In the variant of the Olympiad from the small mekhmat of Moscow State University for students in grades 4-5. Mathematics tutors working with gifted children, teachers of mathematical circles and electives. To solve all problems, it is enough to have the skills of action with natural numbers. No equations need to be made.

1) Cheburashka went to the birthday party of Crocodile Gena. When he had gone exactly half of the way, he suddenly remembered that he had forgotten a gift at home and returned. Therefore, I was late to Gene by 20 minutes. In how many minutes could Cheburashka come to Gena if he did not have to return?

2) Grandmother baked pancakes. My granddaughter came home from school and immediately began to eat them. While she eats 3 pancakes, Grandma only manages to bake two. At the very beginning, there were 17 pancakes on the plate. How many pancakes did the granddaughter eat if she left the table when there were only 7 pancakes left on the plate?

3) 4 "A" class played snowballs. At the end of the battle, it turned out that the number of boys who were hit by snowballs is equal to the number of girls who were not hit by snowballs. Who is more in the class: slain by snowballs or girls?

4) Vova was treated to sweets. First he was given one piece of candy. After eating the first candy, he ate them according to this principle: he ate a candy, and if he liked it, then he was additionally given two more new ones, if he did not like it, he ate the next one. Vova did not like the last 25th candy eaten. He didn't have any more candy. How many candies did he like?

5) Cat Murka eats a jar of Whiskas in 6 minutes, and cat Vaska - 2 times faster. How long will it take them to eat Whiskas together?

Math tutor's comment: a task from among the basic ones for the 6th grade program, however, the tutor can safely include it in the didactics of his classes in the 4th grade, because it can be solved without involving ordinary fractions. Think how.

6) Six math school teachers were asked how many light bulbs are on the ceiling in their math classroom. We received the following answers: the first teacher said: more than one, the second teacher said: more than two, the third said: more than three, the fourth said: more than four. The fifth answered: less than four, and the sixth: less than three. How many light bulbs are in the classroom if exactly half of the math teachers told the truth?

7) Carlson has three coins in three pockets - one is gold, the other is silver, and the third is bronze. The coin that is in the largest pocket is not gold, the silver coin is not in the middle pocket. And in the smallest pocket is not a gold or silver coin. Which one is where?

8) Only one + sign and only one = sign should be inserted into the entry 15628733029 to get the correct equality. Find at least one such option.

9) A cake costs the same as two bagels, and three bagels cost the same as two pieces of cake. Which is more expensive - two bagels or three pieces of cake?

10) There are 5 blue balls and 7 green balls in a box. The best mathematician in the class - Anton, with his eyes closed, takes one ball out of the box and puts it in the basket. How many balls does the mind need to get to ensure that there are 2 green balls in the basket?

11) The figure shows the development of the cube. The numbers 1 and 2 are marked on it.
Write the remaining numbers in the cells: 3, 4, 5 and 6 so that the sum of the numbers placed on any two opposite faces is equal to seven.

Small mekhmat - school

This page contains methodological developments created by leading teachers of the Small Mekhmat for conducting mathematical circles in general education schools. The work is carried out with the financial support of the Moscow Department of Education.

Each methodological development consists of a manual for the leader of the circle, accompanied by a set of tasks for schoolchildren, ready for printing. The tasks are laid out in A5 format and placed two per A4 page. Some of them need to be printed on two sides - in this case they are placed on adjacent pages in the file. All materials are presented as PDF files.

Dear students! The materials published here are intended mainly for teachers who lead mathematical circles. Passing off solutions to problems from the methodological commentary as one’s own during the class is a gross deception, and first of all, not of the teacher, but of oneself. Remember that the true value is a independently solved problem, and not a "plus" in the statement! At the same time, leaflets with tasks, solutions and comments to them can be used for self-education (for example, in cases where the circle is not available). This is highly welcome.

Methodological developments

The methodological developments of the "Small Mekhmat - School" are represented by two lines. The first line is designed for a 4-year course, the second - for a 2½-year course. We recommend that you choose one of the lines and follow it in ascending order of classes (while matching the classes high school largely arbitrary).

• for grade 5 (30 lessons), compiled by D. A. Korobitsyn and G. K. Zhukov: assignments for schoolchildren, a guide for teachers;
• for grade 6 (30 lessons), compiled by D. A. Korobitsyn and G. K. Zhukov: assignments for schoolchildren + answers and comments;
• for grade 7 (30 lessons), compiled by E. A. Astashov, Ya. A. Veryovkin, A. A. Deich, S. M. Saulin, A. V. Feklina:
  • 1st part: tasks for schoolchildren; manual for teachers;
  • 2nd part: tasks for schoolchildren; guide for teachers .
• for grades 8-9:
  • 1st half of the year (15 lessons), compiled by E. A. Astashov and D. A. Udimov: assignments for schoolchildren, a guide for teachers.
  • 2nd half of the year (15 lessons), compiled by E. A. Astashov, Y. A. Veryovkin, O. A. Manzhina and D. A. Udimov: tasks for schoolchildren, a guide for teachers.

