Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
A table of \(4\times4\) cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.
Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.
In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard \(8\times 8\) draughts board, so that each king is attacking at least one other?
Prove that for every convex polyhedron there are two faces with the same number of sides.
A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.
Let \(M\) be the point of intersection of the medians of the triangle \(ABC\), and \(O\) an arbitrary point on a plane. Prove that \[OM^2 = 1/3 (OA^2 + OB^2 + OC^2) - 1/9 (AB^2 + BC^2 + AC^2).\]
Three non-coplanar vectors are given. Is it possible to find a fourth vector perpendicular to the three vectors given?
Find the volume of an inclined triangular prism whose base is an equilateral triangle with sides equal to a if the side edge of the prism is equal to the side of the base and is inclined to the plane of the base at an angle of \(60^{\circ}\).
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?