A king divided his kingdom into 19 counties who are governed by 19 lords. Could it be that each lord’s county has one, five or nine neighbouring counties?
Can there be exactly 100 roads in a state in which three roads leave each city?
John, coming back from Disneyland, told me that there are seven islands on the enchanted lake, each of which is lead to by one, three or five bridges. Is it true that at least one of these bridges necessarily leads to the shore of the lake?
Prove that the number of people who have ever lived on Earth and who shook hands an odd number of times is even.
Is it possible to draw 9 segments on a plane so that each intersects exactly three others?
Prove that a graph with \(n\) vertices, the degree of each of which is at least \(\frac{n-1}{2}\), is connected.
In the Far East, the only type of transport is a carpet-plane. From the capital there are 21 carpet-planes, from the city of Dalny there is one carpet-plane, and from all of the other cities there are 20. Prove that you can fly from the capital to Dalny (possibly with interchanges).
a) A piece of wire that is 120 cm long is given. Is it possible, without breaking the wire, to make a cube frame with sides of 10 cm?
b) What is the smallest number of times it will be necessary to break the wire in order to still produce the required frame?
On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.