Problems

Several guests are sitting at a round table. Some of them are familiar with each other; mutually acquainted. All the acquaintances of any guest (counting himself) sit around the table at regular intervals. (For another person, these gaps may be different.) It is known that any two have at least one common acquaintance. Prove that all guests are familiar with each other.

On the selection to the government of the planet of liars and truth tellers $12$ candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the $12$th, who said: “And now $12$ times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?

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?

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?

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).

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?

A $99\times 99$ chequered table is given, each cell of which is painted black or white. It is allowed (at the same time) to repaint all of the cells of a certain column or row in the colour of the majority of cells in that row or column. Is it always possible to have that all of the cells in the table are painted in the same colour?

10 guests came to a party and each left a pair of shoes in the corridor (all guests have the same shoes). All pairs of shoes are of different sizes. The guests began to disperse one by one, putting on any pair of shoes that they could fit into (that is, each guest could wear a pair of shoes no smaller than his own). At some point, it was discovered that none of the remaining guests could find a pair of shoes so that they could leave. What was the maximum number of remaining guests?