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.
One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.
How many packets of gum, on average, should I buy to collect the full collection prizes?
On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.
Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.
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?
Find the number of zeros in which the number \(11^{100} - 1\) ends.
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?
Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).
\(2n\) 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.
The planet has \(n\) residents, some are liars and some are truth tellers. Each resident said: “Among the remaining residents of the island, more than half are liars.” How many liars are on the island?
You are given a table of size \(m \times n\) (\(m, n > 1\)). In it, the centers of all cells are marked. What is the largest number of marked centers that can be chosen so that no three of them are the vertices of a right triangle?