Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Information on the KöMaL Problem Solving Contests for 2020–2021

We ask every contestant to read this document carefully, even if you participated in our contest last year.

This year we announce contests in mathematics, physics and informatics, altogether in 21 categories with different difficulties. Each contest lasts for 9 months, from September 2020 until the beginning of June 2021. Solutions to problems proposed in every month can be submitted before early next month. An Award Ceremony will take place next fall during the annual KöMaL Ifjúsági Conference.

Participation in our contests in 2020–2021 is again free of charge. However, you can support KöMaL by subscribing to our journal as an individual or as an institution. Monetary donations are also gratefully accepted.

## Entering a contest

Every primary or secondary school student is eligible to participate in the contests.

However, according to the General Data Protection Regulation (GDPR) of the European Union, we need to have a parental consent to handle data for contestants under 16 years. Our data and information policy can be found at https://www.komal.hu/info/adatkezeles.h.shtml.

### Registration

After a successful registration, you can provide us some further data (e.g. your teacher's name, or your address of correspondence to receive certificates), and you can also let us know whether or not we can make (i) your detailed scores or (ii) your participation in a specific contest publicly available.

If you have already registered, no new registration is necessary. You can use your existing password, but you will still need to review and update your personal settings.

You formally enter a particular contest when we receive your first submission in that category.

It is also possible to enter a contest only later during the school year.

Important! We will not consider submissions without prior registration in the contest. You must register first.

Each participant in the KöMaL contest has a grade number between 1 and 12 (which may differ from their grade in school). A 12th-grade contestant is someone who has just started their last school year before the matriculation examination; 11th- or 10th-grade contestants are expected to finish their secondary education in 2022 or 2023.

After a successful registration you cannot modify your personal data (name, school's name, grade, email) by yourself. To change any of these, you need to contact us by email.

You should not repeat a successful registration. Multiple registrations will always cause problems (e.g. your name appearing twice in the contest but with half points).

### Portrait galler

If you wish your portrait to be included in the online results of a contest, please send us a photo by email. Light and uniform backgrounds are preferred. Use sufficiently high resolution because we often need to resize or crop images.

## Contests in mathematics

There are four different contest types in mathematics. They are, in increasing difficulty, type K, C, B and A problems. A student may participate in more than one contest type, however, types K and B cannot be chosen simultaneously. Contestants of grade 9 should declare which contest type K or B they would like to choose (in your personal settings on our webpage).

As a general rule, for every participant we eonly evaluate one solution to each problem.

We of course welcome problem generalizations, different solution techniques, observations, or new problems to propose. We often publish these or acknowledge them by special awards (not related to the contest).

### Type K contest—joint contest of ABACUS and KöMaL for 9th graders in mathematics}

This contest can be chosen exclusively by students of grade 9. We recommend this contest type for students who are not yet familiar with KöMaL. In each of the 7 months from September to March, we propose 5 problems, 3 of which also appear in the ABACUS contest. Each problem is worth 6 points.

Primary school students of grades 3–8 can participate in the ABACUS contest as usual.

### Type C contest–exercises in mathematics

We recommend type C exercises for those who find type B or A problems too difficult or unusual. Type C exercises are quite similar to (or just slightly more difficult than) problems appearing in a standard mathematics curriculum. We recommend this category to you if you plan to take an advanced-level matriculation exam in mathematics.

Certain type C exercises can be solved by primary-school students as well, however, some other exercises contain material taught only in grades 11 12. We propose 7 problems in each month, and each problem is worth at most 5 points. Students of grade at most 10 can submit solutions to Problems 1 5, while students of grade 11 12 can submit solutions to Problems 3 7.

Type C exercises are evaluated for three age groups: for grades 1 8, 9 10, and 11 12.

### Type B contest–problems in mathematics

In type B contest, there are 8 new problems in each month. However, only at most six of your solutions will be taken into account in the contest in any month, and disqualified solutions (see below for a definition), if any, are taken into account first. Therefore you can make good progress in the contest without solving all 8 problems by selecting and focusing on your favorite topics.

The ordering of the problems within each problem set corresponds to the school curriculum: easier problems–recommended for younger contestants–are listed first. Points for a problem (usually between 3 and 6) also indicate difficulty.

Type B problems are evaluated for five age groups: for grades below 9, for grades 9, 10, 11, and 12.

