Математика и Информатика

https://doi.org/10.53656/math2022-5-2-sec

2022/5, стр. 423 - 434

SECURITY ANALYSIS ON CONTENT MANAGEMENT SYSTEMS

Lilyana Petkova
OrcID: 0000-0002-2859-8302
WoSID: AEE-3308-2022
SCOPUS: 57435516700
E-mail: lilyanapetkova92@gmail.com
CEO&.NET Developer at “LiLuzeNet Ltd”
2700 Blagoevgrad Bulgaria
Vasilisa Pavlova
OrcID: 0000-0003-3094-155X
WoSID: H-4596-2015
E-mail: vasypavv@gmail.com
South-West University “Neofit Rilski” (SWU)
Technical Faculty
66 Ivan Mihaylov Blvd.
2700 Blagoevgrad Bulgaria

Резюме: This paper is dedicated to the challenges of the use of the most popular content management systems (CMS) in software development. Fundamental information about the selected CMS platforms and vulnerability analysis are introduced. The review is made on CMS like Umbraco, Sitecore, WordPress and Drupal categorized in two groups defined by the technology used for development. And as the IT world changes a lot these brings one constant battle against threats. Therefore, this article will add some vulnerabilities analysis of the selected systems since 2014. Results were grouped by common vulnerabilities of the selected platforms and such specific ones.

Ключови думи: CMS; web application; security; vulnerability

1. Introduction

Content Management System (CMS) is a web-based application that helps multiple users with different permission levels to build and manage content online even without web programing knowledge. But on the other hand, they can be even used in education as the visual presentation of objects as part of the Object-Oriented Programming (OOP)1) module in Informatics. Every page created by the CMS can be accepted as a complex object which is composed of different sub objects. This by its definition is what the OOP is for – simplification by breaking it down into its simpler parts.

The market share provides us with different options which technically reviewed have their advantages and disadvantages. That can be visible from the description of the use of each of the reviewed systems.

This article is meant to provide an analysis of several CMS over the years and their vulnerability level according to international insights. The selected CMS are chosen according to personal opinion and to international insights!

In preparation of the article, the reports begin with a market share of the use of various content management systems among more than 10000 web applications. Part of personal experience, the analyses are structured in two categories based on the used technology for building the application: Microsoft technology – Umbraco and Sitecore, and PHP2) – WordPress and Drupal, which as part of the research defines them as the most popular among the selected technology.

As the IT world changes a lot these brings one constant battle against threats. Therefore, this article will add a vulnerabilities’ report of the selected systems prepared as a comparison of the number and the common vulnerability for the selected CMS per year for the period of 2014 up until 2022. The data is collected through reports from the most popular worldwide insights like CVE and CVSS (Tal 2019).

2. Relevance of the problem

As CMSs become so popular, people would not pay attention to security, they only focus on having the best website to present to people with great design, pictures, and simple functionality. Hackers will find it easier to hack a simple website where there are no simple security functions to protect the website. Website owners always think that hackers have no interest in their websites since they are not considered as a large corporation as other websites, so they don’t implement any security measures because it will cost more to implement in a website. However, hackers might not target the user website specifically, but it can be targeted from an automated system run by the hackers to hack many websites (Vasek, Wadleigh & Moore 2016).

Up until 2022, approximately 800 million web applications use a CMS, which is around 69% of all applications (Walsh 2022). And by 2026, the CMS market will reach more than 120 billion (Galov 2022). It is easy to think that CMSs are used by hobby bloggers, but those numbers prove the opposite. The CMS provides relatively easy development and maintain, which makes them quite popular choice in quickly providing the best solution to the client.

As software development is evolving with each new year providing new platforms and languages for development, for each programming language there are various content management systems.

Even though over the past decade the number of CMSs evolves, the analysis shows that since 2003, the year WordPress CMS was launched (WordPress 2015), it still dominates the CMS market share (see Figure 1). Another open-source CMS which holds around 10% of the CMS market share, is Drupal, which even though was created a couple of years before WordPress is still way down with those numbers. But according to the analysis those are the most popular content management systems using the PHP as programming basis.

