Problem 6. Among 49 schoolchildren, each is acquainted with at least 25 others. Prove that they can be divided into groups of 2 or 3 people such that everyone in a group is acquainted with everyone else in their group.
points
(A. V. Shapovalov)
Problem 1280
Official solution
Solution. Let's assume that initially all 49 students are standing in the corridor, and we will gradually let them into the classroom.
We will do this in such a way that at any moment in time, the children in the classroom are divided into the required groups.
Let's say that a student named Fyodor is standing in the corridor. If he is acquainted with any other student standing in the corridor, we will simply let them both into the classroom.
Otherwise, all of Fyodor's acquaintances are already in the classroom. Since there are fewer than 50 students in the classroom, they are divided into fewer than 25 groups. Therefore, among Fyodor's acquaintances, there are at least two who are in the same group. If this is a group of 2 students, we will let Fyodor into the classroom and add him to this group. If it is a group of 3 students, we will ask one of Fyodor's acquaintances to form a group with him, and leave the remaining students in a pair.
By gradually letting students into the classroom, we will achieve the goal of dividing all students into the required groups.
The XV Oral City Mathematical Olympiad for grades 6-7 will take place on March 19, 2017.
Students who have received a diploma of prizewinner or a commendation at least at one of the following mathematical competitions are invited to the Olympiad:
- Mathematical Festival (21.02.16 or 19.02.17),
- XIV City Oral Olympiad (20.03.16),
- Winter Archimedes Tournament (17.01.16 or 15.01.17),
- Spring Archimedes Tournament for 5th grade (individual competition 03.04.16),
- II (municipal) stage of the All-Russian School Olympiad for 7th grade (11.12.2016).
Pre-registration for the Olympiad is required by March 12.
For more details, visit olympiads.mccme.ru/ustn/
!
Monthly illustrated magazine for students in grades 5-8
!
In the magazine, you will find interesting articles and problems in mathematics, linguistics, physics, and other natural sciences, and you can participate in a mathematical competition and a Russian language competition!
## Did you know:
- Why do tides occur on Earth?
- How do fireflies turn their light on and off?
- Why are the dimensions of A4 paper what they are?
- How to make a mini-robot
- What is probability theory?
- Why do tea and coffee invigorate?
Find answers to these and many other questions in the magazine "Kvantik"!
All "Kvantik" products - magazines, almanacs, posters, puzzle calendars - can be purchased at the online store kvantik.ru, as well as at the "Mathematical Book" store
at the address: Moscow, Bolshoy Vlasyevsky Pereulok, 11.
You can subscribe to "Kvantik" at any Post Office branches using two catalogs and online at the website vipishi.ru using the MAP catalog:
Catalog "NEWSPAPERS. MAGAZINES" of the "ROSSPECHT" agency:
index 80478 (for a year) index 84252 (for half a year or several months)
"Catalog of Russian Press" (MAP):
index 11348 (for a year)
index 11346 (for half a year or several months)
## Information on enrollment in grades 5-8 with in-depth study of mathematics in 2017
| School | Phone, URL | Address | Grades |
| :---: | :---: | :---: | :---: |
| 2 | (499) 137-17-69 www.sch2.ru | Fotieva St., 18 (m. "Oktyabrskaya") | 6-7, additional in 8 |
| 54 | (499) 245-99-72 (499) 245-54-25 moscowschool54.ru | Dovatora St., 5/9 (m. "Sportivnaya") | 8 |
| 57 | (495) 691-85-72 (495) 691-54-58 sch57.ru | Mal. Znamensky Pereulok, 7/10, bld. 5 (m. "Borovitskaya") | 8 |
| 179 | (495) 692-48-51 www.179.ru | Bol. Dmitrovka St., 5/6, bld. 7 (m. "Okhotny Ryad") | 7-8 |
| 218 | (499) 976-19-85 school218.ru | Dmitrovskoye Shosse, 5a (m. "Dmitrovskaya") | 8 (IUP) |
| 444 | (495) 465-23-52 (495) 465-60-52 schv444.mskobr.ru | Nizh. Pervomayskaya St., 14 (m. "Pervomayskaya") | 5 |
| 1329 | sch1329.mskobr.ru | Nikulin St., 10 (m. "Yugo-Zapadnaya") | 5, 6, 8, additional in 6-8 |
| 1534 | gym1534.ru | Kedrova St., 11 (m. "Akademicheskaya") | 5, additional in 7, 8 |
| 1543 | (495) 433-16-44 (495) 434-26-58 www.1543.ru | 26 Baku Commissars St., 3, bld. 5 (m. "Yugo-Zapadnaya") | 8 |
| 2007 | (495) 716-29-35 fmsh2007.ru | Gorchakova St., 9, bld. 1 (m. "Ul. Gorchakova") | 5, additional in 6-8 |
| Kurchatovskaya | (499) 194-10-44 kurchat.mskobr.ru | Marshal Vasilevsky St., 9, bld. 1 (m. "Shchukinskaya") | 5 |
Information provided by the schools to the MCCME. Published free of charge. Detailed information on enrollment in these and other classes is available at schools.mccme.ru
Up-to-date information on Olympiads: www.olimpiada.ru[^0]
[^0]: Mathematical Festival page
(problems, solutions, winners) www.mccme.ru/matprazdnik/