Once upon a time there were twenty spies. Each of them wrote an accusation against ten of his colleagues. Prove that at least ten pairs of spies have told on each other.
Fred chose 2017 (not necessarily different) natural numbers \(a_1, a_2, \dots , a_{2017}\) and plays by himself in the following game. Initially, he has an unlimited supply of stones and 2017 large empty boxes. In one move Fred adds a1 stones to any box (at his choice), in any of the remaining boxes (of his choice) – \(a_2\) stones, ..., finally, in the remaining box – \(a_{2017}\) stones. His purpose is to ensure that eventually all the boxes have an equal number of stones. Could he have chosen the numbers so that the goal could be achieved in 43 moves, but is impossible for a smaller non-zero number of moves?
In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals (it could be that some pairs will repeat). The prize is given to those who win both matches. Find:
a) the smallest possible number of tournament winners;
b) the mathematical expectation of the number of tournament winners.
In each cell of a board of size \(5\times5\) a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?
An after school club is attended by 4 boys from class 7A, and four from class 7B. Of those who attended three were named Ben, three were named Will, and two were named Tom.
Is it possible for it to be the case that each boy had at least one namesake classmate who attended the club?
Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of \(d\) meters. What is the smallest value of \(d\) for which the photographer will be able to make a successful photograph sooner or later?
We took several positive numbers and constructed the following sequence: \(a_1\) is the sum of the initial numbers, \(a_2\) is the sum of the squares of the original numbers, \(a_3\) is the sum of the cubes of the original numbers, and so on.
a) Could it happen that up to \(a_5\) the sequence decreases (\(a_1> a_2> a_3> a_4> a_5\)), and starting with \(a_5\) – it increases (\(a_5 < a_6 < a_7 <\dots\))?
b) Could it be the other way around: before \(a_5\) the sequence increases, and starting with \(a_5\) – decreases?
A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of \(n\) cells. We will call the natural number \(n\) jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one \(n > 50\) that is not jumpable.
One hundred gnomes weighing each 1, 2, 3, ..., 100 pounds, gathered on the left bank of a river. They cannot swim, but on the same shore is a rowing boat with a carrying capacity of 100 pounds. Because of the current, it’s hard to swim back, so each gnome has enough power to row from the right bank to the left one no more than once (it’s enough for any one of the gnomes to row in the boat, the rower does not change during one voyage). Will all gnomes cross to the right bank?
Prove that for any positive integer \(n\), it is always possible to find a number, consisting of the digits \(1\) and \(2,\) that is divisible by \(2^n\). (For example, \(2\) is divisible by \(2\), \(12\) is divisible by \(4,\) \(112\) is divisible by \(8,\) \(2112\) is divisible by \(16\) and so on...).