### Type A contest–advanced problems in mathematics

Type A problems are more challenging, and recommended for students preparing for international competitions or planning to become researchers.

There are 2 or 3 new problems in each month, each is worth at most 7 points. There are no separate age groups for type A problems, everybody competes together.

## Contests in physics

There are three different contests in physics: types M, G and P. A student may participate in more than one type of contest, however, types G and P cannot be chosen simultaneously. Contestants of grade 10 or less should declare which type G or P they choose (see your personal settings on our webpage).

We welcome problem generalizations, different solution techniques, observations, or new problems to propose. We often publish these or acknowledge them by special awards (not related to the contest).

### Type M contest—experimental problems in physics

In each month, a measurement problem is proposed. This can be submitted by a contestant of any age. Each M problem is worth 6 points.

To carry out an experiment, one can ask someone (a family member, classmate, or a friend) for assistance. Both the contestant's and the assistant's data should appear in the header of the documentation of the experiment. In this competition, it is also allowed for two contestants to cooperate and carry out an experiment together. Members of the pair can have different grades or schools, and they should register for the type M contest independently. In the case of two contestants taking measurements together, the lab report should be uploaded by each of them individually, under his or her own name, and the the document header should show both names, schools, grades and emails every month. Points for the documentation will be awarded to both members of the pair.

### Type G contest–simpler theoretical problems in physics

Type G problems are for students of grade 10 or below. These problems are recommended for contestants who find type P problems too difficult or unusual. Most type G exercises are similar to problems appearing in a regular curriculum, so here one can make good progress without advanced problem-solving skills. If you can handle type G problems confidently, then you are probably prepared to face type P problems.

In each month, 4 type G problems are proposed together with their maximum points. You can freely choose which problems to solve, but only at most three of your solutions will be taken into account in the contest in any month, and disqualified solutions (see below for a definition), if any, are taken into account first. Type G problems are evaluated for three age groups: for grades below 9, for grades 9, and 10.

### Type P contest–theoretical problems in physics

In each month, typically 10 theoretical problems are proposed together with the maximum obtainable points for each. These problems are not arranged in order of difficulty, rather, in an order corresponding to students' ages. You can freely select any problem to solve. If you are in grade 1–8, then at most 3 solutions will be taken into account in the contest, while if you are in grade 9&ndash12, then at most 5 solutions will be counted. In any case, disqualified solutions (see below for a definition), if any, are taken into account first.

Type P problems are evaluated for 5 age groups: for grades below 9, for grades 9, 10, 11, and 12.

## Contests in informatics

### Type I contest–application of informatics and programming exercises

In each month we propose 3 type I problems and one problem of type I/S. Some of these problems can be solved by primary school students, but most of them assume familiarity with concepts typically taught in secondary schools. By solving these problems you can prepare for other contests in informatics or for taking an advanced-level matriculation exam. In every month, 3 of your solutions with the highest scores (out of the possible 4) are taken into account in the type I contest.

Type I problems are programming exercises and problems about the application of informatics. One of the problems is similar to a matriculation exam problem (regarding content and format), and is denoted by the symbol (É). These problems are recommended for students preparing for this type of exam.

Problems of type I/S have level of difficulty between type I and type S. The necessary concepts and algorithms for these can be found on the webpages http://tehetseg.inf.elte.hu/nemes and https://www.oktatas.hu/pub_bin/dload/kozoktatas/tanulmanyi_versenyek/oktv/oktv2020_2021_vk/116_informatika_2021.pdf.

### Type S contest— advanced programming problems

This contest consists of two problems: a more difficult type S problem and a type I/S problem (simultaneously appearing in the type I contest). These problem types are recommended for students preparing for other programming contests. The necessary concepts and algorithms are found on the webpage of the International Olympiad in Informatics https://people.ksp.sk/~misof/ioi-syllabus/. When evaluating type S and I/S problems, we check not only the correctness of the output, but also the efficiency of your algorithm, i.e., whether it can produce an output for a large input within the given time constraints.

## Publishing the problems and exercises

New problems are proposed monthly from September until May in the printed journal issues and on our webpage.

On our webpage, the new problems appear on the 28th of every month (except for September). However, if you have subscribed to our journal, you can access the new problems from the day following the deadline for the previous issue. If you have subscribed to KöMaL, enter your subscription code in your personal settings on our webpage. The subscription code is located on a label on the front page of the September issue.

Even if you are not participating in a contest (for example, due to your age), you can still have access to the new problems after a successful registration and entering your subscription code.

