A new airline "Capitals Direct" has direct flights operating on the following routes (in both directions): Paris - London, Paris - Lisbon, Rome - London, Rome - Madrid, Berlin - Helsinki, Berlin - Amsterdam, Amsterdam - Prague. These are the only flights that the company offers. I would like to travel from Paris to Amsterdam. I cannot buy a direct ticket for sure, but can "Capitals Direct" offer me a connecting flight?
Can one arrange numbers from \(1\) to \(9\) in a row so that each pair of consecutive numbers forms a two-digit multiple of \(7\) or a multiple of \(13\)?
In the Royal Grammar School all Year \(9\) students were gathered in the Queen’s Hall for an important announcement. They have been waiting for it for a while and everyone had enough time to greet every other student with a handshake. Assuming there are \(100\) Year \(9\) students at the school, how many handshakes were made before the announcement?
In \(2149\) a regular transport connection between nine planets of the Solar System was introduced. Space capsules are flying between the following pairs of planets: Earth – Mercury, Pluto – Venus, Earth – Pluto, Pluto – Mercury, Mercury – Venus, Uranus – Neptune, Neptune – Saturn, Saturn – Jupiter, Jupiter – Mars, and Mars – Uranus. Is it possible to travel from Earth to Mars by using this type of transport with possible changes at other planets?
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 \(13\)?
Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?
Prove that there is a vertex in the tree from which exactly one edge emerges (such a vertex is called a hanging top).
Given \(n\) points that are connected by segments so that each point is connected to some other and there are no two points that would be connected in two different ways. Prove that the total number of segments is \(n - 1\).
A system of points connected by segments is called “connected” if from each point one can go to any other one along these segments. Is it possible to connect five points to a connected system so that when erasing any segment, exactly two connected points systems are formed that are not related to each other? (We assume that in the intersection of the segments, the transition from one of them to another is impossible).
A group of \(2n\) people were gathered together, of whom each person knew no less than \(n\) of the other people present. Prove that it is possible to select 4 people and seat them around a table so that each person sits next to people they know. (\(n \geq 2\))