Preparing for Online Competition Mathematics with a Computer


Preparing for the Online Competition "Mathematics with a Computer"

Trainig Program (click to open or close)

THEME: Preparing for online Competition "Mathematics with a Computer"


ANNOTATION: The course uses developed virtual learning environments for math education and themes from past online competitions. The methodology is related to working with ready to use products, modifying them, finding solutions with specified accuracy, dipping in a research atmosphere, experiencing different approaches and interactive learning methods. Emphasis is also placed on participation in online competitions. The specialized mathematical software used is free. The lectures / exercises ratio is 1: 3.

TARGET GROUP: Teachers in Mathematics, Informatics, Information Technologies, Primary Teachers.


Understanding the rules of online competition "Mathematics with a Computer".

Learning digital resources to prepare for online competition "Mathematics with a Computer" and a methodology to prepare for successful participation in them.

Forming knowledge and skills to prepare and organize learning activities in a research style by combining classical tools and dynamic constructions with specialized software.

Developing skills for effective search, retrieval, selection and evaluation of resources related to dynamic constructions and mathematical education.


Attendance training (8 hours) and distance learning (8 hours)


Satisfaction with the training

Applying of acquired skills

Participation in educational network activities

DURATION: 16 hours



Attendance training: printed materials, hands-on toys, resources in,,

Distance learning: a virtual learning environment at a site

Self-assessment poll (click to open or close)


For what time can you solve the equation `|2x+3| = 4x^6-14x^3+x+0,5`?

  • up to  10 min
  • 10 min  ÷ 1 h
  • 1h ÷  8 hours
  • It is not relevant to me
  • up to  10 min
  • 10 min  ÷ 1 h
  • 1h ÷  8 hours
  • It is not relevant to me

Have you created a dynamic design with specialized software?



How many times have you tested a student allowing him to use any resources?



A radial sector of angle α sheet is cut and removed from a circular sheet From the remaining part, an open vessel is formed in the shape of a straight circular cone. What is the value of α for which the volume of the resulting vessel is maximum? Record your guess.




Problem 1. Review the "Mathematics with a Computer" competition rules. Record specific features that distinguish the "Mathematics with a Computer" competition from:


1.1. The municipal round of the National Olympiad in Mathematics;

1.2. Easter Mathematical Competition.

"Mathematics with a Computer" - competition rules (click to open or close)

“Mathematics with a computer  – competition rules

The "Mathematics with a Computer" competition is for third to twelfth grade students. The participants are divided into five age groups, each comprising two successive school classes. It takes place in two rounds.

The first round has two editions in the school year - in December and April. The competition is open for anyone who wants to participate in it.

The second round takes place in September or early October. Invited are only high-school students who have achieved best results in the two first round editions and / or in the “Theme of the Month” competition (see below).

To participate in the two editions of the first round, pre-registration is required in the portal Each registered student chooses the place from where he will participate in the competition, as well as the technique he / she will use - a computer, a laptop or a tablet. At predetermined day and time each registered participant gets 60 minutes access to a worksheet with ten tasks. For the easier tasks, possible answers are offered, and the participant is expected to choose one or more answers based on mathematical reasoning and calculations. For the rest of the tasks, participants are expected to enter a decimal number (usually with a precision of one hundredth or one thousandth). This is done in a dedicated field for the entry of the answer. In order to solve such a task, the participant has to do his / her own research using a computer. Tasks are accompanied by a computer file (generated usually by a GeoGebra system) that allows the student to explore the same or similar task. The participant must adapt the file for his purposes and to find a sufficiently precise answer. The answer is judged by its proximity to the solution found by the jury and / or by the author of the task. There is no limit to the use of resources - textbooks, tutorials, the Internet, including help from friends. The goal is to put participants in real-world conditions and to cultivate skills for using a variety of resources, including looking for information and using help from a problem solving expert.

The second round of the competition is "attendance" type - the participants are together - in one place, at the same time. They can again use textbooks, tutorials, the Internet. There is a limitation on the use of human resources. The working time used by the participantis is also important (when several participants have equal number of points the prize goes to the one who used shorter time for the solution of the problems). Hence everyone has to decide which more productive tool to use when solving the task.


Problem 2. Play with one of the themes of a "Mathematics with a Computer" competition.

04.2015 - 4 grade

12.2014 – 7 grade

12.2015 - 10 grade

The rest of the themes from past competitions can be found at:

2.1. Divide the tasks from the theme you have studied according the answer?

Open answer

Single choice answer  

Multiple choice answer





2.2. Fill in the next table with the numbers of the appropriate problems/tasks from the competition you have studied:


A task known from the math school course, possibly with "awkward/difficult" data.



A task that can be solved with the knowledge of the math school course for the relevant age group, but after a lot of efforts and for a time period longer than the one allowed in the competition. Using an auxiliary file makes it easier to solve the problem.




A task that cannot be solved with the knowledge of the math school course but with the help of a computer can be answered with sufficient precision.





Help - Problem 2 (click to open or close)

For the answers to the tasks in the competition "Mathematics with a Computer"


According to the type of answer, the tasks of the "Mathematics with a Computer" competition are divided into three groups:

  • Tasks with an optional answer, with a choice of exactly one answer from n-given options.


Eligible answers in this case are marked with round buttons. When an answer is given, the corresponding button is marked with a black dot.

