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## mathematics

### 1. Series 25. Year - 2. struggling swimmer

A man wants to swim across a river which flows at a speed of 2 km/h. He is able to swim at a speed of 1 km/h. What is the optimal trajectory and direction he should take so that his trip is the least exhausting? Where and at what time will he reach the other bank? How would the situation change if his aim was the shortest possible trajectory? The width of the river is $d=10\;\mathrm{m}$.

Petr

### 5. Series 24. Year - S. aviation

• How would you draw (using only drawing-compass and a ruler) the Zhukovsky profil?
• Draw the streamlines around the Zhukovsky profile. Choose the parameters $d/l$ and $m/l$ so that they have real world justification.
• What is the lifting force acting on a square board? On a semicircular board?
• Draw the profile of a wing that corresponds to the Kármán–Trefftz transform.

Jakub

### 1. Series 24. Year - S. complex warm-up

• Convince yourselves that the nth roots of a complex number of modulus one lie on a regular $n-gon$ and solve the Bombelli equation $x^{3}-15x-4$ = 0. (see the text for hints)
• Express the identities concerning sin(α+β) and cos(α+β) using the complex exponential.
• Show that we were allowed to neglect the higher powers in deriving the Bernoulli limit, i.e. show that it was legitimate to add the o(1/$N)$ term inside the parentheses.
• Use the little-o notation to solve the problem of small oscillations around equilibrium point in Yukawa potential $V$ = $k \exp(x/λ) /$ $x$.
• Prove that the Chebyshev polynomials cos($n$ arccos $x)$ are really polynomials.

Hint: Let's have a unit complex number $z$ with real part $x$. Then, the expression is equal to the real part of $z^{n}$.

Jakub Michálek a Lukáš Ledvina

### 5. Series 22. Year - 1. turning a carpet

Choosing two different vectors in a plane and shifting the origin, an infinite grid of nodes can be created (see picture). Using the same approach in 3D will make a crystal lattice. If such grid is shifted by one of the vectors, we will get identical grid. Also rotation of the grid by some angle will generate identical grid. Find out all angles which can be basis for rotationaly symetrical grid and draw how such grid looks like.

### 1. Series 22. Year - 3. do not cradle me

Kathy is on a swing (a plank suspended on 2 ropes). At high displacements she kneels down, in lowest point she gets up. This movements she periodically repeats. Ration of distance of centre of gravity from the rotation axis at kneeling down and at standing is 2^{1 ⁄ 12} ≈ 1,06. How many times Kathy must swing to double the amplitude of swinging?

Z asijské olympiády přinesl Honza Prachař

### 5. Series 21. Year - S. sequence, hot orifice and white dwarf

• Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

• The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

• Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.

### 4. Series 21. Year - 1. bees and geometry

When you look at honeycomb you can admire its periodic structure. In a cut the walls cell form regular hexagons and fill whole plane.

Why do the bees make cells as hexagons? Why not for example rectangle of pentagon?

Zadal Honza Prachař inspirován knihou Matematika kolem nás.

### 4. Series 21. Year - P. project 5

Suggest a shape of the most fairness cube of 5-sides. We mean to find such 5-sided object, where the probability of stopping on each side is same for all sides.

Vymysleli Aleš Podolník a Marek Scholz.

### 4. Series 21. Year - S. quantum harmonic oscillator

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called coherent stat of harmonic oscillator and wave packet should oscillate around origin with angular frequency √( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

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