What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?
All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.
A regular hexagon with sides of length \(5\) is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.
Solve the equation \((x + 1)^3 = x^3\).
Can the cells of a \(5 \times 5\) board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?
Five teams participated in a football tournament. Each team had to play exactly one match with each of the other teams. Due to financial difficulties, the organisers cancelled some of the games. As a result, it turned out that all teams scored a different number of points and no team scored zero points. What is the smallest number of games that could be played in the tournament, if three points were awarded for a victory, one for a draw and zero for a defeat?
Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.
Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.
A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.
The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.
Find the equation of the vertical line dividing the area of this figure in half.