When an answer is changed, the previous mark is automatically removed, i.e. only the last answer remains marked.

  • Tasks with a multiple choice answer, with the possibility of choosing k answers of n options, where k is less or equal to n. The evaluation is according to the correct checking / unchecking of each of the given choices. For each correct answer given and for each unselected wrong answer, a point is given. This type of evaluation is suitable for tasks with multiple solutions.


In this type the buttons of the answers are square. The text "You can specify more than one answer" is also displayed.

  • Tasks with an answer that is a decimal number. A specific element is the indication of the accuracy with which the answer is to be recorded. There is an indication that a decimal point is used instead of the standard decimal comma, but a decimal result is also actually reported even if the decimal comma is used. The evaluation of these “free answers” is of a type “target”. A maximum number of points are granted for a correct answer. Points are also given when the answer is approximately correct. The number of points depends on the proximity of the answer to the correct one.


To solve these tasks, it is appropriate to use the resources provided (GeoGebra files), usually by changing values ​​- with a slider, by moving an object, for example a point, by specifying a parameter value, etc. Thus, the capabilities of the software are also learned and trained. Of course, students who know the software environment can directly input data and get results. Participants can also use other computing tools to solve the task. The working time is important when two or more participants have equal number of points so everyone has to decide which more productive tool to use when solving the task.


The auxiliary file in this task is quite simple. It consists of a slider with which the value of the angle αcan be changed. To solve the task, it is necessary to set the new rate measure of 160º and use the received value of the function under consideration to find and select the correct answer:


Knowing of the calculation tools shortens the time for task solving, and allows for organizing and conducting research, both in and out of school.

2.3. How can you use the theme reviewed or its elements with your students?


Problem 3. Specify some variants for solving the task

3.1. Find the area of `ΔABC` with coordinates of the vertices`A(-5,58; 2,6), B(-3,2; -1,78), C(3,06; 5,8)`

Help - Problem 3 (click to open or close)

Task 3. Specify options for solving the task

Find the area of `ΔABC` with coordinates of the vertices `A(-5,58; 2,6), B(-3,2; -1,78), C(3,06; 5,8)`


Some Ideas:

If you use the file at

or ,

You can change the current position of the points by:

  • moving them (but it is usually difficult to achieve the desired accuracy)


  • save the new coordinates in the algebraic window (do not forget to press Enter after the change)


  • save the new coordinates in the main menu settings of the three vertices.

You can enter in the settings menu by right-clicking on the object or the toolbar.


Note that a decimal point is used to record a decimal number.

You can also build the triangle yourself by entering the points in the command line with their coordinates, and build a triangle with vertices the three points (don’t forget to press Enter).


In the algebraic window, you can observe the value of the triangle area, but you can also display that value on the screen. One way is to use the area button.

A second option is to specify in triangle settings that the object properties are to be displayed.

3.2. From a circular sheet a radial sector with angle α is cut and removed. From the rest part of the sheet, an open vessel is formed in the shape of a straight circular cone. What is the value of α for which the volume of the resulting vessel is maximal?

You can use the files:


Problem 4. The "Quad-corner prism with maximum volume" at the address

is based on tasks from the "Mathematics with a Computer" competition.


4.1. Play with it.

4.2. How can you expanding it.

4.3. Specify opportunities for using resources from the theme with your students.

Recommended literature:

Petar S. Kenderov, Toni K. Chehlarova. Extending The Class Of Mathematical Problems Solvable In School Serdica J. Computing 9 (2015), No. 3–4, 191–206 Serdica Journal of Computing Bulgarian Academy of Sciences Institute of Mathematics and Informatics

Кендеров, П., Т. Чехларова. Състезание Математика с компютър и изследователски подход в образованието по математика. Макрос. 2016. ISBN 978-954-561-422-4, 128 с.


Problem 5. The theme "Poliomino in a Numeric Table" at the address

is related to problems from the “Mathematics with a Computer” competition.

5.1. Play with it.

5.2. How can you expand it.

5.3. Specify opportunities for using resources from the theme with your students.


Problem 6. Suggest options for organizing a research tasks for students by using the "Mathematics with a Computer" competition

6.1. Problem 8 for 3rd grade from December 2014

6.2. Problem 9 for 3rd grade from December 2014

6.3. Problem 10 for 10th and 11th grade from April 2015:


Problem 7. Record digital skills that are formed or refined when participating in "Mathematics with a Computer" competition. Note which of them require preliminary training of your students.

Digital skills

Need of preparation





















Help - Problem 7 (click to open or close)

Activities in preparing and participating in a "Mathematics with a Computer" competition related to digital competence:

  • registration
  • open the worksheet
  • using prepared dynamic files
  • software installation
  • show decisions (input of data)
  • modifying the dynamic files provided
  • Internet search
  • use of electronic resources
  • sending responses
  • sharing


Problem 8. Create a dynamic file that is appropriate for the following task from “Mathematics with a Computer” competition:

8.1. Find the abscissa of the orthocenter of the triangle with vertices `A(-4,5; 12,3), B(-8,12; 5,75), C(4,08; 5,26)`


8.2. Find the smallest solution of the equation  `|x+1| = x^5-3x^3+x^2-2,1x+0,5`




Probelm 9. Create a problem for “Mathematics with a Computer” competition.

Problem 10. Suggest ideas for motivating students to take part in “Mathematics with a Computer” competition.