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.
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.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
Prove that if \(a, b, c\) are odd numbers, then at least one of the numbers \(ab-1\), \(bc-1\), \(ca-1\) is divisible by 4.
When boarding a plane, a line of \(n\) passengers was formed, each of whom has a ticket for one of the \(n\) places. The first in the line is a crazy old man. He runs onto the plane and sits down in a random place (perhaps, his own). Then passengers take turns to take their seats, and in the case that their place is already occupied, they sit randomly on one of the vacant seats. What is the probability that the last passenger will take his assigned seat?
We are given 101 natural numbers whose sum is equal to 200. Prove that we can always pick some of these numbers so that the sum of the picked numbers is 100.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).