Another concurrent programming option in the development industry is Microsoft technology. CMSs using that technology are also part of the market share (see Figure 1). According to it, the numbers are significantly lower than the PHP ones, but the practice shows that Microsoft technology usage grows, and its advantages are better than the popular PHP. According to the market share, two of the most popular CMS using Microsoft are Umbraco and Sitecore. The first one is opensource and the other is paid software property of Sitecore Corporation (initially called Pentia A/S).

Figure 1. CMS Market Share among the Top 10000 Web Application

But the never stop innovating IT industry requires the development to be constantly aware of the new “trends” in cybersecurity, which makes us wander of the vulnerabilities level of the CMS technology (Schryen 2009).

From Figure 2 we can view the number of vulnerabilities for the period from 2014 until 2022 for the selected CMSs. On a first view, it is clear that one of the systems (the most popular as shown at Figure 1) becomes less vulnerable through the years, while the other three have a constant turnover. But analyzing the numbers, the CMS with decreased number is still in the list of vulnerability due to its popularity. This still makes it the most vulnerable.

Figure 2. Number Of Vulnerabilities from 2014 – 2022

3. Microsoft CMS

Microsoft CMS included in this category are the kind of CMS that use Microsoft .NET platform in implementing the necessary functionality. With the help of the report from Figure 1, in the Microsoft CMS we have chosen: Umbraco and Sitecore.

3.1. Umbraco CMS

Umbraco is an open-source CMS built on Microsoft’s .NET framework using ASP.NET and is written in C#. It was born in 1999 but version 1.0 was announced in 2003 by Niels Hartvig. And on February 16th, 2005, with Umbraco 2.0 it became fully open-source CMS. The latest major version, Umbraco 9, was launched on September 28th, 2021. Umbraco 9 saw a transition from ASP.NET to the .NET Core framework (Umbraco community, n.d.).

In 2005, Umbraco started hosting Codegarden – the official annual Umbraco developer conference. It evolved from 23 developers to more than 2000 attendees from all around the world.

In 2009, Umbraco launched a dedicated Umbraco Community site, called Our Umbraco. It serves as a platform for the online developer community to find technical documentation about Umbraco products, exchange knowledge, or ask other developers about Umbraco-related questions (Umbraco community, n.d.).

In 2013, the Umbraco source code became available on GitHub, making Umbraco more open-source-friendly and making it easier for the global community of developers to create issues and to make pull requests.

Besides the free CMS, Umbraco’s offer of products expanded in 2015 to include Umbraco Cloud, a product providing secure and updated hosting through Microsoft Azure. Umbraco Heartcore, the headless CMS solution released in 2019, is running on Umbraco Cloud (Wahlberg & Sterling 2011).

Umbraco CMS can be used for complex solutions. It provides third-party integration, powerful audit trail, rollback and scheduling functionality. Its use is not bound by any templating which makes it flexible in design and functionality but on the other hand it means longer effort in development and in-depth planning.

This CMS has a powerful built-in search engine and user management. Another positive point is the availability of page preview in different devices which makes the content management more mobile. Umbraco also supports multi-sites and multilingual.

According to the market share (see Figure 1), 1.8% of the web applications are using Umbraco CMS, but the developer community is very supportive and active and with each year passed it expands very fast. Also, from a security point of view it is less vulnerable than the other open-source CMS selected in the article.

According to CVE reports from 2014 to 2022 (see Figure 2), Umbraco shows minimal quality of vulnerabilities (CVE details n.d.). The biggest number in that period was from year 2020 which in our personal opinion can be because of the massive digital activities due to Covid-19 crisis and because of the major changes from that year in the Umbraco organization – the appearance of the new Umbraco CMS version and all additional features.

To sum up the flaws as visible on Figure 3, 36% of the vulnerabilities are from gaps in the executed code and another 36% from deficiency in protection against cross site scripting (XSS)2) attacks. The other damages can be done from gaining information (9%), directory tranversal4) (9%) and what is called a cross-site request forgery (CSRF)5).

