е средата на диагонала , а , , и са ортоганалните проекции на съответно върху правите , , и . Ако и са средите съответно на отсечките и , да се докаже, че точките , и лежат на една права.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 3, 2018

Задача 1. Да се реши уравнението . Росен Николаев, Дико Суружон, Варна Решение. Въвеждаме означението , където . Съгласно това означение разлежданото уравнение придобива вида не е решение на уравнението. Затова са възможни само случаите 1) и 2) . Разглеж- даме двата случая поотделно. Случай 1): при е изпълнено равенството . Тогава имаме:

Книжка 1
PROBLEM 6. FROM IMO’2018

Sava Grozdev, Veselin Nenkov

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 2, 2018

Задача 1. Да се намери най-малкото естествено число , при което куба с целочислени дължини на ръбовете в сантиметри имат сума на обемите, рав- на на Христо Лесов, Казанлък Решение: тъй като , то не е куб на ес- тествено число и затова . Разглеждаме последователно случаите за . 1) При разглеждаме естествени числа и , за които са изпълнени релациите и . Тогава то , т.е. . Освен това откъдето , т.е. .Така получихме, че . Лесно се проверява, че при и няма естествен

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намерят всички цели числа , за които

2018 година
Книжка 6
„ЭНЦИКЛОПЕДИЯ ЗАМЕЧАТЕЛЬНЫХ ПЛОСКИХ КРИВЫХ“ – МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ В РАМКАХ MITE

Роза Атамуратова, Михаил Алфёров, Марина Белорукова, Веселин Ненков, Валерий Майер, Генадий Клековкин, Раиса Овчинникова, Мария Шабанова, Александр Ястребов

A NEW MEANING OF THE NOTION “EXPANSION OF A NUMBER”

Rosen Nikolaev, Tanka Milkova, Radan Miryanov

Книжка 5
ИТОГИ ПРОВЕДЕНИЯ ВТОРОЙ МЕЖДУНАРОДНОЙ ОЛИМПИАДЬI ПО ФИНАНСОВОЙ И АКТУАРНОЙ МАТЕМАТИКЕ СРЕДИ ШКОЛЬНИКОВ И СТУДЕНТОВ

Сава Гроздев, Росен Николаев, Мария Шабанова, Лариса Форкунова, Нина Патронова

LEARNING AND ASSESSMENT BASED ON GAMIFIED E-COURSE IN MOODLE

Mariya Gachkova, Martin Takev, Elena Somova

УЛИТКА ПАСКАЛЯ

Дарья Коптева, Ксения Горская

КОМБИНАТОРНИ ЗАДАЧИ, СВЪРЗАНИ С ТРИЪГЪЛНИК

Росен Николаев, Танка Милкова, Катя Чалъкова

Книжка 4
ЗА ПРОСТИТЕ ЧИСЛА

Сава Гроздев, Веселин Ненков

ИНЦЕНТЪР НА ЧЕТИРИЪГЪЛНИК

Станислав Стефанов

ЭПИЦИКЛОИДА

Инкар Аскар, Камила Сарсембаева

ГИПОЦИКЛОИДА

Борислав Борисов, Деян Димитров, Иван Стефанов, Николай Нинов, Теодор Христов

Книжка 3
ПОЛИНОМИ ОТ ТРЕТА СТЕПЕН С КОЛИНЕАРНИ КОРЕНИ

Сава Гроздев, Веселин Ненков

ЧЕТИРИДЕСЕТ И ПЕТА НАЦИОНАЛНА СТУДЕНТСКА ОЛИМПИАДА ПО МАТЕМАТИКА

Сава Гроздев, Росен Николаев, Станислава Стоилова, Веселин Ненков

Книжка 2
TWO INTERESTING INEQUALITIES FOR ACUTE TRIANGLES

Šefket Arslanagić, Amar Bašić

ПЕРФЕКТНА ИЗОГОНАЛНОСТ В ЧЕТИРИЪГЪЛНИК

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

НЯКОИ ТИПОВЕ ЗАДАЧИ СЪС СИМЕТРИЧНИ ЧИСЛА

Росен Николаев, Танка Милкова, Радан Мирянов

Книжка 1
Драги читатели,

където тези проценти са наполовина, в Източна Европа те са около 25%, в

COMPUTER DISCOVERED MATHEMATICS: CONSTRUCTIONS OF MALFATTI SQUARES

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ВРЪЗКИ МЕЖДУ ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ В ЧЕТИРИЪГЪЛНИКА

