What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size
Cut the interval
Eight schoolchildren solved
If each problem is solved by
Prove that the equation
Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any
Given a square trinomial
Two ants crawled along their own closed route on a
We are given a polynomial
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
100 fare evaders want to take a train, consisting of 12 coaches, from the first to the 76th station. They know that at the first station two ticket inspectors will board two coaches. After the 4th station, in the time between each station, one of the ticket inspectors will cross to a neighbouring coach. The ticket inspectors take turns to do this. A fare evader can see a ticket inspector only if the ticket inspector is in the next coach or the next but one coach. At each station each fare evader has time to run along the platform the length of no more than three coaches – for example at a station a fare evader in the 7th coach can run to any coach between the 4th and 10th inclusive and board it. What is the largest number of fare evaders that can travel their entire journey without ever ending up in the same coach as one of the ticket inspectors, no matter how the ticket inspectors choose to move? The fare evaders have no information about the ticket inspectors beyond that which is given here, and they agree their strategy before boarding.