On a plane, there are given 2004 points. The distances between every pair of points is noted. Prove that among these noted distances at least thirty numbers are different.
How can one measure out 15 minutes, using an hourglass of 7 minutes and 11 minutes?
Prove that, if \(b=a-1\), then \[(a+b)(a^2 +b^2)(a^4 +b^4)\dotsb(a^{32} +b^{32})=a^{64} -b^{64}.\]
Solve the equation \(xy = x + y\) in integers.
There are three sets of dominoes of different colours. How can you put the dominoes from all three sets into a chain (according to the rules of the game) so that every two neighbouring dominoes are of a different colour?
On every cell of a \(9 \times 9\) board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.
Prove that there will be at least 9 unoccupied cells.
In the TV series “The Secret of Santa Barbara” there are 20 characters. Each episode contains one of the events: some character discovers the Mystery, some character discovers that someone knows the Mystery, some character discovers that someone does not know the Mystery. What is the maximum number of episodes that this tv series can last?
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
Two boys play the following game: they take turns placing rooks on a chessboard. The one who wins is the one whose last move leaves all the board cells filled. Who wins if both try to play with the best possible strategy?
On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.