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 \(13\)?
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.
In the Land of Linguists live \(m\) people, who have opportunity to speak \(n\) languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that \(k\) is the maximum number of people, any two of whom can talk without interpreters. It turned out that \(11n \leq k \leq m/2\). Prove that then there are at least \(mn\) pairs of people in the country who will not be able to talk without interpreters.
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 not 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 four roads coming out of each city.