[Problem F42—F47; M36—M40]

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PROBLEM WORKER

Chapter 1

Fig. 1.

F42. The rope will be thrown over the block, with part of the rope lying on the table and part on the floor. After the rope was released, it began to move. Find the speed of steady uniform motion of the rope. The height of the table is h.

F43. Find the pressure at the center of a liquid of radius R if the liquid is incompressible and has a density ρ.

Weighted Physics Olympiad, 1969

F44. Two dolphins swim towards each other. One of them makes a sound of frequency ω. What frequency does the other dolphin hear if the speeds of the dolphins relative to the water are the same and equal to v? Speed ​​of sound in water and.

Weighted Physics Olympiad, 1968

F45. A spherical capacitor filled with a dielectric and charged to a certain potential difference is discharged through its dielectric. What will be the magnetic field of the discharge currents in the space between the spheres?

F46. A cube falls onto a ball from a height of 1 m, and then jumps almost 1 m. To what height does the ball jump? To what height will the ball jump if exactly the same ball falls on it from a height of 1 m?

G. L. Komkin

F47. If the ABC process (Fig. 1) is performed on an ideal gas, then the amount of heat Q is imparted to it. What amount of heat is imparted to the gas during the ADC process?

Moscow City Physical Olympiad, 1970

M36. Prove that it is impossible to arrange seven lines and seven points on a plane so that three lines pass through each point and three points lie on each line.

M37. A certain number is inscribed in each cell of an infinite sheet of checkered paper so that the sum of the numbers in any square (the sides of which follow the grid lines) does not exceed one in absolute value. Prove that then there exists a number c such that the sum of the numbers in any rectangle (whose sides go along the lines

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mesh) no more than $c$; in other words, that the sums of numbers in all rectangular iks are limited. Prove that it is possible to take $c=4$. Perhaps you can incriminate this estimate (for example, prove that our statement is true for $c=3$ or even for $c=2$)?
Yu. I. Ionin

M38. A circle constructed at the height $AD$ of a right triangle $ABC$ as a diameter intersects leg $AB$ at point $K$, and leg $AC$ at point $M$. Line segment $KM$ intersects height $AD$ at point $L$. It is known that the segments $AK$, $AL$ and $AM$ form a geometric progression (i.e. $AK/AL=AL/AM$). Find the acute angles of triangle $ABC$ (Fig. 2).
L. M. Lopovok

Fig. 2.

M39. Prove that non-negative integers $x, y$ satisfy the equation
$$x^2 - mxy + y^2 = 1$$
(where $m$ is a given integer greater than one) if and only if $x$ and $y$ are adjacent numbers of the sequence

$$\varphi_0 = 0, \quad \varphi_1 = 1, \quad \varphi_2 = m,$$
$$\varphi_3 = m^2 - 1, \quad \varphi_4 = m^3 - 2m,$$
$$\varphi_5 = m^4 - 3m^2 + 1, \quad \dots,$$
in which $\varphi_{k+1} = m\varphi_k - \varphi_{k-1}$ for all $k \geqslant 1$.

For example, all solutions to the equation $x^2 - 3xy + y^2 = 1$ for the number $0 \leqslant x < y$ are pairs $(0, 1)$; $(1, 3)$; $(3, 8)$; $(8, 21)$ adjacent members of the sequence $0, 1, 3, 8, 21, 55, \dots$ defined by the conditions $\varphi_0 = 0, \quad \varphi_1 = 1, \quad \varphi_{k+1} = 3\varphi_k - \varphi_{k-1}$ for all $k \geqslant 1$. (This red fact was used in the work of Yu. Matiyasevich related to Hilbert’s 10th problem, which was discussed in the previous issue «Kvant».)

M40. Find the amount
$$1 \cdot n + 2 \cdot (n - 1) + 3 \cdot (n - 2) + \dots$$
$$\dots + (n - 1) \cdot 2 + n \cdot 1.$$

Try solving the following, more general problem: find the sum
$$S_n, k = [1 \cdot 2 \dots k] \times$$
$$\times [n(n - 1) \dots (n - k + 1)] +$$
$$+ [2 \cdot 3 \dots (k + 1)] \times$$
$$\times [(n - 1)(n - 2) \dots (n - k)] +$$
$$+ [3 \cdot 4 \dots (k + 2)] \times$$
$$\times [(n - 2)(n - 1) \dots (n - k - 1)] + \dots$$
$$\dots + [(n - k)(n - k + 1) \dots (n - 1)] \times$$
$$\times [(k + 1)k \dots 2] +$$
$$+ [(n - k + 1)(n - k + 2) \dots n] \times$$
$$\times [k(k - 1) \dots 1].$$

V. N. Berezin