Problem M96—M100; F108—F112

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TASKS

Solutions to problems from the “Kvant Problem Book” can be sent no later than one and a half spaces after the corresponding issue of the journal goes out of print. The solution to each problem must be written on a separate sheet(s); at the end of each decision you must indicate your last name, first name and patronymic; six-digit postal code, home address; the class and school you attend.

M96. It is known about five positive numbers that if from the sum of any three of them we subtract the sum of the remaining two, then the difference will be positive. Prove that the product of all ten such differences does not exceed the square of the product of the given heel of numbers.
S. T. Berkolaiko

M97. In the trapezoid $ABCD$ with bases $AB=a$ and $CD=b$ there is a segment $A_1B_1$ connecting the midpoints of the diagonals. In the resulting trapezoid $A_1B_1CD$, the segment $A_2B_2$ is again drawn, connecting the midpoints of the diagonals, and so on (Fig. 1). Is it possible for a number to appear twice in the sequence of segment lengths $AB, A_1B_1, A_2B_2, \dots$? Will this sequence be monotonic (increasing or decreasing)? Is she aiming for some kind of limit?
A. L. Rosenmal

M98. Prove that in the table

1
1 1 1
1 2 3 2 1
1 3 6 7 6 3 1

where each number is equal to the sum of the three above it, each line (starting from the third) has an even number. Does each line (except the first two) contain a number divisible by three?

M99. In a triangle, $ABC$ side $AC$ is the largest. Prove that for any point $M$ of the plane $AM+CM$ is at least $BM$. In what cases is equality possible?

N. B. Vasiliev

Fig. 1.

Fig. 2.

M100. Prove that the sum of 45 numbers $tg 1^\circ + tg 5^\circ + tg 9^\circ + \dots + tg 173^\circ + tg 177^\circ$ is 45.
V. P. Beshkarev

F108. Ten ants decided to remove a straw from the table that was lying on it. How should they build if the force with which each of the ants can pull away the straw is slightly less than $1/10$ the frictional force acting on the straw when it is shuffled around the table? An ant cannot lift a straw either.
G. L. Komkin

F109. In experiment, it was found that the temperature of 142 °C ice water in a light state, suspended in the middle of the room, rose by 4 °C in half an hour. When there is ice in the vessel, it took 10 hours to pull it out. Based on this experiment, what is the specific heat of fusion of ice? The specific heat of water is 1 cal/e.

F101. If the voltmeter is connected in parallel to the upper resistance $r_1$ (Fig. 2), then it will show 6 $\theta$, if in parallel to the lower resistance $r_2$, then 4 $\theta$, and if it is connected to points $A$ and $B$, then it will show 12 $\theta$. What is the actual voltage drop across the resistances?

F111. There is a homogeneous cord with an explosive substance. The speed of propagation of the explosion reaction along the cord is $v$, the speed of propagation of the blast wave through the air is $c$. Find the shape of the line along which you need to place the cord so that the waves from all points of the cord arrive at a given point simultaneously.

Is it possible to do the same with an explosive surface and get a descending spherical wave with a high energy density?

F112. There is a stack of $k$ plane-parallel plates whose refractive indices are $n_1, n_2, \dots, n_k$. The thickness of each plate is $d$. How much will a ray incident on a plate with a refractive index of $n_1$ at an angle of $\alpha$ to the surface of the plate be displaced after passing through a stack of plates?