Problems

Prove that the set of all real numbers is not countable.

Prove the triangle inequality: in any triangle \(ABC\) the side \(AB < AC+ BC\).

In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than \(5\) planes?

Find all \(n\) such that a closed system of \(n\) gears in a plane can rotate. We call a system closed if the first gear wheel is connected to the second and the \(n\)th, the second is connected to the first and the third, the third is connected to the second and the fourth, the fourth is connected to the third and the fifth, and so on until the \(n\)th is connected to the \(n-1\)th and the first. In the picture, we have a closed system of three gears.

Show that \(\sqrt[3]{3}\) is irrational.

Suppose you meet a person inhabiting this planet and they ask you “Am I a Goop?" What would you conclude?

On this planet you meet a couple called Tom and Betty. You hear Tom ask someone: “Are Betty and I both Goops?"
What kind is Betty?

You learn that one of the aliens living on this planet is a wizard. You learnt that by overhearing a certain question being asked on the planet. What question could that have been?

Suppose you meet a person inhabiting this planet and they ask you “Am I a Crick?" What would you conclude?

You meet two friends, Katja and Anja. Katja once asked Anja “Is at least one of us a Goop?"
What kinds are Katja and Anja?