Problems

Is it possible to cover a \(6 \times 6\) board with the \(L\)-tetraminos without overlapping? The pieces can be flipped and turned.

Is it possible to cover a \((4n+2) \times (4n+2)\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.

Is it possible to cover a \(4n \times 4n\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.

There are \(100\) people standing in line, and one of them is Arthur. Everyone in the line is either a knight, who always tells the truth, or a liar who always lies. Everyone except Arthur said, "There are exactly two liars between Arthur and me." How many liars are there in this line, if it is known that Arthur is a knight?

Find all the prime numbers \(p\) such that there exist natural numbers \(x\) and \(y\) for which \(p^x = y^3 + 1\).

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?