Станислав Стефанов, Веселин Ненков

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 2. Да се докаже, че всяка от симедианите в триъгълник с лице разделя триъгълника на два триъгълника, лицата на които са корени на урав- нението където и са дължините на прилежащите на симедианата страни на три- ъгълника. Милен Найденов, Варна Задача 3. Четириъгълникът е описан около окръжност с център , като продълженията на страните му и се пресичат в точка . Ако е втората пресечна точка на описаните окръжности на триъгълниците и , да се докаже, че Хаим Х

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 2, 2017

Задача 1. Да се определи дали съществуват естествени числа и , при които стойността на израза е: а) куб на естествено число; б) сбор от кубовете на две естествени числа; в) сбор от кубовете на три естествени числа. Христо Лесов, Казанлък Решение: при и имаме . Следова- телно случай а) има положителен отговор. Тъй като при число- то се дели на , то при и имаме е естестве- но число. Следователно всяко число от разглеждания вид при деление на дава ос

2017 година
Книжка 6
A SURVEY OF MATHEMATICS DISCOVERED BY COMPUTERS. PART 2

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ТРИ ИНВАРИАНТЫ В ОДНУ ЗАДА

Ксения Горская, Дарья Коптева, Асхат Ермекбаев, Арман Жетиру, Азат Бермухамедов, Салтанат Кошер, Лили Стефанова, Ирина Христова, Александра Йовкова

GAMES WITH

Aldiyar Zhumashov

SOME NUMERICAL SQUARE ROOTS (PART TWO)

Rosen Nikolaev, Tanka Milkova, Yordan Petkov

ЗАНИМАТЕЛНИ ЗАДАЧИ ПО ТЕМАТА „КАРТИННА ГАЛЕРИЯ“

Мирослав Стоимиров, Ирина Вутова

Книжка 5
ВТОРОЙ МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ УЧАЩИХСЯ В РАМКАХ MITE

Мария Шабанова, Марина Белорукова, Роза Атамуратова, Веселин Ненков

SOME NUMERICAL SEQUENCES CONCERNING SQUARE ROOTS (PART ONE)

Rosen Nikolaev, Tanka Milkova, Yordan Petkov

Книжка 4
ГЕНЕРАТОР НА ТЕСТОВЕ

Ангел Ангелов, Веселин Дзивев

INTERESTING PROOFS OF SOME ALGEBRAIC INEQUALITIES

Šefket Arslanagić, Faruk Zejnulahi

PROBLEMS ON THE BROCARD CIRCLE

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ПРИЛОЖЕНИЕ НА ЛИНЕЙНАТА АЛГЕБРА В ИКОНОМИКАТА

Велика Кунева, Захаринка Ангелова

СКОРОСТТА НА СВЕТЛИНАТА

Сава Гроздев, Веселин Ненков

Книжка 3
НЯКОЛКО ПРИЛОЖЕНИЯ НА ТЕОРЕМАТА НА МЕНЕЛАЙ ЗА ВПИСАНИ ОКРЪЖНОСТИ

Александра Йовкова, Ирина Христова, Лили Стефанова

НАЦИОНАЛНА СТУДЕНТСКА ОЛИМПИАДА ПО МАТЕМАТИКА

Сава Гроздев, Росен Николаев, Веселин Ненков

СПОМЕН ЗА ПРОФЕСОР АНТОН ШОУРЕК

Александра Трифонова

Книжка 2
ИЗКУСТВЕНА ИМУННА СИСТЕМА

Йоанна Илиева, Селин Шемсиева, Светлана Вълчева, Сюзан Феимова

ВТОРИ КОЛЕДЕН ЛИНГВИСТИЧЕН ТУРНИР

Иван Держански, Веселин Златилов

Книжка 1
ГЕОМЕТРИЯ НА ЧЕТИРИЪГЪЛНИКА, ТОЧКА НА МИКЕЛ, ИНВЕРСНА ИЗОГОНАЛНОСТ

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

2016 година
Книжка 6
ПЕРВЫЙ МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ УЧАЩИХСЯ В РАМКАХ MITE

