Problems

At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.

a) Give an example of how this could occur.

b) Prove that the number of heads of the first dragon in all potential examples is the same.

A pharmacist has three weights, with which he measured out and gave 100 g of iodine to one buyer, 101 g of honey to another, and 102 g of hydrogen peroxide to the third. He always placed the weights on one side of the scales, and the goods on the other. Could it be that each weight used is lighter than 90 grams?

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

In a certain kingdom there were 32 knights. Some of them were vassals of others (a vassal can have only one suzerain, and the suzerain is always richer than his vassal). A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

(In the kingdom the following law is enacted: “the vassal of my vassal is not my vassal”).

Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.

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?

A box contains 111 red, blue, green, and white marbles. It is known that if we remove 100 marbles from the box, without looking, we will always have removed at least one marble of each colour. What is the minimum number of marbles we need to remove to guarantee that we have removed marbles of 3 different colours?

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.)

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.

In each cell of a board of size \(5\times5\) a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?