# 6th Round of 20th year

Series

### 1. ... three cylinder and friction(0 points)

Find the condition of equilibrium (no movement) for the 3 cylinders in figure 1. The cylinders are made of material of density $ρ$, the radius of bottom cylinders is $R$, the radius of top cylinder is $r$. The coefficient of friction is same for all surfaces.

### 2. ... strange atmosphere( points)

The refractive index of atmosphere of a planet of radius $R$ is can be describe by following equation: $n~$=$~n$_{0}$~-~αh$. Calculate, how high above the surface of the planet $h$ the laser beam will follow circular trajectory around the planet.

### 3. ... the square the Joker(0 points)

The electric circuit in the image is made from the infinite number of wire-frame-squares. The next square is exactly sqrt(2) smaller then the previous. The resistance of used wire is $R$ per one side of the largest square. What is electrical resistance between most left and most right point?

### 4. ... binary star( points)

The magnitude of binary star is changing with 4-day period with this sequence:

side minimum: m = 3.5

maximum: m = 3.3

main minimum: m = 4.2

maximum: m = 3.3.

The larger part of binary star has higher temperature then its escort. Assuming, that the Earth is in the orbital plane of binary star, calculate magnitudes of each component and ratio of its linear dimensions. Relation between magnitude $m$ and irradiation $E$ is given by

$m~$=$~-~$2$,$5 log ( $E$ ⁄ $E$_{0} ) $,$

where $E$_{0} is some defined value.

### P. ... how the UFO-man is looking?( points)

In this problem question think about and animal, which communicate at radio frequency (10 Hz –100 MHz). Suggest an biological equivalent of necessary radio-electronic components.

### E. ... blowing experiment( points)

Measure what is the maximum pressure (and underpressure) which can be made by your mouth by blowing and sucking air.

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