Attention eighth graders! Correspondence mathematics school announces admission of students

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Only high school students are admitted to ZMSH. Schoolchildren living in Moscow, Leningrad and their suburbs are not accepted into ZMSH. School classes will begin on September 1.

ZMSH has two courses. Students who successfully complete school will receive a certificate of completion. School education is free.

In this issue «Kvant» there are tasks that serve as an introductory test for correspondence mathematics schools at Moscow State University and Leningrad State University. Those who want to enter the ZMSH must send the solution to these problems no later than March 10, 1972. After checking your work (around July 1972), you will be informed whether you have been accepted into the ZMSH.

Although some of the introductory problems look different from regular school problems, they do not require any additional knowledge of mathematics to solve.

In order to be accepted at school, it is not necessary to solve every problem without exception. When evaluating work, it will take into account not only the number of problems solved, but also the quality of the solution. The solution to each problem must be justified. The answer without any further explanation may not be read. If there are several possible answers to a problem, then all of them must be indicated.

Work must be completed in Russian in a student’s notepad in Klatek. Entry papers will not be processed.

Please do not roll the terradi into a tube when shipping. In the envelope along with the letter you need to put a piece of paper measuring 14x6 cm with your postal address written in it (we will stick it on the envelope when we send the answer).

Glue a sheet of checkered paper onto the cover of the terradi, draw it and fill it out according to the following sample (otherwise your work will not be checked!):

Region:
Last name, first name, year of birth:
School (full name):
Class:
Last name, first name, absent mathematics teacher:
Place of work and position of the parent:

Full postal address:

Inspection results (to be filled in by the inspector)

№ 1 2 3 4 5 6 7 8 9 10 11 12

Schoolchildren living in the Arkhangelsk, Vologda, Kaliningrad, Leningrad, Murmansk, Novgorod, Pskov regions, Komi and Karelian ASSR, Belarus, Latvian, Lithuanian and Estonian SSR must send work to the address: Leningrad, P-228, st. Savushkina, 61, Specialized School at Leningrad State University. Zaonaya school. To the competition.

Schoolchildren living in other regions and republics of the USSR must send their work to the following address: Moscow, V-234, Moscow State University, Mechanical Engineering, ZMSh. To the competition.

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Problem conditions
introductory work control
in 3MSh 1972

1. A motorcycle race is held on the ring road. The start and finish are in the same place. What is the smallest number of stages in this race if the length of the ring road is 330 km, and the length of each stage is 75 km? (The road is one-way.)
2. In a triangle, the centers of the inscribed and circumscribed circles are symmetrical to one of the sides. Find the angles of the triangle.
3. Let $x_1$ and $x_2$ be the roots of the equation $x^2+ax+bc=0$, and $x_1$ and $x_3$ the roots of the equation $x^2+cx+ab=0$. Prove that $x_2$ and $x_3$ are roots of the equation $x^2+bx+ac=0$ if $ac$ is not equal to $bc$.
4. Two people play this game. In front of him, several minutes are drawn on paper in a chain. Each person takes turns transporting one or two neighboring minutes onto the plane. The one who crosses the last minute wins. Who will win when running the game: the beginner or his partner and how much he needs for this game, if at the beginning it is drawn: a) 7 minutes; b) 8 minutes; c) n minutes?
5. In triangle ABC, bissertrix AK and median AM are drawn. What can the ratio of the sides AB and AC be equal to if it is known that one of the segments BM, MK, KC is equal to half the sum of the other two?
6. In what year was Matvey born, if in 1972 he turned that age, what is the sum of the digits of the year of his birth?
7. Is there at least one number $a$ such that both numbers $a+V\sqrt{15}$ and $\frac{1}{a}-V\sqrt{15}$ are integers?
8. $AC$ and $BD$ are two mutually perpendicular chords of a circle. A perpendicular dropped onto line $CD$ from point $A$ intersects line $BD$ at point $M$, and a perpendicular dropped onto line $CD$ from point $B$ intersects line $AC$ at point $K$. Prove that $ABKM$ is a rhombus.
9. One of the brothers’ heels broke the window. Andrey said: “It’s either Vitya or Tolya.” Vitya said: “It was not Yura who did this.” Tolya said: “You both are telling lies.” Dima said: “No, one of them told the truth, and the other didn’t.” Yura said: “No, Dima, you’re wrong.” Their father, who of course can be trusted, is sure that at least three brothers told the truth. Who broke the window?
10. Among all the triangles whose sum of medians is 3 cm, find the triangle with the largest sum of heights.
11. Come up with four non-negative integers so that each number from 1 to 81 can be represented as the sum of four numbers - one from each triple.
12. Given an equilateral triangle ABC. About the point $M$ it is known that $\angle AMC = 30^\circ$, $\angle BMA = 40^\circ$. $\angle ABM > 90^\circ$. Find $\angle MBC$.