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## mathematics

### (10 points)6. Series 31. Year - S. Matrices and populations

Mirek and Lukáš fill matrices with atto-foxes.

### (8 points)5. Series 31. Year - 5. sneaky dribblet

Let's take a rounded drop of radius $ r_0 $ made of water of density $ \rho \_v $ which coincidentally falls in the mist in the homogeneous gravity field $g$. Consider a suitable mist with special assumptions. It consists of air of density $\rho \_{vzd}$ and water droplets with an average density of $ rho\_r $ and we consider that the droplets are dispersed evenly. If a drop falls through some volume of such mist, it collects all the water that is in that volume. Only air is left in this place. What is the dependence of the mass of the drop on the distance traveled in such a fog?

**Bonus:** Solve the motion equations.

Karal wanted to assign something with changing mass.

### (10 points)5. Series 31. Year - S. Differential equations are growing well

Mirek and Lukáš have already grown their algebra, now they have different seeds.

### (6 points)4. Series 31. Year - 3. weirdly shaped glass

We have a cylindrical glass with a small hole at the bottom of the glass. The surface area of the hole is $S$. The glass is filled with water and the water flows into a second glass by itself. The second glass has no holes. What shape should the second glass have so that the water level grows linearly inside it? The glass is supposed to have cylindrical symmetry.

**Bonus:** The bottom of both glasses is at the same high and the glasses are connected by the hole.

Karel was watching how the glass is being filled.

### (7 points)4. Series 31. Year - 4. solve it yourself

We have a black box with three outputs (A, B, and C). We know that it consists of $n$ resistors with the same resistance but we don't know the circuit diagram. So we measure the resistance between each pair of outputs $R\_{AB} = 3 \mathrm{\Omega }$, $R\_{BC} = 5 \mathrm{\Omega }$ a $R\_{CA} = 6 \mathrm{\Omega }$. Your task is to find the minimum possible $n$ and calculate the corresponding resistance of one resistor.

Matěj solved it quickly.

### (3 points)3. Series 31. Year - 1. slowed down

Let's suppose a camera with a frame rate of 24 frames per second (consider evenly spaced and perfectly sharp shots). We record a flight of a helicopter with the rotor rotation velocity of $2 900 \mathrm {cycles/min}$. Then the record is played. What is the apparent rotational velocity of the rotor in the record?

### (10 points)3. Series 31. Year - S. going for a walk with integrals

* We are sorry, this task is not yet translated… *

Mirek and Lukáš random-walk to school.

### (7 points)2. Series 31. Year - 5. raining glass

A worker brought a bag of marbles to a skyscraper construction, to show off in front of his colleagues. But, what an unlucky accident – the marbles pour out and start falling through the scaffolding towards the ground. The scaffolding consists of different levels separated by height $h$. The floor of each level is made out of identical metal grid in which the holes constitute $k \%$ out of the whole grid area. Consider a simplified model of marbles falling through the scaffolding, in which if marble lands in the hole of the grid it goes through unobstructed and if it lands on the solid part of the grid its velocity drops to $0$ and starts to fall down again immediately (i.e. the size of the marbles is insignificant with respect to the size of the holes in the scaffolding and the marbles don't bounce upon landing, instead they stop and immediately roll down into a hole and continue with their fall). Ignore any potential collisions between marbles themselves. If we assume the marbles pour out of the bag with a constant mass flow of $Q$, what is the force on each level of the scaffolding, when the situation comes to a steady state?

Mirek wanted to transfer Ohm's law into mechanics.

### (10 points)2. Series 31. Year - P. ooh Oganesson

What properties does the $118^{\rm th}$ element in the Periodic table have? Alternatively, what sort of properties would it have, had it been stable? Discuss at least three physical qualities.

Karel wanted to have something on extrapolation.