Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?
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?
How many six-digit numbers exist, the numbers of which are either all odd or all even?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
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?