Figure 3.Umbraco CMS Vulnerabilities

Some of the international companies using Umbraco CMS are: Veterans Crisis Line, Marie Curie Library, International Tennis Federation, and Johnnie Walker.

As part of main work experience, Umbraco CMS will be further reviewed in future researches.

3.2. Sitecore CMS

Sitecore is a customer experience management company that provides web content management, and multichannel marketing automation software founded in the 1990s in Denmark. By 2001 it had evolved into the Sitecore CMS. Today, it is one of the world’s most notable digital experience platforms (DXP) (Roberti 2007).

Sitecore differentiates the following products (Sitecore Manual - A guide to using the college CMS 2020):

– Sitecore Experience Manager (XM) – a content management system that powers omnichannel content delivery.

– Sitecore Experience Platform (XP) – a marketing automation solution that creates personalized customer experiences.

– Sitecore Experience Commerce (XC) – an e-commerce platform that leverages marketing automation to create a shopping experience.

Sitecore CMS can be used for very complex solutions. It provides third-party integration, powerful audit trail, rollback and scheduling functionality. Its use is also not bound to any templating but on it requires longer effort in development and in-depth planning (even longer than Umbraco CMS).

The CMS has also powerful built-in search engine and user management and also supports multi-sites and multilingual.

According to the market share (see Figure 1), 1.7% of the web applications are using Sitecore CMS which is less than the use of Umbraco CMS. One of the reasons might be the cost of the CMS as it is not an open-source platform.

From security perspective seen in Figure 2, Sitecore CMS shows minimal number of vulnerabilities. And again, the year 2020 shows higher number of flaws. From all those flaws as visible on Figure 4, the vulnerabilities are due to the executed code and 35% from lack of protection against XSS attacks. The other damages are from directory traversal (15%) and CSRF (10%), or by bypassing something from the application.

Figure 4. Sitecore CMS Vulnerabilities

Some of the international companies using Sitecore CMS are:

- L’Oreal

- Puma

- United Airlines

- Microsoft

4. PHP CMS

PHP CMS group includes CMS which uses PHP language in its functionality. PHP CMS selected in this analysis are the most popular ones according to Figure 1. CMS Market Share Among the top 10000 web application: WordPress and Drupal.

4.1. WordPress CMS

WordPress is an open source CMS, which allows the users to build dynamic websites and blogs. WordPress is the most popular blogging system on the web and allows updating, customizing and managing the website from its back-end CMS and components (WordPress 2015).

WordPress CMS provides more affordable solutions on a low-cost basis. It is easy to learn and use without of the box functionality. Due to its popularity, the developer community is bigger than all the others CMS’ communities. WordPress has over 5000 themes and over 50000 plugins. And as part of the Google suits of products, the SEO perspective is very powerful. As part of the provided functionality, this CMS has a good audit trail and scheduling features (Zamościński & Kozieł 2020).

But as the coin has two sides, the disadvantages are more than the advantages. It over relies on third-party plugins and uses a huge number of resources which makes the design very restricted, and the performance is overweighted. Any customization comes at extra cost. For multilanguage sites, this CMS is not an option as it does not provide multi-sites solutions.

Figure 5. WordPress CMS Vulnerabilities

One of the biggest disadvantages of WordPress CMS is its vulnerability. Its popularity helps in finding solutions in the community of users and developers, but the security level is very low. According to CVE reports from 2014 to 2022 (see Figure 2), WordPress is the CMS that shows most flaws in its software which provides hackers an opportunity to gain access to a system or network, even though through the years the number of vulnerabilities decreases (CVE details n.d.).

According to the CVE report visible on Figure 5, the common vulnerabilities so far are as follows: executed code (17%), XSS attacks (37%), bypassing something (11%), directory traversal (3%). But there are a lot more to be aware of. Such as SQL injections6) (9%), gaining information (9%), CSRF (6%), denial of service (5%), file inclusion (1%), gain privilege (1%) and HTTP response splitting (1%). (Zamościński & Kozieł 2020)

According to the market share, more than 33% among 10000 application used WordPress CMS which placed it the most popular among the selected CMS in this research (see Figure 1). But based on the vulnerability report, this makes us wonder how much the content used in applications with WordPress is secured and how much it is exposed of being harmed.

