Physics and mathematics boarding schools

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PHYSICS AND MATHEMATICS BOARDING SCHOOLS

Six years ago, physics and mathematics integrative schools were created at the four largest universities in the country. Their task is to help teenagers from rural areas and peripheral cities find their way to mathematical and physical science. Of course, no one promises everyone to be accepted into the integraat and that they will become scientists. But here they train people who, to the best of their ability, will participate in the science program.

The Leningrad Integraat accepts children from the Latvian, Lithuanian and Estonian SSR and the rural-western regions of the RSFSR, the Novosibirsk - from the regions of Siberia and the Central Azatian resource areas, the Kiev - from Ukraine and Moldavin. The Moscow Integraat accepts students from the central regions of the RSFSR and Belarus.

The selection of candidates and exams for integrats begin in the regions simultaneously with physics and mathematics Olympiads (usually during the spring holidays). Exact data and admission procedures can be found in the individual public education departments.

Let’s take a closer look at the Moscow Integraat. It accepts students in grades 9 and 10. Most of his students, through a general competition, enter the mechanics, mathematics or physics department of Moscow University or the Moscow Institute of Physics and Technology. Many students of the first graduating class have already entered the department, and many others have published integrated independent works as students.

We rejoice if any of our students show signs of being particularly talented, but we do not strive to catch only exceptional talents. All applicants are required to have only a passion for science and a willingness to work much harder than is sometimes customary for high school students, who find learning relatively easy and are more interested in football or dancing. However, both our students and young teachers are involved in football and tourism. Between the ninth and tenth grade, children voluntarily go to work on state farms in the Caucasus and Crimea and travel after working weeks.

Lectures at Integraat are given by professors and teachers from Moscow University and the Institute of Physics and Technology, and classes are largely taught by teachers, graduate students and the most capable students of these types. The equipment of physics laboratories (there are many of them, in different areas of physics) makes it possible to conduct server work, up to and including performing relatively simple tasks for scientific institutes.

There is quite a lot of competition in the Integraats, but the level of requirements is not so high that every loving mathematician and physics student would not hesitate to try to get into them (see, for example, the tasks given on the next page for those entering the 9th and 10th grades at the Moscow Integraat in 1969). In the integrations themselves, in addition to qualified teaching, you will find a friendly environment of capable and enthusiastic young mathematicians and physicists - an environment in which it is especially pleasant and fun to work.

Teachers of the Moscow Integraat
A. N. Kolmogorov, V. A. Gusev,
A. A. Egorov, E. L. Surkov

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TASKS
ON WRITTEN EXAMINATIONS

Option
for those who built
to 9th grade

1. Solve system equation
   $$\begin{cases} x^3 + y^3 = 9, \\ x^2y + y^2x = 6. \end{cases}$$

2. In triangle ABC, points P, Q and R are the bases of the altitudes dropped from vertices A, B and C respectively. Prove that $\angle ABQ = \angle APR$.

3. In what year were people born who were so old in 1969, what is the sum of the numbers of their year of birth?

4. Why is fog, consisting of drops of transparent water, not transparent?

5. There is a glass of water on the outside scale. A plug of volume V and density p floats in water. The cork is pushed under the water with a finger so that it is completely submerged. What will the scales show?

Option
for those who built
to 10th grade

1. Solve system equation
   $$\begin{cases} x(y + z) = 3, \\ y(x + z) = 4, \\ z(x + y) = 5. \end{cases}$$

2. In the trapezoid ABCD with bases AD=a, BC=b (a>b), a segment MN is drawn parallel to the base and dividing the area of ​​the trapezoid in half. Find the length of this segment.

3. Find all pairs of integers x and y satisfying the equation
   $$x^2 - 6xy + 5y^2 = 11.$$

4. At the bottom of one of the communicating vessels lies a piston of weight P. The force of its friction against the walls is equal to F. How much water must be poured into the other vessel for the piston to move to a height h?

5. The box is placed onto the conveyor belt without an initial speed. The belt moves with speed v. Friction coefficient k. How far will the box travel relative to the tape until its speed becomes equal to v?

6. Find the sum of the squares of the roots of the equation without solving it:
   $$x^2 + px + q = 0.$$

7. Which of the two numbers is greater: $2^{300}$ or $3^{200}$?

8. The graph of the quadratic trinomial y=ax²+bx+c is given. What are the signs of the numbers a, b and c?

9. Prove that the sum of the medians of a triangle is less than the perimeter and greater than $\frac{3}{4}$ the perimeter of this triangle.

10. Trapezine bases a and b (a>b). Find the length of the segment parallel to the bases of the trapezoids, passing through the point of intersection of the diagonals and enclosed between its sides.

11. Prove that circles constructed on the sides of a convex quadrilateral as diameters will completely cover the quadrilateral.

12. Two circles of radii r₁ and r₂ are tangent to a given line. Find the locus of the points of intersection of their common internal tangents, provided that the circles can move along a given straight line in an arbitrary manner.

13. A piece of ice floats in a glass of water with a piece of pork embedded inside. The ice has melted. How will the water level in the vessel change?

14. The cannon fires at an angle α. Initial projectile speed v. A screen is placed at a distance L from the gun, from which the projectile is elastically reflected. At what distance from the gun will it fall?