• for grades 5-6:
  • 1st half of the year (15 lessons), compiled by A. L. Kanunnikov, S. L. Kuznetsov and I. I. Osipov: tasks for schoolchildren, a guide for teachers;
  • 2nd half of the year (15 lessons), compiled by I. I. Osipov: assignments for schoolchildren, a guide for teachers.
• for grades 6-7:
  • 1st half of the year (15 lessons), compiled by N. P. Strelkova and S. L. Kuznetsov: tasks for schoolchildren, a guide for teachers;
  • 2nd half of the year (15 lessons), compiled by S. L. Kuznetsov and A. A. Onoprienko: assignments for schoolchildren, a guide for teachers, application: the game "mathematical abacus".
• for 8 classes, compiled by A. L. Kanunnikov:
  • 1st semester (15 lessons):

Did you ask about the Maly Mekhmat of Moscow State University? Please!
We heard a lot about the Small Mekhmat.
Heard, but did not even think to get there. It seemed that children-geniuses go to the circle, that they do something magical with children there and they begin to love mathematics even more.
Let me just say that it turned out to be completely different. As usual, many people confuse cause and effect.
The reason is the love of mathematics. And the circle is a consequence of this great and bright love for this exact science.
So about mugs. The photo is a witness to the excellent mood after the circles and the fact that mathematicians are not at all boring nerds.

Official information from the websites of Moscow State University
Every year from October to April, the Small Mekhmat operates - free math circles at Faculty of Mechanics and Mathematics. Classes are held every Saturday from 16:40 in the Main building of Moscow State University (I supplement - or 2 building)
There are circles for schoolchildren in grades 1-11 at the Small Mekhmat.
Classes are held on Saturdays from September to December and from February to May. The classes are free.
Traditionally, any student who wished could come to the Small Mekhmat and participate in any circle, starting from any lesson. We try to keep this tradition alive. However, the number of applicants is growing rapidly every year, so we had to introduce a mandatory pre-registration(for all newly coming schoolchildren, including those who studied in previous academic years), as well as competitive selection for some circles.
The circles are taught by students, graduate students and teachers of the Faculty of Mechanics and Mathematics of Moscow University, as well as other faculties of Moscow State University and higher educational institutions mathematical and related areas, graduate students and researchers of academic institutions, workers in science-intensive industries, leaders of the circle-Olympiad movement.
At the circles of the Small Mekhmat, schoolchildren get acquainted with interesting mathematical problems, get used to logically rigorous reasoning, comprehend the beauty and harmony of mathematics. The topics of circles are very diverse and, as a rule, have little to do with the school curriculum in mathematics.

Now for the circle formats. In fact, there are group circles and there are streaming ones. What is it all about?

Lecture circles by Spivak and Pronina are held in large auditoriums.
There is no selection for these circles. Mathematicians do not fall away on their own. All you need to do is register and go. There are a lot of children. We did not attend such lectures. According to parents, it is very interesting: a lot of theory about mathematical topics, tasks. They praise both Spivak and Pronina. BUT!! Since the auditoriums are large, there are many children and it is often noisy. Therefore, the high motivation of children comes to the fore. If your child is diligent, highly motivated and loves math, you can try this format.

Now about group circles.
There is a selection for these mugs .
I have to write an entrance exam. Tasks are different. The time is limited to one hour.. This year, if my memory serves me, the exam was on September 25th. We found out about joining the circle on the first day in the evening, after posting information from one of our friends on Facebook. Signed up. After some time, we received a letter - an invitation to the exam. The record closed very quickly. When we arrived we were a little shocked. A huge, simply unrealistically huge crowd of children / parents / grandparents (that's real, families came and pressed the child)! We quickly found our audience. The exam lasted an hour. Parents, of course, communicated. Someone reported that if the child did not enter, all the way to mathematics is closed forever. Someone reported that they have been going for years since the 1st grade, attending all summer and winter schools. Of course, we believed in Tim, but we clearly understood that it would be difficult for him with such rivals. Yes, even considering the two-year difference. We did not have special hopes for admission - because we did not know the level of the children writing the exam. When the children began to come out in batches after 40 minutes from the beginning and inform their parents that everything had been decided, we somehow realized that there were few chances, so we took the exam philosophically. Tim came out exactly an hour later (but he never goes out earlier) and said that he had decided, but not all. Then we finally calmed down. In the end, they consoled themselves with the fact that it was inconvenient to travel far, and “but Saturday will remain free”
After that we went home.