Some of the internationals companies using WordPress CMS are: The Walt Disney Company, Sony Music, BBC America, The New York Times Company, Mercedes-Benz, TechCrunch, The White House, University of Washington, and PlayStation Blog.

4.2. Drupal CMS

Drupal is a flexible CMS based on the LAMP stack, with a modular design allowing features to be added and removed by installing and uninstalling modules, and allowing the entire look and feel of the website to be changed by installing and uninstalling themes. The base Drupal download, known as Drupal Core, contains the PHP scripts needed to run the basic CMS functionality, several optional modules and themes, and many JavaScript, CSS, and image assets. Many additional modules and themes can be downloaded from the Drupal.org website (Hodgdon 2015 – 2017).

According to the market share, 9.6% of the word-wide application used Drupal CMS which placed it the second most popular among the selected CMS in this research (see Figure 1).

This CMS provides solutions to complex problems with more than 2500 themes and 46 000 modules to use in development like newsletters, podcasting components, etc. But in comparison to WordPress, it is harder to learn and use and needs more time to develop an application using Drupal CMS (Alghofaili 2018).

Drupal developers maintain their own security team responsible for evaluating security warning, searching for vulnerabilities, etc. The CMS has a section from where the users can be informed for the current security state (Meike, Sametinger & Wiesauer 2009). Drupal CMS is considered more secure than WordPress, but proportionally less due to the difference in their popularity.

Figure 6. Drupal CMS Vulnerabilities

From Figure 2 it is visible than in the period from 2014 – 2022 the numberof vulnerabilities in applications using Drupal CMS increases. From Figure6 the common vulnerabilities can be seen too: executed code (15%), XSS attacks (41%), bypassing something (14%), directory traversal (1%), CSRF(8%). But again, we can see some additional ones: SQL injections (6%), gaining information (8%), denial of service (3%), gain privilege (3%) and HTTP response splitting (1%).

Some of the international companies using Drupal CMS are: Nasdaq; The University of Colorado; The Principal Financial Group; Tesla; NPR; NBC; Pfizer; The Grammy.

5. Conclusions

Choosing the right technology is always hiding a big risk. But all depends on the selected building technology and the necessary requirements. Not always the paid one is the best, nor the most popular. In this article, we analyzed two groups of systems separated by the technology used in the development. But with the analyses we are hoping of giving more clearness on the best system to work with or even educate in school or university.

Regarding the purpose of the research, we can conclude that Microsoft CMS provides more secure and flexible solutions (as part of the advantages and disadvantages description for each system) for developing a web application which is even visible from the beginning of the present research on Figure 2. And as demonstrated on Figure 7, .NET platforms have a constant of only 5 common vulnerabilities compared to the PHP systems which have not only those 5 but also some other threats which go over 20% of the whole.

And among the selected systems in current review, we can conclude that the most preferable according to description and the vulnerability’s report is Umbraco CMS, and the most vulnerable and less mobile (as presented in the description of the system) is WordPress CMS. That is also visible on Figure 7. CMS common vulnerabilities report 26% of the threats on WordPress are from another group of flaws compared to the 0% on Umbraco CMS.

Figure 7. CMS common Vulnerabilities Report

Even though the technologies are constantly innovating, the better one is the one which is constantly up-to-date to every new technology in the IT industry. In further research we are going to present the use of such system in the education process as a method for converting the complex programming in something easy and simple.

NOTES

1. Object-Oriented Programming (OOP) – a programming approach of decomposition of a problem into a number of entities called objects and builds data and functions around their objects (Ericson 2009).

