You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?
a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
In a city, there are 15 telephones. Can I connect them with wires so that each phone is connected exactly with five others?
There are 30 people in the class. Can it be that 9 of them have 3 friends (in this class), 11 have 4 friends, and 10 have 5 friends?
In the city Smallville there are 15 telephones. Can they be connected by wires so that there are four phones, each of which is connected to three others, eight phones, each of which is connected to six, and three phones, each of which is connected to five others?
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?