\(N\) points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the \(k\) colours. Prove that if \(N > \lfloor k!e\rfloor\), then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.
Let it be known that all the roots of some equation \(x^3 + px^2 + qx + r = 0\) are positive. What additional condition must be satisfied by its coefficients \(p, q\) and \(r\) in order for it to be possible to form a triangle from segments whose lengths are equal to these roots?
Prove that amongst any 7 different numbers it is always possible to choose two of them, \(x\) and \(y\), so that the following inequality was true: \[0 < \frac{x-y}{1+xy} < \frac{1}{\sqrt3}.\]
The frog jumps over the vertices of the hexagon \(ABCDEF\), each time moving to one of the neighbouring vertices.
a) How many ways can it get from \(A\) to \(C\) in \(n\) jumps?
b) The same question, but on condition that it cannot jump to \(D\)?
c) Let the frog’s path begin at the vertex \(A\), and at the vertex \(D\) there is a mine. Every second it makes another jump. What is the probability that it will still be alive in \(n\) seconds?
d)* What is the average life expectancy of such frogs?
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
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.
In a one-on-one tournament 10 chess players participate. What is the least number of rounds after which the single winner could have already been determined? (In each round, the participants are broken up into pairs. Win – 1 point, draw – 0.5 points, defeat – 0).
A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.
A chequered strip of \(1 \times N\) is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.
Which of the players can always win (no matter how their opponent played)?
On a ring road at regular intervals there are 25 posts, each with a policeman. The police are numbered in some order from 1 to 25. It is required that they cross the road so that there is a policeman on each post, but so that number 2 was clockwise behind number 1, number 3 was clockwise behind number 2, and so on. Prove that if you organised the transition so that the total distance travelled was the smallest, then one of the policemen will remain at his original post.