5th Round of 20th year

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1. ... dead pianist( points)

The pianist falls from the skyscraper. During the fall he is playing tone A on piano. Exactly $k$ floor lower is window cleaning man. What is value $k$, if the last thing the window cleaning man heard is tone Ais (half-tone higher then A)? Speed of sound is 347 m·s^{ − 1}, height of one floor is 3,1 m.

2. ... captain Kork again in action( points)

Diary of captain Kork: Star time 51824,2. Future of Star flotilla is endangered again. Romulan's are trying to destroy us, they are attacking us using battleship Karusel equipped with rotating laser gun. Doctor Spok decided that the direct contact is impossible and therefore we have to move as far as possible from the enemy battleship.

However we have no available scientist on the board and you have to figure out how to make a manoeuvre to escape the enemy. Our space ship Enterprise has spherical shape of diameter $R$, at beginning is at distance $r$_{0} from enemy ship. The laser gun of Karusel is rotating at angular speed $ω$. The laser gun works also as a detector of targets (us), and after detectiong the target it fires in the same direction as the target was seen. What is the minimal speed our ship has to have in order to escape to safety?

3. ... resistor sequence( points)

Lets assume you are a director of company which started as first to produce resistors for mass market. From some marketing research it was discovered that demand for resistors is uniformly distributed between 1 Ω –10 MΩ. For technical reasons you can manufacture only finite number of types of resistors, lets say 169.

If customer demands resistor of value $R$_{p} and you offer him resistor of value $R$_{n}, his dissatisfaction will be given by ( 1$~-~R$_{p} ⁄ $R$_{n} ) ². The question is what are optimal 169 values of resistors to make maximum customer satisfaction. To make it more simple assume that first and last resistors from you portfolio must have values 1 Ω  and 10 MΩ.

4. ... Buffalo Bill and Jessie James( points)

Buffalo Bill is searching for Jessie James, well known bandit. Finaly he found him and in town Clay County they meet in fighting. Buffalo noticed a barrel full of petrol located in between both of them on a trolley. But how to get barrel near to Jessie (to light it up)?

Jessie made a hole in the barrel with his gun in 9/10 of height and the petrol is flowing from the barrel. Buffalo hits the barrel exactly in the middle and shoots again. Calculate what is the acceleration of the barrel and trolley at the beginning and how it depend on the position of the second hole from Bill's revolver. Assume the mass of bullet is negligible.

What height is optimal for petrol to splash to longest distance?

P. ... what type of windows?( points)

One of organizers recently upgraded his windows in his house. They are now double-glazed. The space between two glass tables can be filled by inert gas or vacuum. Suggest a method to distinguish between these two types windows (without destroying windows).

E. ... left-handed world( points)

Measure optical activity of glucose solution as it depends on glucose concentration. Optical activity is rotation of polarisation direction of polarised light when passing through the solution. It is directly proportional to the optical path through the solution and depends on wavelength. Try to find/invent/remember how we explain optical activity at molecular level.

Measurement of optical activity is used for measuring sugar concentration in solutions. Is this method reliable? Has each sugar the same optical activity?

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