New admission to the correspondence department of the Small Faculty of Mechanics and Mathematics

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A new example for the correspondence department of the Small Mechanical Engineering Department

The Small Faculty of Mechanics and Mathematics - a mathematical school at the Faculty of Mechanics and Mathematics of Moscow State University - announces admission to the correspondence department for seventh grade students of general education schools. Admission to the Small Faculty of Mechanics and Mathematics (MMMF) is based on the results of solving the problems of the introductory work published below.

The main task of the Small Faculty is to introduce mathematics, deepen knowledge within the framework of the school curriculum, as well as expand the mathematical horizons of secondary school students.

Part-time classes begin in September. Education at the Small Mechanism is free. The duration of training is three years. Correspondence students who have particularly successfully completed the 8th or 9th grades are recommended for admission to Physics and Mathematics School No. 18 at Moscow State University. Students who successfully complete all tasks receive a certificate of completion of MMMF.

The teachers at the correspondence department of MMMF are graduate students and students of the Faculty of Mechanics and Mathematics. The development of thematic brochures is carried out by the MMMF methodological council, consisting of professors and teachers of the Faculty of Mechanics and Mathematics.

It is advisable to enroll in the Small Mechanical Engineering Department and send solutions to the problems of the introductory work to MMMF no later than April 15 (all tasks do not have to be solved). The work must be done in the school’s checkered terrace. On the cover of the terradi, stick a piece of paper with the following information:

1. resources, region, region;
2. family name, name of the student;
3. school and class (full name);
4. full home address (including post office code);

Problems of introductory work at the Small Faculty of Mechanics and Mathematics in 1989

1. The cube-shaped tank holds 15 m³ of water. The same amount of water goes into a 2.5 m high tank with a square base. Compare the sides of the bases of the first and second tank.

2. Find $x_1^3 + x_2^3$ without calculating $x_1$ and $x_2$, where $x_1$ and $x_2$ are the roots of the equation $x^2 + px + q = 0$.

3. Ilya Muromets, Alyosha Popovich, Dobrynya Nikitich and Mikula Selyaninovich struck several Mountain Serpents with arrows. Each of the kites contained exactly 3 arrows, and these were arrows of different types. Ilya Muromets had the largest graying - 6, and Alyosha Popovets had the least amount - 3. How many Serpents were shot?

4. The two play the next game. There are 1988 points marked on the plane,

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Being the vertices of a regular 1988-gon. The players take turns connecting two vertices of the polygon with a segment, also observing the rules: you cannot connect points, at least one of which is already connected to the other, and you cannot intersect already drawn segments. The one who cannot make another move according to these rules loses. How should you play to win? Who will win if played correctly?

5. Prove that for any natural number $k$ the number $k^3+5k$ is divisible by 3.

6. In the acute triangle $ABC$ the altitudes $AK$ and $CL$ are drawn. Find the area of ​​triangle $BKL$ if the area of ​​triangle $ABC$ is $S$ and $\cos \angle B=k$.

7. Is the number a perfect square?

$$N = \frac{11...1}{2n \text{ цифр}} - \frac{22...2}{n \text{ цифр}}?$$

8. In a triangle, the lengths of two sides are known: $a$ and $b$. What must be the length of the triangle so that the largest angle of the triangle has the smallest number?

9. Inside the regular triangle $ABC$ lies the point $O$. It is known that $\angle AOB=113^\circ, \angle BOC=123^\circ$. Find the angles of a triangle whose sides are equal to the segments $OA, OB, OC$.

10. On a cell cot, which has the shape of a regular 6-gon with side 1, 7 mice landed. Are there always 2 mice, the distance between which does not exceed 1?

ХIX International Physical Olympiad
(For the beginning, see p. 59)

Thus, Soviet schoolchildren were fully prepared to perform the experimental task “Study of the magnetic field using the ballistic galvanometer method” at the XVI International Olympid, the optical task at the previous Olympid, and the theoretical task “Maxwell’s Pendulum” at the last Olympid.

The Olympiad participants themselves (Soviet schoolchildren) unanimously noted that they performed below their capabilities. One of the reasons is fatigue from the summer harvest. It may be advisable to establish a week’s rest after training before the start of the Olympics. According to the Olympians, the verification of solutions to the problems of the working commission was carried out to a high degree of objectivity; the problems themselves were not very difficult, although interesting.

Some of the dissatisfaction of our team members did not achieve a high performance result was expressed by the amazing beauty of the Alpine mountains and lakes of Upper Austria. Visits to the “city of Mozart” of Salzburg and the beautiful capital of Austria, Vienna, left lasting impressions. The members of the Soviet team will not forget the care and cordiality with which they were surrounded by the hotel owners and especially constantly accompanied by their translators Olga Garant.

At the final meeting of the International Commission, the head of the PPR team, A. Kotlitsky, on behalf of his government, invited the participants and guests of the Olympids to the anniversary XX Olympids in Poland. Interestingly, it was there that the first International Physics Olympiad for schoolchildren took place in 1967.

We wish success to future Soviet Olympids!