[Problem M31—M35; F36—F41]

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All mathematics problems in this issue are taken from those presented at the last Moscow Mathematical Olympiad; some wording should be changed.

F36. A drop of liquid lies on the flat side of a hemispherical glass plate. Show how the refractive index of a liquid can be determined from observing total internal reflection. The refractive index of the stack is also unknown and needs to be determined.

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F37. In two vertically located cylinders, the cross-sectional areas of which are $S_1$ and $S_2$, there are two irreversible pistons connected by a thin irreversible thread of length $l$ (Fig. 1). The space between the pistons is filled with water. Find the tension of the thread if the ends of the vessels are open to the atmosphere. The density of water is $\rho$.

F38. The wide elbow of a U-shaped mercury manometer has a larger diameter than the narrow one. To which elbow should a pressure measurement report scale be attached to ensure greater measurement accuracy?

F39. A planet of radius $R$ and mass $M$ is surrounded by an equally dense atmosphere consisting of gas with molecular weight $\mu$. What is the temperature of the atmosphere on the surface of the planet if the altitude of the atmosphere is $H$?

Physics Olympiad of Moscow State University, 1967

F40. At what potential difference between the electrodes does a neon light bulb light up if the ionization energy of neon is $I=21,5$ eV, and the average distance between two successive collisions of the electrode with gas atoms is 0.4 mc? The electrodes look like large plates located at a distance of $d=3$ mm from each other.

F41. How many times is the security on a moonlit night during a full moon less than on a sunny day, with the same height of the Moon and Sun above the horizon? Assume that the illuminated resulting harrier uniformly scatters light into space. The radius of the Moon is taken to be 2000 km, and the distance from the Moon to the Earth is 400,000 km.

M31. A square sheet of paper is cut straight into two parts. One of the resulting parts is again cut into two parts, and this is done many times. What is the minimum number of cuts that must be made so that among the resulting parts there will be an equal number of dimensions above?

I. Bernishtein

M32. All cells of the table $100 \times 100$ contain pluses. It is allowed to simultaneously change the signs in all cells of one row or one column. Is it possible, by continuing the operations several times, to obtain a table with exactly 1970 minutes?

A. Zelevinsky

M33. There is a natural number $n>1000$. Let’s take the remainders from dividing the number $2^n$ by the numbers 1, 2 3, . . . , $n$ and find the sum of all these remainders. Prove that this sum is greater than $2n$.

A. Kuinirenko

M34. Prove that if a natural number is divisible by 10 101 010 101, then in its decimal notation at least six digits are different from zero.

A. Tolpieo

M35. A certain 19-facet is described near a sphere of radius 10. Prove that you will find two points on its surface, the distance between which is greater than 21.

A. Kuinirenko