It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
Prove that in a game of noughts and crosses on a \(3\times 3\) grid, if the first player uses the right strategy then the second player cannot win.
Vincent makes small weights. He made 4 weights which should have masses (in grams) of 1, 3, 4 and 7, respectively. However, he made a mistake and one of these weights has the wrong mass. By weighing them twice using balance scales (without the use of weights other than those mentioned) can he find which weight has the wrong mass?
There are some coins on a table. One of these coins is fake (has a different weight than a real coin). By weighing them twice using balance scales, determine whether the fake coin is lighter or heavier than a real coin (you don’t need to find the fake coin) if the number of coins is: a) 100; b) 99; c) 98?
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.
Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.
27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?
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.
On a board of size \(8 \times 8\), two in turn colour the cells so that there are no corners of three coloured squares. The player who can’t make a move loses. Who wins with the right strategy?
On a plane there are 100 sheep-points and one wolf-point. In one move, the wolf moves by no more than 1, after which one of the sheep moves by a distance of no more than 1, after that the wolf again moves, etc. At any initial location of the points, will a wolf be able to catch one of the sheep?