\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
A gang contains 50 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters has taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 8 different raids.
Solve the equation \(xy = x + y\) in integers.
10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “\(+\)” or “\(-\)” between them so that the resulting algebraic sum is divisible by 1001.
Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.
Let \(f\) be a continuous function defined on the interval \([0; 1]\) such that \(f (0) = f (1) = 0\). Prove that on the segment \([0; 1]\) there are 2 points at a distance of 0.1 at which the function \(f 4(x)\) takes equal values.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
Calculate \(\int_0^{\pi/2} (\sin^2 (\sin x) + \cos^2 (\cos x))\,dx\).
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).