2. PHP: Hypertext Preprocessor is a general-purpose scripting language for web development (PHP 2017 – 2022).

3. Cross Site Scripting (XSS) attack is a type a script injection into trusted web applications. Usually, the malicious script is sent through forms and can harm in different ways the end user – by manipulating the page content, accessing cookies, sensitive information, etc. (Singh 2018; OWASP, Cross-site scripting n.d.).

4. Directory traversal flaw aims to access files and directories (configuration or system files or source code) stored outside the web root folder (OWASP, Path Traversal n.d.).

5. Cross-Site Request Forgery (CSRF) is an attack which forces authenticated users to submit a request to the application against which they are authenticated. This kind of attacks can compromise the entire web application (OWASP, Cross Site Request Forgery 2022).

6. SQL injection exploit can manipulate the database data by executing SQLInsert/Update/ Delete queries, shutting down database server, and in some cases commanding the operating system. (OWASP, SQL Injection n.d.).

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

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

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2020 година
Книжка 6
ABSTRACT DATA TYPES

Lasko M. Laskov

Книжка 5
GAMIFICATION IN CLOUD-BASED COLLABORATIVE LEARNING

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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

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Sava Grozdev, Veselin Nenkov

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

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Sava Grozdev, Hiroshi Okumura, Deko Dekov

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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

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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

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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

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Книжка 5
ДЪЛБОКО КОПИЕ В C++ И JAVA

Христина Костадинова, Марияна Райкова

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

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РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 6, 2018

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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

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

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DIDACTIC POTENTIAL OF REMOTE CONTESTS IN COMPUTER SCIENCE

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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

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Борислав Борисов, Деян Димитров, Николай Нинов, Теодор Христов

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Веселин Ненков, Станислав Стефанов, Хаим Хаимов

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Sava Grozdev, Veselin Nenkov

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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

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Rosen Nikolaev, Tanka Milkova, Radan Miryanov

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

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Росен Николаев, Танка Милкова, Катя Чалъкова

Книжка 4
ЗА ПРОСТИТЕ ЧИСЛА

Сава Гроздев, Веселин Ненков

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Станислав Стефанов

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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

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Росен Николаев, Танка Милкова, Радан Мирянов

Книжка 1
Драги читатели,

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

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Sava Grozdev, Hiroshi Okumura, Deko Dekov

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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

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SOME NUMERICAL SQUARE ROOTS (PART TWO)

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ЗАНИМАТЕЛНИ ЗАДАЧИ ПО ТЕМАТА „КАРТИННА ГАЛЕРИЯ“

Мирослав Стоимиров, Ирина Вутова

Книжка 5
ВТОРОЙ МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ УЧАЩИХСЯ В РАМКАХ MITE

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Rosen Nikolaev, Tanka Milkova, Yordan Petkov

Книжка 4
ГЕНЕРАТОР НА ТЕСТОВЕ

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INTERESTING PROOFS OF SOME ALGEBRAIC INEQUALITIES

Šefket Arslanagić, Faruk Zejnulahi

PROBLEMS ON THE BROCARD CIRCLE

Sava Grozdev, Hiroshi Okumura, Deko Dekov

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Велика Кунева, Захаринка Ангелова

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Сава Гроздев, Веселин Ненков

Книжка 3
НЯКОЛКО ПРИЛОЖЕНИЯ НА ТЕОРЕМАТА НА МЕНЕЛАЙ ЗА ВПИСАНИ ОКРЪЖНОСТИ

Александра Йовкова, Ирина Христова, Лили Стефанова

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Сава Гроздев, Росен Николаев, Веселин Ненков

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Александра Трифонова

Книжка 2
ИЗКУСТВЕНА ИМУННА СИСТЕМА

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Иван Держански, Веселин Златилов

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

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

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FUZZY LOGIC

Reinhard Magenreuter

GENETIC ALGORITHM

Reinhard Magenreuter

Книжка 5
NEURAL NETWORKS

Reinhard Magenreuter

Книжка 4
АКТИВНО, УЧАСТВАЩО НАБЛЮДЕНИЕ – ТИП ИНТЕРВЮ

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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

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Румяна Маврова, Пенка Рангелова

Книжка 1
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