A couple of weeks later, we received an invitation to enroll Timur in the circle. They scored a total of about 120 people (this is the third class). Timur apparently solved a sufficient number of tasks for enrollment. Children were divided into several groups of 20-30 people.
Psychologically, many children were lost on Ezkamen - therefore, it is necessary to set the child up for the crowd, excitement, crampedness and noise. With us, Timur is used to different conditions, so he is not particularly lost.

Now about the format of classes.
Classes are held once a week in the second building (to the left of the main building of Moscow State University) on Saturdays. There are usually fewer children in a group than the average. About 12-15 people go constantly. The rest from time to time: either training, then departures, then competitions, then they fell ill. Pretty comfortable to do. The lesson goes like this: the children are given a list of tasks on a specific topic. Children solve problems. When the problem is solved, the child raises his hand. The teacher comes up to him and accepts. If everything is correct, the problem is accepted and the child solves the next one. If not, the teacher can ask leading questions. And these are not questions, as parents usually like to ask - tips in fact. No. Here they will simply offer to think about ... something that is the same, but from a different angle. As a result, the child will either make a decision on his own, or not.

This is how the whole thing goes on. Who finished everything - is free. But usually there were too many tasks and instead of an hour less than 80 minutes, we did not succeed. That is, the first children come out after 1 hour and 20-25 minutes. Whatever you don't decide, you take home. Everything needs to be done at home. At the beginning of the next task of the work of the previous topic, if there are questions. If there are no questions - the next topic.

So, some conclusions and points and what is valuable in the classroom
1) Way to find a solution. The child works by himself. Leading questions very competently bring the child to the solution, allowing him to look at the problem with different eyes. An algorithm for finding a solution is gradually being formed.
2) Good work on topics. Tasks are on the rise. The selection was made with a large margin: there are tasks that we saw in the collections for grades 5-6-7. That is, no one will come out with the words "lightness". Such deep study gives a deep understanding of the topic, immersion and ownership of the topic.
3) Many criticize, but there is no theory on the topic. Children literally reach everything with their brains. Bring your experience and thoughts. That is, nothing is put into the mouth and chewed. You have to work here.
4) And yes. Main. This is for kids who love math. We see how the children get bored, we saw how they refuse to go. There are those who say that "I can do it at home." Of course yes! You can also at home. But here it turns out more efficiently, taking into account the first points. Once on the website of the Bolshoy Mehmat we read something like “if you want to use mathematics in business, programming, etc., you are not here. Mehmat is for those who love mathematics and numbers by themselves.” That is, there is pure non-applied mathematics and logic. For the Small, this is partly suitable. But all the same, love for numbers and the ability to see beauty in tasks is the main thing that will allow a child to study at the Small Mechanics and Mathematics with joy.
5) We are lucky with our parents. A Whatsapp group was quickly created. That's where the teachers got involved. Thus, issues were quickly resolved. We received prompt assignments on topics and recommendations for solutions. Moreover, it is convenient to solve organizational matters: if you miss a lesson, or, for example, reported about the blocking of the streets, etc. things related to the topic.
6) They celebrated holidays twice: the first before the New Year. Thanks to our teachers - there were sweets and prizes. It was a fun holiday with solving problems in the form of a tournament. The second time on Halloween we had fun with tasks about worms, witches, etc. Well, that is, classes with a human face
7) Purely technically, no one is marked in terms of visiting. But if you do not go, you must understand that you are occupying someone else's place, which a child striving for knowledge did not fall into. It's a matter of everyone's conscience. I think this is wrong. We only missed a few times.
8) After classes, they send (in our group, I know, in others this is not the case) the results. How did the children decide in the group? It is very useful in terms of understanding whether your child is lagging behind, how much this topic is clear to him. This allows you to quickly pull up topics.
9) Well, it’s just great for a child to see what a large university looks like from the inside. Feel all these corridors, the bustle of student life.
10) And of course, eat in the canteenJWhere without her? Moreover, the food at Moscow State University is inexpensive and very tasty.

In addition, there is a summer and winter school on the small Mekhmat. Two weeks of immersion in mathematics. All for children!

The results are probably hard to evaluate. We still do a lot of work at home. But of course, the level has grown a lot. And in terms of understanding the tasks, and in terms of analyzing the conditions. And this gives a new round of growth of interest. What is important. For the future.

Recruitment for circles of the Small Mekhmat in 2015

Classes of circles of the Small Mekhmat in the new academic year start September 19, 2015.
To participate in classes e-registration required: https://register.mmmf.msu.ru . Registration will be open until 14:00 Friday 18 September 2015.
Class schedule for circles: http://mmmf.msu.ru/time-table/timetable.html.

