Is it possible to find a way of arranging numbers from \(0\) to \(9\) in a row so that each pair of consecutive numbers adds up to a multiple of \(5\), \(7\), or \(13\)?
A conference was attended by a finite group of scientists, some of whom are friends. It turned out that every two scientists, who have an equal number of friends at the conference, do not have friends in common. Prove that there is a scientist who has exactly one friend among the conference attendees.
Let’s prove the following statement: every graph without isolated vertices is connected.
Proof We use the induction on the number of vertices. Clearly the statement is true for graphs with \(2\) vertices. Now, assume we have proven the statement for graphs with up to \(n\) vertices.
Take a graph with \(n\) vertices by induction hypothesis it must be connected. Let’s add a non-isolated vertex to it. As this vertex is not isolated, it is connected to one of the other \(n\) vertices. But then the whole graph of \(n+1\) vertices is connected!
Let’s compute the infinite sum: \[1+2 + 4 + 8 + 16 + ... + 2^n + ... = c\] Observe that \(1+2+4+8+... = 1 + 2(1+2+4+8+16+...)\), namely \(c = 1+2c\), then it follows that \[c = 1+2+4+8+... = -1.\]
There are \(16\) cities in the kingdom. Prove that it is possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most five roads coming out of each city.
Show that a bipartite graph with \(n\) vertices cannot have more than \(\frac{n^2}{4}\) edges.