One subscription code can only be used by a single person.

## Content and format of the submissions

Please have a look at some sample solutions in our earlier issues or on our website. They indicate the required format and expected level of detail for your submissions.

### Presenting your solutions to problems in mathematics and theoretical physics

When you write down your solution to a problem, the reader should be able to follow each of your steps easily. Try to be as concise as possible by carefully formulating your solution steps. This may take a lot of time, so do not postpone it to the last minute.

You will obtain the maximum score only for a full solution; the final result alone is insufficient. You need to prove your propositions in general. You can however freely use theorems that appear in your standard curriculum by citing only the name of the theorem (and without giving explicit references or a proof). You can refer to well-known theorems (e.g. Menelaus's theorem or Hölder's inequality) only by their names, in other cases you need to state the theorem properly and cite a reference (book title and page number, or internet address). When you cite a theorem, you should also show why this particular theorem is applicable in your case, and how exactly the theorem is used in your proof.

From time to time it turns out that a proposed ''new'' problem (or its equivalent form or generalization) has already been solved and published elsewhere. We prefer that your problem-solving skills improve rather than only use search engines. Hence no points will be awarded if your solution only contains a link or reference, or only shows that the new problem is a special case or corollary of a theorem published elsewhere; your submission should contain the detailed solution steps.

In any case, when you cite something by consulting articles, books or the internet, your submission should explicitly list all the references or sources you used.

In the case of physics problems, the wording of the problem may not contain all data required for the solution. The values of some material constants or some geographical or astronomical quantities can be looked up in printed charts or on the internet.

It is always a good idea to decompose a more complicated solution into some simpler steps by using paragraphs, titles or subtitles. Auxiliary lemmas, formulae or figures are easier to refer to if you number them.

For a geometry problem, it is important to include a figure so that one can follow and check your steps. Without a figure you will not obtain full score. For more complicated figures, you should also provide a description (e.g. let point $\displaystyle P'$ be the reflection of the point $\displaystyle P$ through the line $\displaystyle e$ "). Use sufficiently high resolution when submitting your work electronically. The dimensions of a typical figure can be between 500 and 1000 pixels; it should clearly show each important detail, and possibly fit on a screen.

As a solution of a mathematics problem calculated results with computer programs – including online services such as Wolfram Alpha – are not considered. Similarly, it will not be accepted if you examine too many separate cases (e.g. 30 or more): typically one can significantly reduce the number of cases to be considered by analyzing the problem more carefully.

### Experimental problems in physics

The experiment documentation should contain the principle used in the actual measurement, a draft (or photo) of the equipment arrangement, a sufficient number of data points with sufficient precision (in a table, together with the proper units), a graph visualizing the measured data, and an estimate for the order of magnitude of the measurement errors. Do not use excessively many digits when displaying measured or computed quantities: the output precision should be based on the error estimates as well. The experiment documentation should be concise, but should contain enough details so that anyone can reproduce your experiment. If you have too many data points (e.g. more than 50), it is enough to list only the most typical ones and the average of the rest. If your documentation is more than 6 pages long, you should include a half-page conclusion as well.

### Solutions to problems in informatics

The solution for a type I problem should be submitted in one of the languages Basic, C, C++, C\#, Java, Pascal or Python. You can use any integrated development environment, we reccomend the following: info_emelt_szoftverlista_2020maj.pdf

For type I problems about the application of informatics we reccomend the following: info_emelt_szoftverlista_2020maj.pdf Otherwise, our problem description will contain any further applications that you can use (most of which are freely available software.

For type I/S or type S problems you should use any of the languages Basic, C, C++, C#, Java, Pascal or Python. You should create a documentation describing the theoretical framework for your solution or algorithm, and add comments in your source code so that others can easily understand the purpose of your code pieces or blocks.

You can test your solutions of type I/S or S by using the automatic evaluation system at http://ideone.com.

## Preparing and submitting your solutions

You cannot submit your solutions in email.

For problems in mathematics or physics, you can use the online editor on our website or upload a pdf file.

A submission will be disqualified if a file contains solutions to more than one problem or is not neat enough to read. A submission will also be disqualified if instead of mathematical formulae it contains strings like (1+root5)/2*x or x2+((1+5+2sqrt(5)x2)/4, which are hard to interpret.

