Show that \(2^{2n} - 1\) is always divisible by \(3\), if \(n\) is a positive natural number.
The famous Fibonacci sequence is a sequence of numbers, which starts from two ones, and then each consecutive term is a sum of the previous two. It describes many things in nature. In a symbolic form we can write: \(F_0 = 1, F_1 = 1, F_n = F_{n-1} + F_{n-2}\).
Show that \[F_0+F_1+ F_2 + \dots + F_n = F_{n+2}-1\]
In certain country, there are \(n\) cities. Some of them are connected by roads, roads go in both directions. It is possible to get from any city to any other city using only roads, however, for any pair of cities, there is always only one way to get from one of them to the other, there are no alternative routes.
Show that there are exactly \(n-1\) roads in this country.
If \(x\) is any positive real number and \(n \ge 2\) is a positive natural number, show that \[(1+x)^n > 1+nx\]
Anna and Bob play a game with the following rules: they both receive a positive integer number. They do not know each other’s numbers, but they do know that their numbers come one after another – they do not know which one is larger. (If Anna gets \(n\), Bob gets either \(n-1\) or \(n+1\)). Anna then asks Bob – “do you know what number I have?” If Bob does know, he has to say Anna’s number and he wins the game. If he does not, he has to say that he does not. Then, he asks Anna if she knows his number. If Anna does not know, she asks Bob. This continues until one of them finds out what is the other’s number. Assuming that both Anna and Bob know mathematics sufficiently well to be able to solve this problem, find out who wins the game and how.
For simplicity let’s assume Bob always gets the odd number and Anna always gets the even number - two consecutive numbers have opposite parity!
A real number \(y\) is such that \(y+\frac1{y}\) happens to be an integer number. Show that for any natural \(n\), it is also true that \(y^n + \frac1{y^n}\) is an integer number.
Jane’s birthday cake is square-shaped and has side length 25 cm. Suppose she makes 4 horizontal cuts perpendicular to the vertical edges of the cake and 4 vertical cuts perpendicular to the horizontal edges of the cake. Show that at least one of the pieces has an area of at most 25 cm\(^2\).
Show that given any nine points on a sphere, there is a closed hemisphere that contains six of them. A closed hemiphere is one that contains the equator with respect to the division.
We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.
The kingdom of Rabbitland consists of a finite number of cities. No matter how you split the kingdom into two, there is always a train connection from a city in one part of the divide to a city in the other part of the divide. Show that one can in fact travel from any city to any other, possibly changing trains.