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## mathematics

### (9 points)3. Series 34. Year - 5. smuggling in space

Two spaceships move towards each other on a straight line. The initial distance between them is $d$. The first one moves with the velocity $v_1$, the second with the velocity $v_2$ (in the same reference frame). The first one can reach the maximal acceleration $a_1$, the second one $a_2$ (both regardless of the direction). Their crews want to exchange some „goods“. In order to do that, the spaceships need to meet – i. e. they must be at the same time at the same place and have the same speed. What is the minimal time for them to reach the meeting? Neglect the relativistic effects.

### (8 points)3. Series 33. Year - 4. ladybird on a rubber

Ladybird moves with velocity $4 \mathrm{cm\cdot s^{-1}}$. When we place the ladybird onto a rubber, she comes through it in $10 \mathrm{s}$. What happens when the ladybird starts moving and we start prolonging the rubber the way that its length will be increasing with velocity $5 \mathrm{cm\cdot s^{-1}}$? Is the ladybird able to come through the whole rubber to its end? If yes, how long will it take? Consider that the rubber prolongs uniformly and never breaks.

Matej was watching Vsauce.

### (10 points)2. Series 33. Year - S.

* We are sorry. This type of task is not translated to English. *

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

- Simulate the dynamics of a predator-prey system using Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} &= r\_x x - D\_x xy ,\\ \frac{\d y}{\d t} &= r\_y xy - D\_y y . \end {align*}\] where $x$ and $y$ are the population sizes of prey and predator respectively, the parameters $r\_x$ and $r\_y$ represent the populations’ growth and the parameters $D\_x$ and $D\_y$ represent the shrinking of the populations. Set the parameters to be $r\_x = 0{.}8$, $D\_x0= 1{.}0$, $r\_y = 0{.}75$, $D\_y = 1{.}5$. Run the simulations for several different value pairs for initial population sizes $x = 0{.}5$ and $y = 2{.}0$; $x = 1{.}5$ and $y = 0{.}5$; $x = 1{.}95$ and $y = 0{.}75$. Plot the predator population size as a function of the prey population size. Discuss the results.

**Bonus:**Find the solutions for the same situations analytically (by integrating the differential equations). - Using the competitive Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} = r\_x x \(1 - \(\frac {x + I\_{xy} y}{k\_x}\)\) ,

\frac{\d y}{\d t} = r\_y y \(1 - \(\frac {y + I\_{yx} x}{k\_y}\)\) . \end {align*}\] simulate the dynamics of two competing populations (e.g. hawks and eagles) for the following values of parameters: $r\_h = 0{.}8$, $I\_{he} = 0{.}2$, $k\_h = 2{.}0$, $r\_e = 0{.}6$, $I\_{eh} = 0{.}3$, $k\_e = 1{.}0$. Set the initial population sizes to be $h = 0{.}01$, $e = 1{.}0$. Then, simulate the same situation, but change the interaction coefficients to $I\_{he} = 1{.}5$ a $I\_{eh} = 0{.}6$. Plot the results in one graph - the sizes of populations vs time. Discuss the results. - Verify the importance of pivoting.

Solve the system of linear equations \[\begin{equation*} \begin{pmatrix} 10^{-20} & 1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \end {equation*}\] at first exactly (on paper), then using LU factorization with partial pivoting (you may utilize some Python module, e.g.`scipy.linalg.lu()`

), and finally, solve the system using LU factorization without pivoting. Compare the resultant $\vect {x}$ obtained from the three methods and the results of matrix multiplication $L^{-1}\cdot U$ ($P\cdot L^{-1}\cdot U$ in the case of pivoting). - Consider an infinite parallel-plate capacitor. The gap between plates has a thickness $L=10 \mathrm{cm}$ and the voltage between the plates is $U=5 \mathrm{V}$. Between the plates of the capacitor grounded electrode in the shape of an infinitely long prism with square base of side length $a=2 \mathrm{cm}$, whose center lies $l=6{,}5 \mathrm{cm}$ away from the grounded plane of the original capacitor. The prism is oriented such that one of its short sides is perpendicular to the capacitor plates. Find the distribution of electric potential in the condensator. Since the problem has a translational symmetry in the direction of the infinite side of the prism, it is sufficient to solve it only in the plane parallel to the plates, i.e. it is a 2D problem. Render the potential distribution in this plane. You may utilize the code attached to this task.

