What is the smallest number of cells that can be chosen on a \(15\times15\) board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)
There are 40 identical cords. If you set any cord on fire on one side, it burns, and if you set it alight on the other side, it will not burn. Ahmed arranges the cords in the form of a square (see the figure below, each cord makes up a side of a cell). Then, Helen arranges 12 fuses. Will Ahmed be able to lay out the cords in such a way that Helen will not be able to burn all of them?
In a \(10 \times 10\) square, all of the cells of the upper left \(5 \times 5\) square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)
An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.
Fred chose 2017 (not necessarily different) natural numbers \(a_1, a_2, \dots , a_{2017}\) and plays by himself in the following game. Initially, he has an unlimited supply of stones and 2017 large empty boxes. In one move Fred adds a1 stones to any box (at his choice), in any of the remaining boxes (of his choice) – \(a_2\) stones, ..., finally, in the remaining box – \(a_{2017}\) stones. His purpose is to ensure that eventually all the boxes have an equal number of stones. Could he have chosen the numbers so that the goal could be achieved in 43 moves, but is impossible for a smaller non-zero number of moves?
In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals (it could be that some pairs will repeat). The prize is given to those who win both matches. Find:
a) the smallest possible number of tournament winners;
b) the mathematical expectation of the number of tournament winners.
Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of \(d\) meters. What is the smallest value of \(d\) for which the photographer will be able to make a successful photograph sooner or later?
Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?
a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.
b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?
A group of psychologists developed a test, after which each person gets a mark, the number \(Q\), which is the index of his or her mental abilities (the greater \(Q\), the greater the ability). For the country’s rating, the arithmetic mean of the \(Q\) values of all of the inhabitants of this country is taken.
a) A group of citizens of country \(A\) emigrated to country \(B\). Show that both countries could grow in rating.
b) After that, a group of citizens from country \(B\) (including former ex-migrants from \(A\)) emigrated to country \(A\). Is it possible that the ratings of both countries have grown again?
c) A group of citizens from country \(A\) emigrated to country \(B\), and group of citizens from country \(B\) emigrated to country \(C\). As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of \(C\) moved to \(B\), and part of the residents of \(B\) migrated to \(A\). It turned out that as a result, the ratings of all three countries increased again (compared to those that were after the first move, but before the second). (This is, in any case, what the news agencies of these countries say). Can this be so (if so, how, if not, why)?
(It is assumed that during the considered time, the number of citizens \(Q\) did not change, no one died and no one was born).