It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
A professional tennis player plays at least one match each day for training purposes. However in order to ensure he does not over-exert himself he plays no more than 12 matches a week. Prove that it is possible to find a group of consecutive days during which the player plays a total of 20 matches.
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.
9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.
Prove that for any number \(d\), which is not divisible by \(2\) or by \(5\), there is a number whose decimal notation contains only ones and which is divisible by \(d\).
Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?
On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.
There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.
Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.