**Bonus:**Calculate and render the distribution of the electric field strength $\vect {E}$.

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

- Solve the two-body problem using the Verlet algorithm and the fourth-order Runge-Kutta method (RK4) over several (many) periods. Use a step size large enough for the numerical errors to become significant. Observe the way the errors manifest themselves on the shape of the trajectories.
- Solve for the time-dependent position equation of a damped linear harmonic oscillator described by the equation $\ddot {x}+2\delta \omega \dot {x}+\omega ^2 x=0$, where $\omega $ is the angular velocity and $\delta $ is the damping ratio. Change the parameters around and observe the changes in the oscillator’s motion. For which values of the parameters is damping the fastest?
- Model sedimentation using the method of ballistic deposition \[\begin{equation*} h_i(t+1) = max(\(h_{i-1}(t), h_i(t)+1, h_{i+1}(t)\)) \, , \end {equation*}\] where $h_i$ is the height of i-th column. And study the development of the roughness of the surface $W(t,L)$ (see this year’s series 4, problem S). Initially (for small values of $t$) the roughness is proportional to some power of $t$: $W(t,L) \sim t^{\beta }$. For large values of $t$, however, it is proportional to some (possibly different) power of the grid length $L$. $W(t,L) \sim L^{\alpha }$. Find the powers $\alpha $ and $\beta $. Choose an appropriate step size so that you could study both modes of sedimentation. The length of the surface should be at least $L = 256$. (Warning: the simulations may take several hours.)
- Simulate on a square grid the growth of a tumor using the Eden growth model with the following variation: when a healthy and an infected cell come into contact, the probability of the healthy one being infected is $p_1$ and the probability of the infected one being healed is $p_2$. Initially, try out $p_1 \gg p_2$, the proceed with $p_1 > p_2$ and then with $p_1 < p_2$. At the beginning, let only 5 cells (arranged into the shape of a cross) be infected.

Describe qualitatively what you observe. - Rewrite the attached code for the growth of a fractal (diffusion limited aggregation model) on a hexagonal grid to the growth of a fractal on a square grid and calculate the dimension of the resultant fractal.

**Note:** Using the codes attached to this task is not mandatory, but it is recommended.

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. a walk with integrals

- Propose three different examples of Markov chains, at least one of which is related to physics. Is a random walk without backtracking (a step cannot be time reversed previous step) an example of Markov chain? What about a random walk without a crossing (it can lead to each point at most once)?
- Consider a 2D random walk without backtracking on a square grid beginning at the point $(x,y) = (0,0)$. It is constrained by absorbing states $b_1\colon y = -5$, $b_2\colon y = 10$. Find the probability of the walk ending in $b_1$ rather than in $b_2$.
- Simulate the motion of a brownian particle in 2D and plot the mean distance from the origin as a function of time. Assume a discrete time and a constant step size. (One step takes $\Delta t = \textrm{const} $, and the step size is $\Delta l = \textrm{const} $). A step in any arbitrary direction is possible, i.e. every step is described by it’s length and an angle $\theta \in [0,2\pi )$, while all directions are equally probable. Focus especially on the asymptotic behavior, i.e. the mean distance for $t \gg \Delta t$.
- Error function is defined as \[\begin{equation*} {erf}(x)=\frac {2}{\sqrt {\pi }}\int _0^x \eu ^{-t^2} \d t . \end {equation*}\] Calculate the integral for many different values of $x$ and plot it’s value as a function of $x$. What do you get by numerically deriving this function?
- Look up the definition of Maxwell-Boltzmann probability distribution $f(v)$, i.e. the probability distribution of speeds of particles in an idealized gas. Utilizing MC integration calculate the mean value of speed defined as \[\begin{equation*} \langle v\rangle = \int _0^{\infty } vf(v) \d v , \end {equation*}\] Use the Metropolis-Hastings algorithm for sampling the Maxwell-Boltzmann distribution. Compare the values of particular parameters with the values from literature.

Mirek and Lukáš random-walk to school.