Problem M81—M85; F93—F97

30

Tasks

M81. An arbitrary point $P$ is taken inside the square $A_1 A_2 A_3 A_4$. From vertex $A_1$ a perpendicular is dropped to line $A_2 P$, from vertex $A_2$ - to $A_3 P$, from $A_3$ - to $A_4 P$, from $A_4$ - to $A_1 P$. Prove that all four perpendiculars (or their extensions) intersect at one point.

A. N. Vilenkin

M82. There are several identical automatic vehicles on the ring road. It is known that if all the gasoline available in these automatic vehicles were poured into one, then this car would be able to drive along the entire ring road and return to its original place. Prove that at least one of the cars standing on the road can compress all the stakes, taking gasoline from the rest of the cars along the way.

M83. Prove that the numbers 1, 2, …, $n$ for any $n$ cannot be divided into two groups so that the product of numbers in one group is equal to the product of numbers in the other.

M84. Let $A$ be the base of the perpendicular dropped from the center of the given circle onto the given straight line $l$.

On this line two more points $B$ and $C$ are taken so that

$$AB = AC.$$

Two arbitrary secants are drawn through the points $B$ and $C$, one of which intersects the circle at points $P$ and $Q$, the second - at points $M$ and $N$. Let lines $PM$ and $QN$ intersect line $l$ at points $R$ and $S$. Prove that

$$AR = AS.$$

This problem (or some of its variants) is sometimes called the “butterfly problem”; the origin of this name is clear from figure 1.

M85. Prove that if $a_1, a_2, ..., a_m$ are pairwise distinct natural numbers, none of which is divisible by the square of an integer greater than one, and $b_1, b_2, ..., b_m$ are integers other than zero, then

$$b_1 \sqrt{a_1} + b_2 \sqrt{a_2} + \ldots + b_m \sqrt{a_m} \neq 0.$$

L. N. Vaserstein

31

F93. In the side wall of a vessel filled with a liquid with a refractive index $n$, a hole of small radius $r$ continues. A horizontal thin beam of light is directed along the axis of the hole from the vessel. To what level $h$ above the hole must the liquid flow out so that the light ray emerges from the jet without ever experiencing internal reflection?

Neglect the change in the cross section of the jet, and assume the refractive index of the liquid to be sufficiently large.

I. F. Ginzburg

F94. A cycle is performed over one gram-molecule of an ideal gas (a closed process consisting of two isochores and two isobars (Fig. 2)). The temperatures at points 1 and 3 are $T_1$ and $T_3$, respectively. Determine the work done by the gas per cycle if it is known that points 2 and 4 lie on the same isotherm.

F95. The surface of the river forms an inclined plane. Can a body float freely along a river at a speed exceeding the speed of the current or the maximum speed of the current?

F96. Two identical balls connected by thread. Find the rise height of this system if one of the balls is thrown upward with a speed of $v$.

A. Drozdov

F97. A coil of insulated wire is bent in the shape of a figure eight, the bulb of which has a radius of $r_1 = 1$ cm and $r_2 = 3$ cm. The coil is in a magnetic field with induction $B = 10^4$ ec, perpendicular to the plane of the coil.

The wire insulation is designed for a voltage of 10 e. Will insulation breakdown occur if the magnetic field is suddenly turned off? Field switch-off time $10^{-3}$ sec.

L. G. Aslamaov

Fig. 2.