Evening math school

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The evening mathematics school at the Faculty of Mechanics and Mathematics of Moscow State University (MSH) has been operating for the seventh year. Its main goal is to promote the mathematical development of schoolchildren and expand their mathematical horizons. VMS continues the traditions of mathematical circles that have been working at Moscow University for more than thirty years, and, in essence, is itself a free mathematical circle, covering more than 800 schoolchildren. Admission to all VMS classes is free, and you can join the work of the school throughout the year.

Once a week, 6-9th grade students gather in 26 VMS groups. Classes are conducted in groups by graduate students and students of the Faculty of Mechanics and Mathematics of the university, students of the Moscow Pedagogical Institute. V.I. Lenin, Institute of Steel and other universities. In addition to solving interesting problems, the classes cover various mathematics issues that go beyond the scope of the high school curriculum. Classes are often conducted in the form of a game. Bot example. The manager says: “A cowboy walked into the bar and asked for water. Instead of answering, the owner grabbed his tights and shot at the ceiling. The cowboy thanked him and left. What’s the matter? Are you missing data? Well, I’m ready to truthfully answer some of your questions that apply to this case. Just keep in mind that questions should begin with the words: “Is it true that…” And I will only answer “yes” or “no.” After twenty minutes of questioning and probing, it dawns on someone: “The cowboy got a bone stuck in her throat and jumped out of fright!!!”

Every week, the leading Moskovs give lectures on problems of modern mathematics to students in grades 8-9.

One of the forms of work of the VMS is the “home Olympics”. Every week the children will be given the conditions of 4-5 problems, which will be given two weeks to solve. Written decisions will be shared with the group leadership for review. At the end of the cycle, consisting of 20 tasks, the jury, selected from the group leaders, sums up the results of the competition. At the competitions (with a mark!), all children must demonstrate an understanding of methods for solving competitive problems.

Classes at the Higher School of Music have a good effect on students’ studies. Thus, both Moscow, which hosted the section in the 1969 International Mathematical Olympiad as part of the Soviet team, were students of the Higher School of Music. Many current students of mechanical engineering and physics at Moscow State University and the Institute of Physics and Technology once studied at the Higher School of Medicine. Among the former students of the VMS there are students - chemists, biology, economists, psychologists…

In conclusion, we present some of the competitive problems of the Evening Mathematical School for grades 8-9. Most of them are also available to students in grades 6-7.

Tasks

1. There are 7 identical coins, but among them 5 coins are real and 2 are fake. Real coins weigh 10 e, and counterfeit coins weigh 9.8 e. What is the smallest number of weighings on a cup scale without mushrooms that must be done in order to be sure to identify counterfeit coins?

2. Sasha drew a parallelogram ABCD, marked point M - the middle of side BC, point N - the middle of side CD and went for a walk. The harmful Lila approached the drawing and erased everything except points A, M, N. Help Sasha restore the drawing, that is, find points B, C and D.

3. There are 10 people, dogs and flies in the room. Each person has 2 legs, each dog has 4 and each flour has 6 legs. Everyone in the room has 46 legs. How can this be (Find all possibilities.).

4. Also find two natural numbers whose sum of squares is 16,000. (Indicate all solutions.).

5. The country of Resolventa has a network of cities. Between any two of them there is a direct railway connection, so that from any city to any other you can get along a straight line without passing through other cities. Draw a diagram of this country’s railway communication so that there are as few road crossings as possible. (No more than two roads can intersect in one place.)

6. Given points A and B on the plane. Find the locus of points C on the plane such that triangle ABC is acute.

7. Prove that no power of 2 has four identical digits. Find the power of the number 2 that ends the triangle with the same digits.

8. Serpent Gornych has 2000 heads. The fabulous rich man can knock off 33, 21, 17 or 1 head with one mesh blow, but at the same time the Snake will lose the corresponding 48, 0, 14, 349 heads in return. If all the heads are cut off, no new heads will grow back. Will the rich man be able to satisfy the Serpent?

A. A. Orlov, A. L. Rosenal

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2. $A = \frac{e^2}{d} \approx 14$.

The total energy of an electron in an atom is

$$W = E_{\text{киин}} + E_{\text{пот}} = \frac{mv^2}{2} - \frac{2e^2}{d} = \frac{e^2}{d} - \frac{2e^2}{d} = -\frac{e^2}{d}.$$

3. $T_1 = \frac{T_0}{\left(1 - \frac{P}{p_0 S}\right)\left(1 - \frac{P}{\rho g l S}\right)}$.

4. $a = \frac{mg}{m + H^2 l^2 C}$.

E.m.f. induction occurring in the rod is $E = Hlv$. The charge of the capacitor is $Q = EC = CHlv$, therefore the current in the circuit is $I = \frac{\Delta Q}{\Delta t} = HlCa$ ($a$ is the acceleration of the rod). The magnetic field acts on this current with a force $F = IHl = H^2 l^2 aC$. Now we need to write down the equation of motion of the rod $mg - F = ma$ and solve it for $a$.

To trem in dews in physics

1. The easiest way is to use a calormeter.
2. In the coordinate system associated with the dock, at the moment the chalk hits the dock, it has a speed of $v_1 = u + (-v)$. Since the friction force acting on the chalk and on the dock is directed along the vector $v_1$, it cannot change the direction of the speed of the chalk relative to the dock. When moving, only the relative speed changes. This means that the track on the dock is a straight line making an angle $\alpha$ with the direction of movement of the dock such that

$$tg \alpha = \frac{u}{v}.$$

3. The resistance of the light bulb from a flashlight is small - a few. The resistance of the entire garland is several hundred. Finger resistance is several thousand. When connected in series, the direction differences in the sections of the circuit are proportional to the resistances of the sections; therefore, on your finger, if you put it into the cartridge, you will have almost the entire direction of the network.

To the question “4, 5 and 6”
Can. See figure.

1. Three weigh-ins. It is necessary not only to show in what order the coins should be weighed in order to avoid friction by weighing, but also to prove that two weighings are not enough.

2. You can take advantage of the fact that the middle of the segment $MN$ lies on the diagonal $AC$ and divides it in a ratio of 3:1.

3. Three solutions (not counting one “degenerate” solution, when the number of people is 0).

4. Four solutions. Find out what remainders the number $x^2 + y^2$ gives when divided by 4 depending on the four $x$ and $y$.

5. Nine intersection points.

6. See figure:

7. For example, $2^{39}$ ends with three eights.

8. No. Each time the number of goals changes to a multiple of 3.