Мария Шабанова, Марина Белорукова, Роза Атамуратова, Веселин Ненков

НЕКОТОРЫЕ ТРАЕКТОРИИ, КОТОРЫЕ ОПРЕДЕЛЕНЫ РАВНОБЕДРЕННЫМИ ТРЕУГОЛЬНИКАМИ

Ксения Горская, Дарья Коптева, Даниил Микуров, Еркен Мудебаев, Казбек Мухамбетов, Адилбек Темирханов, Лили Стефанова, Ирина Христова, Радина Иванова

ПСЕВДОЦЕНТЪР И ОРТОЦЕНТЪР – ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ В ЧЕТИРИЪГЪЛНИКА

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

FUZZY LOGIC

Reinhard Magenreuter

GENETIC ALGORITHM

Reinhard Magenreuter

Книжка 5
NEURAL NETWORKS

Reinhard Magenreuter

Книжка 4
АКТИВНО, УЧАСТВАЩО НАБЛЮДЕНИЕ – ТИП ИНТЕРВЮ

Христо Христов, Христо Крушков

ХИПОТЕЗАТА В ОБУЧЕНИЕТО ПО МАТЕМАТИКА

Румяна Маврова, Пенка Рангелова, Елена Тодорова

Книжка 3
ОБОБЩЕНИЕ НА ТЕОРЕМАТА НА ЧЕЗАР КОШНИЦА

Сава Гроздев, Веселин Ненков

Книжка 2
ОЙЛЕР-ВЕН ДИАГРАМИ ИЛИ MZ-КАРТИ В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова, Ирина Вутова, Иван Душков

ОБВЪРЗВАНЕ НА ОБУЧЕНИЕТО ПО АЛГЕБРА И ГЕОМЕТРИЯ

Румяна Маврова, Пенка Рангелова

Книжка 1
STATIONARY NUMBERS

Smaiyl Makyshov

МЕЖДУНАРОДНА ЖАУТИКОВСКА ОЛИМПИАДА

Сава Гроздев, Веселин Ненков

2015 година
Книжка 6
Книжка 5
Книжка 4
Книжка 3
МОТИВАЦИОННИТЕ ЗАДАЧИ В ОБУЧЕНИЕТО ПО МАТЕМАТИКА

Румяна Маврова, Пенка Рангелова, Зара Данаилова-Стойнова

Книжка 2
САМОСТОЯТЕЛНО РЕШАВАНЕ НА ЗАДАЧИ С EXCEL

Пламен Пенев, Диана Стефанова

Книжка 1
ГЕОМЕТРИЧНА КОНСТРУКЦИЯ НА КРИВА НА ЧЕВА

Сава Гроздев, Веселин Ненков

2014 година
Книжка 6
КОНКУРЕНТНОСТ, ПОРОДЕНА ОТ ТАНГЕНТИ

Сава Гроздев, Веселин Ненков

Книжка 5
ИНФОРМАТИКА В ШКОЛАХ РОССИИ

С. А. Бешенков, Э. В. Миндзаева

ОЩЕ ЕВРИСТИКИ С EXCEL

Пламен Пенев

ДВА ПОДХОДА ЗА ИЗУЧАВАНЕ НА УРАВНЕНИЯ В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова, Ирина Вутова

Книжка 4
ОБУЧЕНИЕ В СТИЛ EDUTAINMENT С ИЗПОЛЗВАНЕ НА КОМПЮТЪРНА ГРАФИКА

Христо Крушков, Асен Рахнев, Мариана Крушкова

Книжка 3
ИНВЕРСИЯТА – МЕТОД В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова

СТИМУЛИРАНЕ НА ТВОРЧЕСКА АКТИВНОСТ ПРИ БИЛИНГВИ ЧРЕЗ ДИНАМИЧЕН СОФТУЕР

Сава Гроздев, Диана Стефанова, Калина Василева, Станислава Колева, Радка Тодорова

