Problem Statement of Series 6, Year 39

Marek has come up with a get-rich-quick scheme. He needs the money to buy Blue socks with the FYKOS logo available on the FYKOS e-shop in all sensible sizes at reasonable prices. Marek wants a whole lot, though. He therefore buys gold, starts running, and after reaching sufficient speed, sells it back. Due to relativistic effects, the stationary merchant perceives the gold to be heavier, so Marek makes a profit. How fast does Marek have to run for the scheme to work if $p$ is the buy–sell ratio?

The arrival of daylight saving time is here. And with it, it comes the time to get fit for summer. Pepa heard that the body burns some calories simply by existing. Instead of changing to a more active lifestyle, he decided to keep eating—but slowly. Determine how quickly he would need to eat a slice of toast so that during that time his body expends the same amount of energy as is contained in the slice.

However, Pepa decided that he would like to eat the toast slices at twice the above frequency and is willing to work for it. How many times per minute must he stand up from a chair in order to be calorically balanced again?

David is interested in determining the velocity he would acquire in free space by emptying his bladder of volume $V=0.7\,\mathrm{L}$. When urinating on Earth, the droplets land at a horizontal distance $l=1\,\mathrm{m}$ away from him. The outlet of his urethra is at a height $h = 0.8\,\mathrm{m}$ and is oriented parallel to the ground. His center of mass is located at the same height. David's mass is $80\,\mathrm{kg}$.

A confused hornet flies into Viktor's room. In panic, he immediately runs outside to find a firearm and slams the door behind him. What is the probability that, within 50 seconds—before Viktor prepares the air rifle—the hornet exits through the window and thus saves its life?

For simplicity, assume the hornet moves in straight lines between walls at a speed of $2\,\mathrm{m\cdot s^{-1}}$, and upon reaching a wall, it changes direction randomly. The room has dimensions $3 \times 4 \times 2\,\mathrm{m}$, and the open window occupies one half of one of the shorter walls.

The cap, which we can approximate as a cylinder with a rough surface, has mass $m$, diameter $d = 20\,\mathrm{cm}$, and its axis of rotation is at a fixed distance $d/4$ from the cylinder's axis of symmetry (this offset is caused by the visor). He spun the cap with a sufficient angular velocity $\omega$ and threw it horizontally in front of him.

Due to the Magnus effect caused by wind blowing horizontally (parallel to the ground and to the initial trajectory of the cap) with speed $v$, the cap rose upward and landed on the desired branch.

Create a function that determines whether Matyáš's chosen method is more advantageous than simply standing up. When standing up, his center of mass rises to a height $h/2$; Matyáš has mass $M$. Plot this function graphically.

This problem has an open solution, so be sure to cite all sources used.

Consider a hypothetical opposite of a pulley – let us call it a pushey. Describe, from as many different physical perspectives as possible, how a pushey might work and what properties it could have.

Experimentally determine the volume of an $n$-dimensional sphere, that is, the set of all points in $n$-dimensions that are equidistant from a common center.

To write the solution of the serial problem is rather trivial. Think of a topic and write the serial text instead.