The main goal of the work of the Small Mechanics and Mathematics Faculty (MMMF) is to popularize mathematics among schoolchildren in Moscow and the Moscow region, other regions of the Russian Federation and even the Russian-speaking population outside of Russia. The work of the MMMF is aimed at developing the skills of mathematical thinking and solving non-standard (Olympiad) mathematical problems in middle-aged and older schoolchildren.

Mathematical circles at the Faculty of Mechanics and Mathematics of Moscow State University have been operating for more than 50 years. By the end of the 70s. these circles for students in grades 7–9 worked under the name Evening Mathematical School (“School for Young Mathematicians”). In the mid 70s. Sunday began its work at Mekhmat preparatory school(School "Applicant") to prepare students of the 10th (graduating) grade for admission to universities with advanced requirements in mathematics. In 1978, a Correspondence Mathematical School was established for students in grades 8–10 called “Maly Mekhmat” (before that, the Correspondence Preparatory School for residents of the Moscow Region had been operating for some time), and on December 11, 1981, a unified school for young mathematicians “Maly Mekhmat” was created ”, which consists of three departments: evening, correspondence and department “Applicant”. In 1982, the "Applicant" department was included in the evening department, and the Small Mekhmat found its current structure. In the mid 80s. for some time, a separate stream of programmers was created at the Maly Mekhmat.

The Maly Mekhmat has an evening department with full-time education for schoolchildren in grades 2–11 and a correspondence department with distance learning for schoolchildren in grades 8–11.

Everyone is welcome to the evening section without entrance examinations; classes start on the first Saturday in October, but all students who come during the year are also enrolled. Admission to the correspondence department is made on a competitive basis based on the results of the introductory work carried out in April.

Classes run from October to May. The duration of study is from 1 year to 4 years (for correspondence department) or up to 8 years (in the evening department). Evening classes are held on Saturday afternoons and evenings.

Selected tasks are discussed in some circles of the evening department entrance exams at Moscow State University in the past. One of methodological developments for the correspondence department is also devoted to selected tasks of entrance exams at Moscow State University of past years.

Education in the evening and part-time departments is free.

Mathematical circles

There are circles for schoolchildren in grades 1-11 at the Small Mekhmat.

Circle classes are taking place on Saturdays September to December and February to May. The classes are free.

Traditionally, any student who wished could come to the Small Mekhmat and participate in any circle, starting from any lesson. We try to keep this tradition alive. However, the number of applicants is growing rapidly every year, so we had to introduce mandatory pre-registration (for all newly coming schoolchildren, including those who studied in previous academic years), as well as competitive selection for some circles.

The circles are taught by students, graduate students and teachers of the Faculty of Mechanics and Mathematics of Moscow University, as well as other faculties of Moscow State University and higher educational institutions of mathematical and related areas, graduate students and researchers from academic institutions, workers in high-tech industries, and leaders of the circle-Olympiad movement.

At the circles of the Small Mekhmat, schoolchildren get acquainted with interesting mathematical problems, get used to logically rigorous reasoning, comprehend the beauty and harmony of mathematics. The topics of circles are very diverse and, as a rule, have little to do with the school curriculum in mathematics. In grades 5–8, problems are solved on such classical “circle” topics as the Dirichlet principle, colorings, invariants, divisibility, logic, combinatorics, induction, graphs, etc.

Classes for grades 5-8 are held in groups of 15-30 people. Each student receives a sheet with tasks, solves them and discusses them individually with the teacher. There are several teachers in each classroom at the same time. Key tasks are sorted out at the blackboard. The texts of the leaflets are presented on our website: past years - in the archive, current - by class in the "Materials of classes" section (see left).

The above applies to the "traditional" circles of the Small Mekhmat for grades 5-8: they are presented in the "Schedule" section in the table above. The classes of the circles presented in the table below are arranged somewhat differently. Find out exactly which of the circles is right for you, you can only take part in the classes.

For high school students (grades 9-11), popular lectures on mathematics are read and circle classes are held (you can get acquainted with the teachers of circles and choose the one you like at the presentation of all circles at the beginning of the first lesson). Lectures and circles are spaced apart in time, so schoolchildren in grades 9-11 have the opportunity to attend both a lecture and a circle lesson on the same day. Each lecture is devoted to a separate topic, which, as a rule, has nothing to do with the topics of other lectures, or with the activities of circles. Sometimes we can talk about very difficult ideas and results, but prior knowledge that goes beyond school curriculum, to understand the material of the lectures is not required. Lectures are given by leading scientists and teachers of Moscow and are open to everyone. You can find annotations of all previously read lectures and announcements of the upcoming ones on our website in the "Lectures" section.

As a rule, schoolchildren who are interested in mathematics, after one or two years of studying at the Small Mekhmat, enter mathematical classes and schools and continue to study mathematics there.

Small mekhmat does not set itself the goal of preparing for entrance exams.

We are waiting for you at the Small Mekhmat!