Problems

Find all natural numbers \(n\) for which there exist integers \(a,b,c\) such that \(a+b+c = 0\) and the number \(a^n + b^n + c^n\) is prime.

Find all the prime numbers \(p\) such that the number \(2p^2+1\) is also prime.

Show that a bipartite graph with \(n\) vertices cannot have more than \(\frac{n^2}{4}\) edges.

In a graph \(G\), we call a matching any choice of edges in \(G\) in such a way that all vertices have only one edge among chosen connected to them. A perfect matching is a matching which is arranged on all vertices of the graph.
Let \(G\) be a graph with \(2n\) vertices and all the vertices have degree at least \(n\) (the number of edges exiting the vertex). Prove that one can choose a perfect matching in \(G\).

A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?

Now imagine you got \(10\) new guests arriving to the completely full hotel. What should you do now?

The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers \(1,2,3...\) corresponding to all natural numbers. How to deal with this one?

Imagine you have \(2\) new guests arriving to the full hotel. How do you accommodate them?

What would you do about \(10000\) new guests arriving to the full hotel?

Imagine you have now a general finite number of new guests arriving to the full hotel. What do you do?