### Editing your solutions online by using the Electronic Workbook

The Electronic Workbook is part of our website to directly input or edit your solutions. You can modify your solutions any time before the submission deadline.

Mathematical formulae can be entered by using the TeX system. If you are not familiar with TeX, we recommend you take our short TeX course.

If you prepare your solution on a computer, you can upload the complete file on our website. In the case of mathematics or physics problems, please use PDF format for compatibility.

• Each solution should be put in a different file.
• The upper left corner of each submission should contain in block capitals:
• the type of the problem (A, B, C, K, M, G or P) in red,
• your school's name and location,

Handwritten solutions should be uploaded as a single good quality pdf file.

Make sure that the image is legible, do not use a resolution that is either too high or too low. If you are taking photos, take several shots in natural (scattered) light, and upload the best one. Rotate the image in portrait layout, crop the edges to keep the actual solution only, and resize.

There are several programs and telephone apps for manipulating images. Our tip is CamScanner, which is easy to use for creating a single appropriate pdf file.

### Submission of the solutions to problems in informatics

You can submit your solutions in informatics only through the Electronic Workbook on the KöMaL webpage. If your submission has more than one file, you should create a single folder containing all your files and the documentation. The folder name should be the same as the problem number. Then you should compress this folder and submit it as a single ZIP file. Executable files (e.g. any .exe file created during the development phase) should not be included in the final compressed file.

In the programming problems, the first lines of the source code should contain the following as a comment:

• the number of the problem;
• the full name and grade of the contestant;
• the name of the school and town of the contestant
• the contestant's email;
• the name and version number of the compiler used.

Please note that it is necessary for the inputs and outputs of your solutions to have the appropriate format as specified by the problem description, since your code will be run on various test problems generally in an automated way.

Any queries or issues regarding the problems and solutions in informatics can be sent to .

The submission deadline for mathematics is the 10th day, while for physics and informatics it is the 15th day of the month following the publication of the journal issue. If the 10th or 15th falls on a weekend or on a public holiday, the deadline is the next workday. Please submit your work in time because a few hours before the deadline our servers usually become busy and slower, and internet problems can sometimes also occur. Late submissions will not be accepted.

## Evaluation

The actual status of the contest and the detailed results of the contestants are found on our web page and updated regularly. We also notify our contestants in email about their actual number of points.

### Queries

After your solutions have been submitted and evaluated, you can send a short message or query to the person who marked your solution; this communication takes place within the Electronic Workbook. Solutions to different problems are evaluated by different people, so your query should address only the actual problem.

Use appropriate tone and style that is similar to a face-to-face communication with a teacher or parent.

If you feel that your issue cannot be resolved this way, you can contact the editors. Please send an email to within two weeks.

In general, we suggest you keep a copy of your solutions so that you can later compare them with the official solutions published in the journal. Rarely, items get lost in the post, and we need a copy to settle any possible dispute.

### Disqualification from the contest

Important! The KöMaL Contest problems (except possibly for the ''Type M experimental problems in physics'', see above) are to be solved individually and not in teams. You must work out the solutions on your own. Before the submission deadline, you must not discuss these problems with anyone else, and you must not ask for or accept any hints. Solutions prepared by a team or plagiarized work will be disqualified, including that of the original author. These "solutions" are returned to the corresponding teacher in your school. In more severe cases jeopardizing the contest (e.g. when a problem is discussed on the internet), the affected participants will be disqualified from the contest.

### Publishing the final results

Final results will be published after all submissions have been evaluated. We expect to publish the results on our webpage in August, and in the printed journal in September 2021. Portraits of the most successful contestants will be included in the December 2021 issue of the journal. The MATFUND Hungarian High School Mathematics and Physics Foundation will provide awards to the best students at the annual KöMaL Conference in 2021. Certificates will be sent via regular post.

## Further remarks

By participating in our contest, you give your consent that we may publish your solution anonymously, or an edited version of it with your name.

The editors welcome new and interesting problems to propose, articles, or reports on math circle activities or stimulating school coursework. For problem suggestions please also include your solution. You can contact us via post or visit the Editorial Office in person. Students whose problem suggestions have been accepted will often be acknowledged by special awards at the end of the school year.

As already remarked above, we encourage you to discuss the problems further, after the submission deadline, for example, in the KöMaL Forum on our webpage. We sometimes publish comments or problem generalizations in our journal as well.

We wish you a challenging and successful KöMaL contest for this school year.

The Editors