ПРОГРАМИРАНЕ НА ЧИСЛОВИ РЕДИЦИ

Ивайло Старибратов, Цветана Димитрова

Книжка 2
ФРАКТАЛЬНЫЕ МЕТО

Валерий Секованов, Елена Селезнева, Светлана Шляхтина

Книжка 1
ЕВРИСТИКА С EXCEL

Пламен Пенев

SOME INEQUALITIES IN THE TRIANGLE

Šefket Arslanagić

2013 година
Книжка 6
Книжка 5
МАТЕМАТИЧЕСКИЕ РЕГАТЬI

Александр Блинков

Книжка 4
Книжка 3
АКАДЕМИК ПЕТЪР КЕНДЕРОВ НА 70 ГОДИНИ

чл. кор. Юлиан Ревалски

ОБЛАЧНИ ТЕХНОЛОГИИ И ВЪЗМОЖНОСТИ ЗА ПРИЛОЖЕНИЕ В ОБРАЗОВАНИЕТО

Сава Гроздев, Иванка Марашева, Емил Делинов

СЪСТЕЗАТЕЛНИ ЗАДАЧИ ПО ИНФОРМАТИКА ЗА ГРУПА Е

Ивайло Старибратов, Цветана Димитрова

Книжка 2
ЕКСПЕРИМЕНТАЛНАТА МАТЕМАТИКА В УЧИЛИЩЕ

Сава Гроздев, Борислав Лазаров

МАТЕМАТИКА С КОМПЮТЪР

Сава Гроздев, Деко Деков

ЕЛИПТИЧЕН АРБЕЛОС

Пролет Лазарова

Книжка 1
ФРАГМЕНТИ ОТ ПАМЕТТА

Генчо Скордев

2012 година
Книжка 6
ДВЕ ДИДАКТИЧЕСКИ СТЪЛБИ

Сава Гроздев, Светлозар Дойчев

ТЕОРЕМА НА ПОНСЕЛЕ ЗА ЧЕТИРИЪГЪЛНИЦИ

Сава Гроздев, Веселин Ненков

ИЗЛИЧАНЕ НА ОБЕКТИВНИ ЗНАНИЯ ОТ ИНТЕРНЕТ

Ивайло Пенев, Пламен Пенев

Книжка 5
ДЕСЕТА МЕЖДУНАРОДНА ОЛИМПИАДА ПО ЛИНГВИСТИКА

д–р Иван А. Держански (ИМИ–БАН)

ТЕОРЕМА НА ВАН ОБЕЛ И ПРИЛОЖЕНИЯ

Тодорка Глушкова, Боян Златанов

МАТЕМАТИЧЕСКИ КЛУБ „СИГМА” В СВЕТЛИНАТА НА ПРОЕКТ УСПЕХ

Сава Гроздев, Иванка Марашева, Емил Делинов

I N M E M O R I A M

На 26 септември 2012 г. след продължително боледуване ни напусна проф. дпн Иван Ганчев Донев. Той е първият професор и първият доктор на науките в България по методика на обучението по математика. Роден е на 6 май 1935 г. в с. Страхилово, В. Търновско. След завършване на СУ “Св. Кл. Охридски” става учител по математика в гр. Свищов. Тук той организира първите кръжоци и със- тезания по математика. През 1960 г. Иван Ганчев печели конкурс за асистент в СУ и още през следващата година започ

Книжка 4
Книжка 3
СЛУЧАЙНО СЪРФИРАНЕ В ИНТЕРНЕТ

Евгения Стоименова

Книжка 2
SEEMOUS OLYMPIAD FOR UNIVERSITY STUDENTS

Sava Grozdev, Veselin Nenkov

EUROMATH SCIENTIFIC CONFERENCE

Sava Grozdev, Veselin Nenkov

FIVE WAYS TO SOLVE A PROBLEM FOR A TRIANGLE

Šefket Arslanagić, Dragoljub Milošević

ПРОПОРЦИИ

Валя Георгиева

ПЪТЕШЕСТВИЕ В СВЕТА НА КОМБИНАТОРИКАТА

Росица Керчева, Румяна Иванова

ПОЛЗОТВОРНА ПРОМЯНА

Ивайло Старибратов

Книжка 1
ЗА ЕЛЕКТРОННОТО ОБУЧЕНИЕ

Даниела Дурева (Тупарова)

МАТЕМАТИКАТА E ЗАБАВНА

Веселина Вълканова

СРАВНЯВАНЕ НА ИЗРАЗИ С КВАДРАТНИ КОРЕНИ

Гинка Бизова, Ваня Лалева